MATHRISK

MATHRISK - 2024

2024Activity reportProject-TeamMATHRISK

RNSR: 201221215M

Research center Inria Paris Centre
In partnership with:Ecole Nationale des Ponts et Chaussées, CNRS, Université Gustave Eiffel
Team name: Mathematical Risk handling
In collaboration with:Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS)
Domain:Applied Mathematics, Computation and Simulation
Theme:Stochastic approaches

Keywords

Computer Science and Digital Science

A6. Modeling, simulation and control
A6.1. Methods in mathematical modeling
A6.1.2. Stochastic Modeling
A6.2.1. Numerical analysis of PDE and ODE
A6.2.2. Numerical probability
A6.2.3. Probabilistic methods
A6.4.2. Stochastic control
A8.7. Graph theory
A8.12. Optimal transport

1 Team members, visitors, external collaborators

Research Scientist

Agnès Sulem [Team leader, INRIA, Senior Researcher, HDR]

Faculty Members

Aurélien Alfonsi [ENPC, Professor, CERMICS Laboratory, HDR]
Vlad Bally [Université Gustave Eiffel, Professor, HDR]
Julien Guyon [ENPC, Professor, CERMICS laboratory]
Benjamin Jourdain [ENPC, Professor, CERMICS Laboratory, HDR]
Damien Lamberton [Université Gustave Eiffel, Professor, HDR]

Post-Doctoral Fellows

Guillaume Szulda [ENPC]
Guido Gazzani [ENPC, until Apr 2024]

PhD Students

Hervé Andrès [ENPC, CIFRE, Milliman ]
Faten Ben Said [EDF, CIFRE, ENPC]
Arthur Bourdon [Milliman, CIFRE, from Nov 2024, ENPC]
Elise Devey [INRIA]
François Escolan [ENPC, from Nov 2024, ERC HighLEAP]
Thibault Jeannin [ENPC, from Nov 2024]
Edoardo Lombardo [ENPC/Univ Tor Vegata Roma]
Kexin Shao [INRIA]
Nerea Vadillo Fernandez [AXA Climate, CIFRE, until Jan 2024]

Interns and Apprentices

Ali Aldirani [INRIA, Intern, from Jun 2024 until Aug 2024]
Augustin Chenevois [INRIA, Intern, from May 2024 until Jul 2024]
Mariam Maatoug [INRIA, Intern, from Mar 2024 until Jul 2024]

Administrative Assistants

Derya Gok [INRIA]
Martial Le Henaff [INRIA, from May 2024]

Visiting Scientists

Hamed Amini [FIU, from Apr 2024 until Jul 2024]
Fabio Baschetti [Scuola Normale Superiore di Pisa, until May 2024, (4th year PhD student)]
Arturo Kohatsu Higa [Univ Ritsumeikan d'Asie Pacifique, until Feb 2024]

External Collaborators

Ludovic Goudenège [CNRS, Senior scientist, LaMME, HDR]
Ahmed Kebaier [UNIV EVRY, Professor, HDR]
Jerome Lelong [ENSIMAG, Professor, HDR]
Antonino Zanette [UNIV UDINE, Professor, HDR]

2 Overall objectives

The Inria project team MathRisk team was created in 2013. It is the follow-up of the MathFi project team founded in 2000. MathFi was focused on financial mathematics, in particular on computational methods for pricing and hedging increasingly complex financial products. The 2007 global financial crisis and its “aftermath crisis” has abruptly highlighted the critical importance of a better understanding and management of risk.

The project team MathRisk addresses broad research topics embracing risk management in quantitative finance and insurance and in other related domains as economy and sustainable development. In these contexts, the management of risk appears at different time scales, from high frequency data to long term life insurance management, raising challenging renewed modeling and numerical issues. We aim at both producing advanced mathematical tools, models, algorithms, and software in these domains, and developing collaborations with various institutions involved in risk control. The scientific issues we consider include:

Option pricing and hedging, and risk-management of portfolios in finance and insurance. These remain crucial issues in finance and insurance, with the development of increasingly complex products and various regulatory legislations. Models must take into account the multidimensional features, incompleteness issues, model uncertainties and various market imperfections and defaults. It is also important to understand and capture the joint dynamics of the underlying assets and their volatilities. The insurance activity faces a large class of risk, including financial risk, and is submitted to strict regulatory requirements. We aim at proposing modelling frameworks which catch the main specificity of life insurance contracts.

Systemic risk and contagion modeling. These last years have been shaped by ever more interconnectedness among all aspects of human life. Globalization and economics growth as well as technological progress have led to more complex dependencies worldwide. While these complex networks facilitate physical, capital and informational transmission, they have an inherent potential to create and propagate distress and risk. The financial crisis 2007-2009 has illustrated the significance of network structure on the amplification of initial shocks in the banking system to the level of the global financial system, leading to an economic recession. We are contributing on the issues of systemic risk and financial networks, aiming at developing adequate tools for monitoring financial stability which capture accurately the risks due to a variety of interconnections in the financial system.

(Martingale) Optimal transport. Optimal transport problems arise in a wide range of topics, from economics to physics. In mathematical finance, an additional martingale constraint is considered to take the absence of arbitrage opportunities into account. The minimal and maximal costs provide price bounds robust to model risk, i.e. the risk of using an inadequate model. On the other hand, optimal transport is also useful to analyse mean-field interactions. We are in particular interested in particle approximations of McKean-Vlasov stochastic differential equations (SDEs) and the study of mean-field backward SDEs with applications to systemic risk quantization.

Advanced numerical probability methods and Computational finance. Our project team is very much involved in numerical probability, aiming at pushing numerical methods towards the effective implementation. This numerical orientation is supported by a mathematical expertise which permits a rigorous analysis of the algorithms and provides theoretical support for the study of rates of convergence and the introduction of new tools for the improvement of numerical methods. Financial institutions and insurance companies, submitted to more and more stringent regulatory legislations, such as FRTB or XVA computation, are facing numerical implementation challenges and research focused on numerical efficiency is strongly needed. Overcoming the curse of dimensionality in computational finance is a crucial issue that we address by developing advanced stochastic algorithms and deep learning techniques.

The MathRisk project is strongly devoted to the development of new mathematical methods and numerical algorithms. Mathematical tools include stochastic modeling, stochastic analysis, in particular various aspects of stochastic control and optimal stopping with nonlinear expectations, Malliavin calculus, stochastic optimization, random graphs, (martingale) optimal transport, mean-field systems, numerical probability and generally advanced numerical methods for effective solutions. The numerical platform Premia that MathRisk is developing in collaboration with a consortium of financial institutions, focuses on the computational challenges the recent developments in financial mathematics encompass, in particular risk control in large dimensions.

3 Research program

3.1 Systemic risk in financial networks

After the recent financial crisis, systemic risk has emerged as one of the major research topics in mathematical finance. Interconnected systems are subject to contagion in time of distress. The scope is to understand and model how the bankruptcy of a bank (or a large company) may or not induce other bankruptcies. By contrast with the traditional approach in risk management, the focus is no longer on modeling the risks faced by a single financial institution, but on modeling the complex interrelations between financial institutions and the mechanisms of distress propagation among these.

The mathematical modeling of default contagion, by which an economic shock causing initial losses and default of a few institutions is amplified due to complex linkages, leading to large scale defaults, can be addressed by various techniques, such as network approaches or mean field interaction models.

The goal of our project is to develop a model that captures the dynamics of a complex financial network and to provide methods for the control of default contagion, both by a regulator and by the institutions themselves.

We have contributed in the last years to the research on the control of contagion in financial systems in the framework of random graph models (see PhD thesis of R. Chen 74 and Z. Cao 73).

In 57, 106, 9, we consider a financial network described as a weighted directed graph, in which nodes represent financial institutions and edges the exposures between them. The distress propagation is modeled as an epidemics on this graph. We study the optimal intervention of a lender of last resort who seeks to make equity infusions in a banking system prone to insolvency and to bank runs, under complete and incomplete information of the failure cluster, in order to minimize the contagion effects. The paper 9 provides in particular important insight on the relation between the value of a financial system, connectivity and optimal intervention.

The results show that up to a certain connectivity, the value of the financial system increases with connectivity. However, this is no longer the case if connectivity becomes too large. The natural question remains how to create incentives for the banks to attain an optimal level of connectivity. This is studied in 75, where network formation for a large set of financial institutions represented as nodes is investigated. Linkages are source of income, and at the same time they bear the risk of contagion, which is endogeneous and depends on the strategies of all nodes in the system. The optimal connectivity of the nodes results from a game. Existence of an equilibrium in the system and stability properties is studied. The results suggest that financial stability is best described in terms of the mechanism of network formation than in terms of simple statistics of the network topology like the average connectivity.

In 8, H. Amini (University of Florida), A. Minca (Cornell University) and A. Sulem study Dynamic Contagion Risk Model With Recovery Features. We introduce threshold growth in the classical threshold contagion model, in which nodes have downward jumps when there is a failure of a neighboring node. We are motivated by the application to financial and insurance-reinsurance networks, in which thresholds represent either capital or liquidity. An initial set of nodes fail exogenously and affect the nodes connected to them as they default on financial obligations. If those nodes’ capital or liquidity is insufficient to absorb the losses, they will fail in turn. In other terms, if the number of failed neighbors reaches a node’s threshold, then this node will fail as well, and so on. Since contagion takes time, there is the potential for the capital to recover before the next failure. It is therefore important to introduce a notion of growth. Choosing the configuration model as underlying graph, we prove fluid limits for the baseline model, as well as extensions to the directed case, state-dependent inter-arrival times and the case of growth driven by upward jumps. We then allow nodes to choose their connectivity by trading off link benefits and contagion risk. Existence of an asymptotic equilibrium is shown as well as convergence of the sequence of equilibria on the finite networks. In particular, these results show that systems with higher overall growth may have higher failure probability in equilibrium.

3.2 Stochastic Control, optimal stopping and non-linear backward stochastic differential equations (BSDEs) with jumps

Option pricing in incomplete and nonlinear financial market models with default.

A. Sulem with M.C. Quenez and M. Grigorova have studied option pricing and hedging in nonlinear incomplete financial markets model with default. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio processes follow nonlinear dynamics with a nonlinear driver $f$ , which encodes the imperfections or constraints of the market. A large class of imperfect market models can fit in this framework, including imperfections coming from different borrowing and lending interest rates, taxes on profits from risky investments, or from the trading impact of a large investor seller on the market prices and the default probability. Our market is incomplete, in the sense that not every contingent claim can be replicated by a portfolio. In this framework, we address in 14 the problem of pricing and (super)hedging of European options. By using a dynamic programming approach, we provide a dual formulation of the seller’s superhedging price as the supremum over a suitable set of equivalent probability measures $Q \in 𝒬$ of the non-linear $ℰ_{Q}^{f}$ -expectation under $Q$ of the payoff. We also provide a characterization of this price as the minimal supersolution of a constrained BSDE with default. In 85, we study the superhedging problem for American options with irregular payoffs. We establish a dual formulation of the seller’s price in terms of the value of a non-linear mixed optimal control/stopping problem. We also characterize the seller's price process as the minimal supersolution of a reflected BSDE with constraints. We then prove a duality result for the buyer's price in terms of the value of a non-linear optimal control/stopping game problem. A crucial step in the proofs is to establish a non-linear optional and a non-linear predictable decomposition for processes which are $ℰ_{Q}^{f}$ -strong supermartingales under $Q$ , for all $Q \in 𝒬$ . American option pricing in a non-linear complete market model with default is previously studied in 78. A complete analysis of BSDEs driven by a Brownian motion and a compensated default jump process with intensity process $(λ_{t})$ is achieved in 76. Note that these equations do not correspond to a particular case of BSDEs with Poisson random measure, and are particularly useful in default risk modeling in finance.

Optimal stopping.

The theory of optimal stopping in connection with American option pricing has been extensively studied in recent years. Our contributions in this area concern:

(i) The analysis of the binomial approximation of the American put price in the Black-Scholes model. We proved that the rate of convergence is, up to a logarithmic factor, of the order $1 / n$ , where $n$ is the number of discretization time points 102; (ii) The American put in the Heston stochastic volatility model. We have results about existence and uniqueness for the associated variational inequality, in suitable weighted Sobolev spaces, following up on the work of P. Feehan et al. (2011, 2015, 2016) (cf 104). We also established some qualitative properties of the value function (monotonicity, strict convexity, smoothness) 103. (iii) A probabilistic approach to the smoothness of the free boundary in the optimal stopping of a one-dimensional diffusion (work in collaboration with T. De Angelis)(University of Torino) (see 40),

Stochastic control with jumps.

The 3rd edition of the book Applied Stochastic Control of Jump diffusions (Springer, 2019) by B. Øksendal and A. Sulem 16 contains recent developments within stochastic control and its applications. In particular, there is a new chapter devoted to a comprehensive presentation of financial markets modelled by jump diffusions, one on backward stochastic differential equations and risk measures, and an advanced stochastic control chapter including optimal control of mean-field systems, stochastic differential games and stochastic Hamilton-Jacobi-Bellman equations.

3.3 Volatility Modeling

J. Guyon and co-authors have investigated the modeling of the volatility of financial markets 88, 86, 89. In particular, the (mostly) path-dependent nature of volatility has been shown in 88, an article that has been downloaded 7,000+ times on SSRN. Path-dependent volatility (PDV) provides a new paradigm of volatility modeling, which can be mixed with stochastic volatility (PDSV) to account for the exogenous part of volatility. In 87, J. Guyon has uncovered a remarkable property of the S&P 500 and VIX markets, which he called inversion of convex ordering. In 86, M. El Amrani and J. Guyon have shown that, contrary to a common belief in the mathematical finance community, the term-structure of the at-the-money skew does not follow a power law. In 89, J. Guyon and S. Mustapha have calibrated neural stochastic differential equations jointly to S&P 500 smiles, VIX futures, and VIX smiles.

3.4 Insurance modeling

Asset Liability Management.

Life insurance contracts are popular and involve very large portfolios, for a total amount of trillions of euros in Europe. To manage them in a long run, insurance companies perform Asset and Liability Management (ALM) : it consists in investing the deposit of policyholders in different asset classes such as equity, sovereign bonds, corporate bonds, real estate, while respecting a performance warranty with a profit sharing mechanism for the policyholders. A typical question is how to determine an allocation strategy which maximizes the rewards and satisfies the regulatory constraints. The management of these portfolios is quite involved: the different cash reserves imposed by the regulator, the profit sharing mechanisms, and the way the insurance company determines the crediting rate to its policyholders make the whole dynamics path-dependent and rather intricate. A. Alfonsi et al. have developed in 47 a synthetic model that takes into account the main features of the life insurance business. This model is then used to determine the allocation that minimizes the Solvency Capital Requirement (SCR). In 48, numerical methods based on Multilevel Monte-Carlo algorithms are proposed to calculate the SCR at future dates, which is of practical importance for insurance companies. The standard formula prescribed by the regulator is basically obtained from conditional expected losses given standard shocks that occur in the future.

3.5 (Martingale) Optimal Transport and Mean-field systems

3.5.1 Numerical methods for Optimal transport

Optimal transport problems arise in a wide range of topics, from economics to physics. There exists different methods to solve numerically optimal transport problems. A popular one is the Sinkhorn algorithm which uses an entropy regularization of the cost function and then iterative Bregman projections. Alfonsi et al. 50 have proposed an alternative relaxation that consists in replacing the constraint of matching exactly the marginal laws by constraints of matching some moments. Using Tchakaloff's theorem, it is shown that the optimum is reached by a discrete measure, and the optimal transport is found by using a (stochastic) gradient descent that determines the weights and the points of the discrete measure. The number of points only depends of the number of moments considered, and therefore does not depend on the dimension of the problem. The method has then been developed in 49 in the case of symmetric multimarginal optimal transport problems. These problems arise in quantum chemistry with the Coulomb interaction cost. The problem is in dimension ${(ℝ^{3})}^{M}$ where $M$ is the number of electrons, and the method is particularly relevant since the optimal discrete measure weights only $N + 2$ points, where $N$ is the number of moments constraint on the distribution of each electron. Numerical examples up to $M = 100$ can be thus investigated while existing methods could not go beyond $M \approx 10$ .

3.5.2 Mean-field systems

Mean-field systems and optimal transport.

In 71, O.Bencheikh and B. Jourdain prove that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step $h$ of a system of $N$ interacting particles is $𝒪 (N^{- 1} + h)$ . The challenge was to improve the $𝒪 (N^{- 1 / 2})$ strong rate of convergence in the number of particles. In 72, they prove the same estimation for the Euler discretization of a system interacting particles with mean-field rank based interaction in the drift coefficient. To deal with the initialization error, they investigate in 70 the approximation rate in Wasserstein distance with index $ρ \geq 1$ of a probability measure $μ$ on the real line with finite moment of order $ρ$ by the empirical measure of $N$ deterministic points.

In 100, B. Jourdain and A. Tse propose a generalized version of the central limit theorem for nonlinear functionals of the empirical measure of i.i.d. random variables, provided that the functional satisfies some regularity assumptions for the associated linear functional derivatives of various orders. Using this result to deal with the contribution of the initialization, they check the convergence of fluctuations between the empirical measure of particles in an interacting particle system and its mean-field limiting measure. In 20, R. Flenghi and B. Jourdain pursue their study of the central limit theorem for nonlinear functionals of the empirical measure of random variables by relaxing the i.i.d. assumption to deal with the successive values of an ergodic Markov chain. In 51, A. Alfonsi and B. Jourdain show that any optimal coupling for the quadratic Wasserstein distance $𝒲_{2}^{2} (μ, ν)$ between two probability measures $μ$ and $ν$ on $𝐑^{d}$ is the composition of a martingale coupling with an optimal transport map. They prove that $σ \mapsto 𝒲_{2}^{2} (σ, ν)$ is differentiable at $μ$ in both Lions and the geometric senses iff there is a unique optimal coupling between $μ$ and $ν$ and this coupling is given by a map.

3.5.3 Martingale Optimal Transport

In mathematical finance, optimal transport problems with an additional martingale constraint are considered to handle the model risk, i.e. the risk of using an inadequate model. The Martingale Optimal Transport (MOT) problem introduced in 69 provides model-free hedges and bounds on the prices of exotic options. The market prices of liquid call and put options give the marginal distributions of the underlying asset at each traded maturity. Under the simplifying assumption that the risk-free rate is zero, these probability measures are in increasing convex order, since by Strassen's theorem this property is equivalent to the existence of a martingale measure with the right marginal distributions. For an exotic payoff function of the values of the underlying on the time-grid given by these maturities, the model-free upper-bound (resp. lower-bound) for the price consistent with these marginal distributions is given by the following martingale optimal transport problem : maximize (resp. minimize) the integral of the payoff with respect to the martingale measure over all martingale measures with the right marginal distributions. Super-hedging (resp. sub-hedging) strategies are obtained by solving the dual problem. With J. Corbetta, A. Alfonsi and B. Jourdain 6 have studied sampling methods preserving the convex order for two probability measures $μ$ and $ν$ on $𝐑^{d}$ , with $ν$ dominating $μ$ . Their method is the first generic approach to tackle the martingale optimal transport problem numerically and it can also be applied to several marginals.

Martingale Optimal Transport provides thus bounds for the prices of exotic options that take into account the risk neutral marginal distributions of the underlying assets deduced from the market prices of vanilla options. For these bounds to be robust, the stability of the optimal value with respect to these marginal distributions is needed. Because of the global martingale constraint, stability is far less obvious than in optimal transport (it even fails in multiple dimensions). B. Jourdain has advised the PhD of W. Margheriti devoted to this issue and related problems. He also initiated a collaboration on this topic with M. Beiglböck, one of the founders of MOT theory. In 92, B. Jourdain and W. Margheriti exhibit a new family of martingale couplings between two one-dimensional probability measures $μ$ and $ν$ in the convex order. The integral of $| x - y |$ with respect to each of these couplings is smaller than twice the $𝒲^{1}$ distance between $μ$ and $ν$ . Moreover, for $ρ > 1$ , replacing $| x - y |$ and $𝒲_{1}$ respectively with ${| x - y |}^{ρ}$ and $𝒲_{ρ}^{ρ}$ does not lead to a finite multiplicative constant. In 93, they show that a finite constant is recovered when replacing $𝒲_{ρ}^{ρ}$ with the product of $𝒲_{ρ}$ times the centred $ρ$ -th moment of the second marginal to the power $ρ - 1$ and they study the generalisation of this stability inequality to higher dimension. In 94, they give a direct construction of the projection in adapted Wasserstein distance onto the set of martingale couplings of a coupling between two probability measures on the real line in the convex order which satisfies the barycentre dispersion assumption. Under this assumption, Wiesel had given a clear algorithmic construction of the projection for finitely supported marginals before getting rid of the finite support condition by a rather messy limiting procedure. In 67, with M. Beiglböck and G. Pammer they establish stability of martingale couplings in dimension one : when approximating in Wasserstein distance the two marginals of a martingale coupling by probability measures in the convex order, it is possible to construct a sequence of martingale couplings between these probability measures converging in adapted Wasserstein distance to the original coupling. In 68, they deduce the stability of the Weak Martingale Optimal Transport Problem with respect to the marginal distributions in dimension one which is important since financial data can give only imprecise information on these marginals. As application, this yields the stability of the superreplication bound for VIX futures and of the stretched Brownian motion. In 95, B. Jourdain et al. prove that, in dimension one, contrary to the minimum and maximum in the convex order, the Wasserstein projections of $μ$ (resp. $ν$ ) on the set of probability measures dominated by $ν$ (resp. dominating $μ$ ) in the convex order are Lipschitz continuous in $(μ, ν)$ for the Wasserstein distance. The thesis of K. Shao (advisers: B. Jourdain, A. Sulem) focuses so far on optimal couplings for costs ${| y - x |}^{ρ}$ in dimension one.

Quantization.

In order to exploit the natural links between quantization and convex order in view of numerical methods for (Weak) Martingale Optimal Transport, B. Jourdain has initiated a fruitful collaboration with G. Pagès, one of the leading experts of quantization. For two compactly supported probability measures in the convex order, any stationary quadratic primal quantization of the smaller remains dominated by any dual quantization of the larger. B. Jourdain and G. Pagès prove in 99 that any martingale coupling between the original probability measures can be approximated by a martingale coupling between their quantizations in Wassertein distance with a rate given by the quantization errors but also in the much finer adapted Wassertein distance. In 96, in order to approximate a sequence of more than two probability measures in the convex order by finitely supported probability measures still in the convex order, they propose to alternate transitions according to a martingale Markov kernel mapping a probability measure in the sequence to the next and dual quantization steps. In the case of ARCH models, the noise has to be truncated to enable the dual quantization steps. They exhibit conditions under which the ARCH model with truncated noise is dominated by the original ARCH model in the convex order and also analyse the error of the scheme combining truncation of the noise according to primal quantization with the dual quantization steps. In 98, they prove that for compactly supported one dimensional probability distributions having a log-concave density, $L^{r}$ -optimal dual quantizers are unique at each level $N$ . In the quadratic $r = 2$ case, they propose an algorithm which computes this unique optimal dual quantizer with geometric rate of convergence.

3.5.4 Martingale Schrödinger problems

Calibration problems in finance can be cast as Schrödinger problems. Due to the no-arbitrage condition, martingale Schrödinger problems must be considered. To jointly calibrate S&P 500 (SPX) and VIX options, J. Guyon has introduced dispersion-constrained martingale Schrödinger problems. In 86, he solved for the first time this longstanding puzzle of quantitative finance that has often been described as the Holy Grail of volatility modeling: build a model that jointly and exactly calibrates to the prices of SPX options, VIX futures, and VIX options. He did so using a nonparametric, discrete-time, minimum-entropy approach. He established a strong duality theorem and characterized the absence of joint SPX/VIX arbitrage. The minimum entropy jointly calibrating model is explicit in terms of the dual Schrödinger portfolio, i.e., the maximizer of the dual problems, should it exist, and is numerically computed using an extension of the Sinkhorn algorithm. Numerical experiments show that the algorithm performs very well in both low and high volatility regimes.

3.6 Deep learning for large dimensional financial problems

Neural networks and Machine Learning techniques for high dimensional American options.

The pricing of American option or its Bermudan approximation amounts to solving a backward dynamic programming equation, in which the main difficulty comes from the conditional expectation involved in the computation of the continuation value.

In 105, B. Lapeyre and J. Lelong study neural networks approximations of conditional expectations. They prove the convergence of the well-known Longstaff and Schwartz algorithm when the standard least-square regression on a finite-dimensional vector space is replaced by a neural network approximation, and illustrate the numerical efficiency of the method on several numerical examples. Its stability with respect to a change of parameters as interest rate and volatility is shown. The numerical study proves that training neural network with only a few chosen points in the grid of parameters permits to price efficiently for a whole range of parameters.

In 82, two efficient techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are proposed to compute the price of American basket options. Both techniques are based on Machine Learning, exploited together with binomial trees or with a closed formula for integration. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. In 84, an efficient method is provided to compute the price of multi-asset American options, based on Machine Learning, Monte Carlo simulations and variance reduction techniques. Numerical tests show that the proposed algorithm is fast and reliable, and can handle American options on very large baskets of assets, overcoming the curse of dimensionality issue.

$•$ Machine Learning in the Energy and Commodity Market. Evaluating moving average options is a computational challenge for the energy and commodity market, as the payoff of the option depends on the prices of underlying assets observed on a moving window. An efficient method for pricing Bermudan style moving average options is presented in 83, based on Gaussian Process Regression and Gauss-Hermite quadrature. This method is tested in the Clewlow-Strickland model, the reference framework for modeling prices of energy commodities, the Heston (non-Gaussian) model and the rough-Bergomi model, which involves a double non-Markovian feature, since the whole history of the volatility process impacts the future distribution of the process.

3.7 Advanced numerical probability methods and Computational finance

Our project team is very much involved in numerical probability, aiming at pushing numerical methods towards the effective implementation. This numerical orientation is supported by a mathematical expertise which permits a rigorous analysis of the algorithms and provides theoretical support for the study of rates of convergence and the introduction of new tools for the improvement of numerical methods. This activity in the MathRisk team is strongly related to the development of the Premia software.

3.7.1 Approximation of stochastic differential equations

High order schemes.

The approximation of SDEs and more general Markovian processes is a very active field. One important axis of research is the analysis of the weak error, that is the error between the law of the process and the law of its approximation. A standard way to analyse this is to focus on marginal laws, which boils down to the approximation of semigroups. The weak error of standard approximation schemes such as the Euler scheme has been widely studied, as well as higher order approximations such as those obtained with the Richardson-Romberg extrapolation method.

Stochastic Volterra Equations.

Stochastic Volterra Equations (SVE) provide a wide family of non-Markovian stochastic processes. They have been introduced in the early 80's by Berger and Mizel and have received a recent attention in mathematical finance to model the volatility : it has been noticed that SVEs with a fractional convolution kernel $G (t) = c_{H} t^{H - 1 / 2}$ reproduce some important empirical features. The problem of approximating these equations has been tackled by Zhang 110 and Richard et al. 109 who show under suitable conditions a strong convergence rate of $O (n^{- H -})$ for the Euler scheme, where $n$ is the number of time steps. We almost recover the rate for classical SDEs when $H \to 1 / 2$ . However, an important drawback is that the required computation time is proportional to $n^{2}$ .

Abstract Malliavin calculus and convergence in total variation.

In collaboration with L. Caramellino and G. Poly, V. Bally has settled a Malliavin type calculus for a general class of random variables, which are not supposed to be Gaussian (as it is the case in the standard Malliavin calculus). This is an alternative to the $Γ$ -calculus settled by Bakry, Gentile and Ledoux. The main application is the estimate in total variation distance of the error in general convergence theorems. This is done in 61.

Invariance principles.

As an application of the above methodology, V. Bally et al. have studied several limit theorems of Central Limit type (see 62 and 60). In particular they estimate the total variation distance between random polynomials, and prove a universality principle for the variance of the number of roots of trigonometric polynomials with random coefficients 64).

Analysis of jump type SDEs.

V. Bally, L. Caramellino and A. Kohatsu Higa, study the regularity properties of the law of the solutions of jump type SDE's 58. They use an interpolation criterion (proved in 66) combined with Malliavin calculus for jump processes. They also use a Gaussian approximation of the solution combined with Malliavin calculus for Gaussian random variables. Another approach to the same regularity property, based on a semigroup method has been developed by Bally and Caramellino in 63. An application for the Bolzmann equation is given by V. Bally in 66. In the same line but with different application, the total variation distance between a jump equation and its Gaussian approximation is studied by V. Bally and his PhD student Y. Qin 65 and by V. Bally, V. Rabiet, D. Goreac 64. A general discussion on the link between total variation distance and integration by parts is done in 61. Finally V. Bally et al. estimate in 59 the probability that a diffusion process remains in a tube around a smooth function.

3.7.2 Monte-Carlo and Multi-level Monte-Carlo methods

Error bounds of MLMC.

In 91, B. Jourdain and A. Kebaier are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, they deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. As long as the deviation is below an explicit threshold, they check that the multilevel estimator satisfies a Gaussian-type concentration inequality optimal in terms of the variance.

Approximation of conditional expectations. The approximation of conditional expectations and the computation of expectations involving nested conditional expectations are important topics with a broad range of applications. In risk management, such quantities typically occur in the computation of the regulatory capital such as future Value-at-Risk or CVA. A. Alfonsi et al. 48 have developed a Multilevel Monte-Carlo (MLMC) method to calculate the Solvency Capital Ratio of insurance companies at future dates. The main advantage of the method is that it avoids regression issues and has the same computational complexity as a plain Monte-Carlo method (i.e. a computational time in $O (ε^{- 2})$ to reach a precision of order $ε$ ). In other contexts, one may be interested in approximating conditional expectations. To do so, the classical method consists in considering a parametrized family $φ (α, \cdot)$ of functions, and to minimize the empirical $L^{2}$ -distance $\frac{1}{M} \sum_{k = 1}^{M} {(Y_{i} - φ (α, X_{i}))}^{2}$ between the observations and their prediction. In general, it is assumed to have as many observations as explanatory variables. However, when these variables are sampled, it may be possible to sample $K$ values of $Y$ 's for a given $X_{i}$ and to minimize $\frac{1}{M} \sum_{k = 1}^{M} {(\frac{1}{K} \sum_{k = 1}^{K} Y_{i}^{k} - φ (α, X_{i}))}^{2} .$ A. Alfonsi, J. Lelong and B. Lapeyre 52 have determined the optimal value of $K$ which minimizes the computation time for a given precision. They show that $K$ is large when the family approximates well the conditional expectation. The computational gain can be important, especially if the computational cost of sampling $Y$ given $X$ is small with respect to the cost of sampling $X$ .

3.8 Remarks

We have focused above on the research program of the last four years. We refer to the previous MathRisk activity report for a description of the research done earlier, in particular on Liquidity and Market Microstructure 55, 46, 4, dependence modelling 101, interest rate modeling 44, Robust option pricing in financial markets with imperfections 76, 108, 13, 12, Mean field control and Stochastic Differential Games 107, 90, 112, Stochastic control and optimal stopping (games) under nonlinear expectation 78, 80, 79, 77, robust utility maximization 111, 112, 81, Generalized Malliavin calculus and numerical probability.

4 Application domains

4.1 Financial Mathematics, Insurance

The domains of application are quantitative finance and insurance with emphasis on risk modeling and control. In particular, the project-team Mathrisk focuses on financial modeling and calibration, systemic risk, option pricing and hedging, portfolio optimization, risk measures.

5 Social and environmental responsibility

Our work aims to contribute to a better management of risk in the banking and insurance systems, in particular by the study of systemic risk, climate risk, asset price modeling, stability of financial markets.

6 Highlights of the year

6.1 Awards

Risk award:

Julien Guyon has been awarded Quant of the Year 2025 by Risk, the leading financial risk management magazine.

Article ENPC

Article NYU

Julien Guyon was selected for his overall contributions to modern quantitative finance, in particular volatility modeling, as well as two important papers published in Risk in December 2023 (paper 89 co-authored with Scander Mustapha) and 23 February 2024 (co-authored with Florian Bourgey). Using classical optimization techniques or modern deep learning tools, both articles shed new light on the joint S&P 500/VIX smile calibration problem, a difficult problem sometimes referred to as the Holy Grail of volatility modeling that had eluded quants for many years before Guyon cracked it in 2020.

7 New software, platforms, open data

7.1 New software

7.1.1 PREMIA

Keywords:
Computational finance, Quantum Finance, Monte-Carlo methods, Option pricing, Numerical probability, Machine learning, Numerical algorithm
Scientific Description:
Premia is a numerical platform for computational finance. It is designed for option pricing, hedging and financial model calibration. Premia is developed by the MathRisk project team in collaboration with a consortium of financial institutions. The Premia project keeps track of the most recent advances in the field of computational finance in a well-documented way. It focuses on the implementation of numerical analysis techniques for both probabilistic and deterministic numerical methods. An important feature of the platform Premia is the detailed documentation which provides extended references in option pricing. Premia contains various numerical algorithms: deterministic methods (Finite difference and finite element algorithms for partial differential equations, wavelets, Galerkin, sparse grids ...), stochastic algorithms (Monte-Carlo simulations, quantization methods, Malliavin calculus based methods), tree methods, approximation methods (Laplace transforms, Fast Fourier transforms...) These algorithms are implemented for the evaluation of vanilla and exotic options on equities, interest rate, credit, energy and insurance products. Moreover Premia provides a calibration toolbox for Libor Market model and a toolbox for pricing Credit derivatives. The latest developments of the software address evaluation of financial derivative products, risk management and computations of risk measures required by new financial regulation. They include the implementation of advanced numerical algorithms taking into account model dependence, counterparty credit risk, hybrid features, rough volatility and various nonlinear effects. A big effort has been put these last years on the development and implementation of deep learning techniques using neural network approximations, and Machine Learning algorithms in finance, in particular for high-dimensional American option pricing, high-dimensional PDEs, deep hedging. Moreover Quantum computing in Finance is explored, in particular option pricing using quantum computers.
Functional Description:
Premia is a software designed for quantitative finance, developed by the MathRisk project team in collaboration with a consortium of financial institutions presently composed of Crédit Agricole CIB and NATIXIS. The Premia project keeps track of the most recent advances in computational finance and focuses on the implementation of numerical techniques to solve financial problems. An important feature of the platform Premia is its detailed documentation which provides extended references in computational finance. Premia is a powerful tool to assist Research and Development professional teams in their day-to-day duty. It is also a useful support for academics who wish to perform tests on new algorithms or pricing methods. Besides being a single entry point for accessible overviews and basic implementations of various numerical methods, the aim of the Premia project is: - to elaborate a powerful testing platform for comparing different numerical methods between each other, - to build a link between professional financial teams and academic researchers, - to provide a useful teaching support for Master and PhD students in mathematical finance. The project Premia has started in 1999 and is now considered as a standard reference platform for quantitative finance among the academic mathematical finance community.
Release Contributions:
A big effort has been put these last years on the development and implementation of deep learning techniques using neural network approximations, and Machine Learning algorithms in finance, in particular for high-dimensional American option pricing, high-dimensional PDEs, deep hedging.The latest developments of the software address also the evaluation of financial derivative products, risk management and computations of risk measures by advanced numerical algorithms taking into account model dependence, counterparty credit risk (computations of XVA), hybrid features, rough stochastic volatility models and various new regulations. Nested Monte Carlo strategies with GPU optimizations, and Chebyshev Interpolation method for Parametric Option Pricing have been implemented. We have also developed our activity on insurance contracts, in particular on the computation of risk measures (Value at Risk, Condition Tail Expectation) of variable annuities contracts like GMWB (guaranteed minimum withdrawal benefit) including taxation and customers mortality modeling.
News of the Year:

The new release Premia 26 has been delivered to the Consortium on June 5th 2024. It contains the following new implemented algorithms.

I. Machine Learning, Risk Management, Insurance

- Robust deep hedging. E. Lütkebohmert, T. Schmidt, J. Sester, Quantitative Finance, 22-8, 2022. - Pricing and hedging American-style options with deep learning. S.Becker, P.Cheridito, A.Jentzen, T.Welti, Journal of Risk and Financial Management 13-7, 2020 - Solving high-dimensional optimal stopping problems using deep learning. S.Becker, P.Cheridito, A.Jentzen, T.Welti, European Journal of Applied Mathematics, 32-3, 2021 - Efficient Pricing and Hedging of High Dimensional American Options using Deep Recurrent Networks. A.S.Na J.W.LWan, Quantitative Finance, 23-4, 2023. - A Machine Learning Approach to Adaptive Robust Utility Maximization and Hedging. T.Chen and M.Ludkovski, Siam Journal of Financial Mathematics, 12-3, 2021. - Enhancing Valuation of Variable Annuities in Lévy Models with Stochastic Interest Rate, L.Goudenege, A.Molent, X.Wei A.Zanette

II. Advanced numerical methods for Equity Derivatives: - Quantum computing for Asian option pricing, L. Goudenege - Hedging Bermudan options with the dual representation. A.Alfonsi, A.Kebaier, J.Lelong - Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation. A.Alfonsi - High order approximations of the Cox-Ingersoll-Ross process semigroup using random grids. A.Alfonsi and E.Lombardo, IMA Journal of Numerical Analysis, 2023. - RKHS regularization of singular local stochastic volatility McKean-Vlasov models. C.Bayer, D.Belomestny O.Butkovsky, J.Schoenmakers
URL:
http://www.premia.fr
Publications:
hal-03791594, hal-03436046, hal-01940715, hal-01873346, hal-03810106, hal-03526905, hal-03013606, hal-02183587
Contact:
Agnes Sulem
Participants:
Agnes Sulem, Antonino Zanette, Aurélien Alfonsi, Benjamin Jourdain, Jerome Lelong, Bernard Lapeyre, Ahmed Kebaier, Ludovic Goudenège
Partners:
Ecole des Ponts ParisTech, Université d'Udine

7.2 New platforms

Participants: Julien Guyon.

UEFA draws: draw simulator of the league phase of the 2024-25 UEFA Champions League Available at UEFA drawsø

8 New results

Participants: MathRisk Members.

8.1 Risk processes in stochastic networks

Participants: A. Sulem, H. Amini, Z. Cao, A. Minca.

Central Limit Theorems for Price-Mediated Contagion in Stochastic Financial Networks

In 36, we provide central limit theorems to analyze the combined effects of fire sales and default cascades on systemic risk within stochastic financial networks. The impact of prices is modeled through a specifically defined inverse demand function. Our study presents various limit theorems that delve into the dynamics of total shares sold and the equilibrium pricing of illiquid assets in a streamlined fire sales context. We show that the equilibrium prices of these assets demonstrate asymptotically Gaussian fluctuations. In our numerical experiments, we demonstrate how our central limit theorems can be applied to construct confidence intervals for the magnitude of contagion and the extent of losses due to fire sales.

Ruin Probabilities for Risk Processes in Stochastic Networks

In 35, we study multidimensional Cramér-Lundberg risk processes where agents, located on a large sparse network, receive losses from their neighbors. To reduce the dimensionality of the problem, we introduce classification of agents according to an arbitrary countable set of types. The ruin of any agent triggers losses for all of its neighbours. We consider the case when the loss arrival process induced by the ensemble of ruined agents follows a Poisson process with general intensity function that scales with the network size. When the size of the network goes to infinity, we provide explicit ruin probabilities at the end of the loss propagation process for agents of any type. These limiting probabilities depend, in addition to the agents' types and the network structure, on the loss distribution and the loss arrival process. For a more complex risk processes on open networks, when in addition to the internal networked risk processes the agents receive losses from external users, we provide bounds on ruin probabilities.

8.2 Graphon Mean-field games

Participants: A. Sulem, Z. Cao, H. Amini.

8.2.1 Stochastic Graphon Mean-field Games and approximate Nash Equilibria

Graphon BSDEs with jumps are studied by H. Amini, A. Sulem, and Z. Cao in 18. The use of graphons has emerged recently in order to analyze heterogeneous interaction in mean-field systems and game theory. Existence, uniqueness and stability of solutions under some regularity assumptions are established. We also prove convergence results for interacting mean-field particle systems with inhomogeneous interactions to graphon mean-field BSDE systems.

In 37 we study continuous stochastic games with inhomogeneous mean field interactions on large networks and explore their graphon limits. We consider a model with a continuum of players, where each player's dynamics involve not only mean field interactions but also individual jumps induced by a Poisson random measure. We examine the case of controlled dynamics, with control terms present in the drift, diffusion, and jump components. We introduce the graphon game model based on a graphon controlled stochastic differential equation system with jumps, which can be regarded as the limiting case of a finite game's dynamic system as the number of players goes to infinity. Under some general assumptions, we establish the existence and uniqueness of Markovian graphon equilibria. We then provide convergence results on the state trajectories and their laws, transitioning from finite game systems to graphon systems. We also study approximate equilibria for finite games on large networks, using the graphon equilibrium as a benchmark. The rates of convergence are analyzed under various underlying graphon models and regularity assumptions.

8.2.2 Reinforcement Learning for Graphon Mean-Field Games

Participants: A. Sulem, Z. Cao, H. Amini, K. Shao.

This is an ongoing work in collaboration with Mathieu Laurière (NYU Shangai). We develop theoretical and numerical analysis of extended Graphon Mean Field Games (GMFG) in a discrete-time setting. On the theoretical side, we provide rigorous analysis on the existence of approximated Nash equilibrium of the GMFG system by considering joined state-action distribution, we also refined the proof of existence by categorizing pure policies and mixed policies. On the numerical side, we explore some learning schemes (i.e. reinforcement learning) to study graphon mean field equilibrium.

8.3 Decentraized control problems in energy networks

Participants: A. Sulem, E. Devey, H. Amini, N. Oudjane.

The project aims at solving optimal control problems in which many agents cooperate to minimize a convex cost functional. A distinctive feature of this cost functional is that it includes two-way interactions.

The main goal of this paper is to propose a new perspective on the construction of near-optimal distributed controls, which is non-asymptotic in nature and imposes no structural assumptions. Unlike methods based on Mean Field Control theory, we preserve the heterogeneity of the agents.

We design an alternating-direction optimisation algorithm that constructs near-optimal distributed solutions to the problem and provide a quantitative analysis of its convergence rate.

This research is motivated by an optimisation challenge faced by EDF, which aims to manage flexible producers and consumers by regulating the electricity they supply to or draw from the grid. The problem incorporates power constraints on each line of the electricity grid, leading to graphical constraints. This is ongoing work in collaboration with Nadia Oudjane (EDF R&D, Saclay)

8.4 Optimal stopping and American options

Participants: D. Lamberton, A. Alfonsi, J. Lelong, A. Kebaier, A. Zanette, L. Goudenege.

D. Lamberton and Tiziano De Angelis (University of Torino) are working on the optimal stopping problem of a one dimensional diffusion in finite horizon. They develop a probabilistic approach to the regularity of the associated free boundary problem, and derive a probabilistic proof of the differentiability of the free boundary for the optimal stopping problem of a one-dimensional diffusion in 40. They have new results concerning the second order mixed derivative of the value function (work in progress).

A. Alfonsi, J. Lelong and A. Kebaier are working on a numerical method to price American and Bermudean options based on the dual representation introduced by Rogers (2002) (see 32).

In 22, L. Goudenege and coautors propose an algorithm called Backward Hedging, designed for hedging European and American options while considering transaction costs. Comparisons with the Deep Hedging algorithm in various numerical experiments showcase the efficiency and accuracy of the proposed method.

8.5 Martingale Optimal transport

Participants: B. Jourdain, K. Shao.

For many examples of couples $(μ, ν)$ of probability measures on the real line in the convex order, B. Jourdain and K. Shao observe numerically in 28 that the Hobson and that the Hobson and Neuberger martingale coupling, which maximizes for $ρ = 1$ the integral of ${| y - x |}^{ρ}$ with respect to any martingale coupling between $μ$ and $ν$ , is still a maximizer for $ρ \in (0, 2)$ and a minimizer for $ρ > 2$ . They investigate the theoretical validity of this numerical observation and give rather restrictive sufficient conditions for the property to hold. We also exhibit couples $(μ, ν)$ such that it does not hold. The support of the Hobson and Neuberger coupling is known to satisfy some monotonicity property which we call non-decreasing. B. Jourdain and K. Shao check that the non-decreasing property is preserved for maximizers when $ρ \in (0, 1]$ . In general, there exist distinct non-decreasing martingale couplings, and they find some decomposition of $ν$ which is in one-to-one correspondence with martingale couplings non-decreasing in a generalized sense.

In 27, they complete the analysis of the Martingale Wasserstein Inequality by checking that this inequality fails in dimension $d \geq 2$ when the integrability parameter $ρ$ belongs to $[1, 2)$ while a stronger Maximal Martingale Wasserstein Inequality holds whatever the dimension $d$ when $ρ \geq 2$ .

While many questions in robust finance can be posed in the martingale optimal transport framework or its weak extension, others like the subreplication price of VIX futures, the robust pricing of American options or the construction of shadow couplings necessitate additional information to be incorporated into the optimization problem beyond that of the underlying asset. In 26, B. Jourdain and G. , B. Jourdain and G. Pammer (Gratz University) take into account this extra information by introducing an additional parameter to the weak martingale optimal transport problem. They prove the stability of the resulting problem with respect to the risk neutral marginal distributions of the underlying asset. Finally, they deduce stability of the three previously mentioned motivating examples.

8.6 Convex order

Participants: B. Jourdain, G. Pagès.

Motivated by the study of the propagation of convexity by semi-groups of stochastic differential equations and convex comparison between the distributions of solutions of two such equations, B. Jourdain and G. Pagès study in 43 the comparison for the convex order between a Gaussian distribution and a Gaussian mixture. They give and discuss intrinsic necessary and sufficient conditions for convex ordering. On the examples that they have worked out, the two conditions appear to be closely related. In 25, they address convex ordering for stochastic Volterra equations and their Euler schemes.

8.7 Volatility modeling in Insurance and Finance

Participants: B. Jourdain, H. Andrès, J. Guyon, A. Alfonsi.

In 39, H. Andrès and B. Jourdain show the existence and uniqueness of a continuous solution to a path-dependent volatility model introduced by Guyon and Lekeufack (2023) 88 to model the price of an equity index and its spot volatility. The considered model for the trend and activity features can be written as a Stochastic Volterra Equation (SVE) with non-convolutional and non-bounded kernels as well as non-Lipschitz coefficients. They first prove the existence and uniqueness of a solution to the SVE under integrability and regularity assumptions on the two kernels and under a condition on the second kernel weighting the past squared returns which ensures that the activity feature is bounded from below by a positive constant. Then, assuming in addition that the kernel weighting the past returns is of exponential type and that an inequality relating the logarithmic derivatives of the two kernels with respect to their second variables is satisfied, they show the positivity of the volatility process which is obtained as a non-linear function of the SVE's solution. They show numerically that the choice of an exponential kernel for the kernel weighting the past returns has little impact on the quality of model calibration compared to other choices and the inequality involving the logarithmic derivatives is satisfied by the calibrated kernels. These results extend those of Nutz and Valdevenito (2023).

Real-world economic scenarios provide stochastic forecasts of economic variables like interest rates, equity, stocks or indices, inflation, etc. and are widely used in the insurance sector for a variety of applications (see 29). In 19, H. Andrès, A. Boumezoued and B. Jourdain have studied the use of signatures to contribute to provide a formal mathematical setup for defining such scenarios.

Path-dependent volatility is a new paradigm for volatility modeling that has attracted a lot of attention in the markets, both as a risk-neutral pricing model and as a model able to generate realistic real-world scenarios 38.

In 17, A. Alfonsi and N. Vadillo propose a new stochastic volatility model for the average daily temperature in the context of the valuation of temperature derivatives in the insurance and energy market (see also the PhD of N. Vadillo 30).

J. Guyon and J. Lekeufack 88 learn from data that volatility is mostly path-dependent: up to 90% of the variance of the implied volatility of equity indexes is explained endogenously by past index returns, and up to 65% for (noisy estimates of) future daily realized volatility. The path-dependency that we uncover is remarkably simple: a linear combination of a weighted sum of past daily returns and the square root of a weighted sum of past daily squared returns with different time-shifted power-law weights capturing both short and long memory. This simple model, which is homogeneous in volatility, is shown to consistently outperform existing models across equity indexes and train/test sets for both implied and realized volatility. It suggests a simple continuous-time path-dependent volatility (PDV) model that may be fed historical or risk-neutral parameters. The weights can be approximated by superpositions of exponential kernels to produce Markovian models. In particular, J. Guyon and J. Lekeufack propose a 4-factor Markovian PDV model which captures all the important stylized facts of volatility, produces very realistic price and (rough-like) volatility paths, and jointly fits SPX and VIX smiles remarkably well. They thus show that a continuous-time Markovian parametric stochastic volatility (actually, PDV) model can practically solve the joint SPX/VIX smile calibration problem.

Using two years of S&P 500, Eurostoxx 50, and DAX data, M. El Amrani and J. Guyon 86, empirically investigate the term-structure of the at-the-money-forward (ATM) skew of equity indexes. While a power law (2 parameters) captures the term-structure well away from short maturities, the power law fit deteriorates considerably when short maturities are included. By contrast, 3-parameter shapes that look like power laws but do not blow up at vanishing maturity, such as time-shifted or capped power laws, are shown to fit well regardless of whether short maturities are included or not. Their study suggests that the term-structure of equity ATM skew has a power-law shape for maturities above 1 month but has a different behavior, and in particular may not blow up, for shorter maturities. The 3-parameter shapes are derived from non-Markovian variance curve models using the Bergomi-Guyon expansion. A simple 4-parameter term-structure similarly derived from the (Markovian) two-factor Bergomi model is also considered and provides even better fits. The extrapolated zero-maturity skew, far from being infinite, is distributed around a typical value of 1.5 (in absolute value).

J. Guyon and S. Mustapha 89 calibrate neural stochastic differential equations jointly to S&P 500 smiles, VIX futures, and VIX smiles. Drifts and volatilities are modeled as neural networks. Minimizing a suitable loss allows them to fit market data for multiple S&P 500 and VIX maturities. A one-factor Markovian stochastic local volatility model is shown to fit both smiles and VIX futures within bid-ask spreads. The joint calibration actually makes it a pure path-dependent volatility model, confirming the findings in [Guyon, 2022, The VIX Future in Bergomi Models: Fast Approximation Formulas and Joint Calibration with S&P 500 Skew].

8.8 Pricing and calibration of path-dependent volatility models

Participants: J. Guyon, G Gazzani.

G. Gazzani and J. Guyon consider a stochastic volatility model where the dynamics of the volatility process are described by a linear combination of a (exponentially) weighted sum of past daily returns and the square root of a weighted sum of past daily squared returns in the spirit of 88.They discuss the influence of an additional parameter that allows to reproduce the implied volatility smiles of SPX an VIX options within a 4-factor Markovian model (4FPDV) 42. The empirical nature of this class of path-dependent volatility models (PDVs) comes with computational challenges, especially in relation to VIX options pricing and calibration. To address these challenges, they propose an accurate neural network approximation of the VIX leveraging on the markovianity of the 4FPDV. This approximation is subsequently used to tackle the joint calibration problem of SPX and VIX options. They additionally discuss a local volatility extension of the 4FPDV, in order to exactly calibrate market smiles.

8.9 Numerical probability

Participants: A. Alfonsi, B. Jourdain, A. Kebaier, V. Bally, O. Bencheikh, B. Jourdain, J. Lelong, A. Zanette, L. Goudenège, A. Molent.

8.9.1 Approximations of Stochastic Differential Equations (SDEs)

High order schemes for the weak error for the CIR and Heston processes.

A. Alfonsi and E. Lombardo have developed in 53 high order schemes for the weak error for the CIR process, based on the construction proposed in a recent paper by A. Alfonsi and V. Bally. We keep on this analysis to extend these results to the Heston model (see 33).

Weak error analysis

B. Jourdain investigate with several co-authors the convergence to the uniform distribution of vectors of partial sumsmodulo one with a common factor in 21, the Euler-Maruyama Scheme applied to Diffusion Processes in 24, and the weal approximation of stable-driven SDEs with Lebesgue drift in 41.

Approximation of Stochastic Volterra Equations (SVE).

In 54, A. Alfonsi studies the stochastic invariance in a convex domain of SVEs. He also provides a second order approximation scheme for SVEs with multiexponential kernels which stay in some convex domain, and this is used for the multi-exponential Heston model. With E. Lombardo, he studies stochastic Volterra equations with jumps and non-Lipschitz coefficients in 34. A. Alfonsi and A. Kebaier study the weak error for the approximation of Stochastic Volterra Equations and processes with rough paths.

8.9.2 Abstract Malliavin calculus and convergence in total variation

In 65, V. Bally and his PhD student Yifen Qin obtain total variation distance result between a jump-equation and its Gaussian approximation by Malliavin calculus techniques.

They approximate the invarient measure of a Markov process, solution of a stochastic equation with jumps by using a Euler scheme with decreasing step introduced by D Lamberton and G Pages in the early 2000 in the case of diffusion processes driven by Brownian motion. The novelty here is that They deal with jump processes. Under appropiate non degeneracy hypothesis, they have estimated the error in total variation distance and also proved convergence of the density functions.

A. Alfonsi, V. Bally and A Kohatzu Higa (Ritzumikan University) are working on a continuation of the above mentioned work on the approximation of the invarient measure for some non linear stochastic differential equations of Mc Kean Vlasov and Bozmann type.

8.9.3 Sewing Lemma

A. Alfonsi and V. Bally have proposed a new approach based on the sewing lemma on the Wasserstein Space to study existence and uniqueness of solutions of the Boltzmann equation 45. They are now working in 31 with L. Caramellino (Roma Univ) to extend their results by using the stochastic sewing lemma recently proposed by Khoa Lê (2020).

8.10 Deep learning for large dimensional financial problems

Participants: A. Zanette, L. Goudenège, A. Molent, A. Kebaier.

We pursue the development of Machine Learning an Deep learnig techniques in particular for McKean-Vlasov models of singular stochastic volatility, robust utility maximization, and high-dimensional optimal stopping problems. The corresponding algorithms are implemented in the Premia software.

8.11 Quantum Computing in Finance

Participants: A. Zanette, L. Goudenège, A Espa, A. Molent, A. Sulem.

We are constructing a pricing framework using the Qiskit framework. Comparison of efficiency with other techniques has been done.

9 Bilateral contracts and grants with industry

9.1 Bilateral contracts with industry

Consortium PREMIA, Crédit Agricole Corporate Investment Bank (CA - CIB ) - INRIA
CIFRE agreement ENPC/EDF PhD thesis of Faten Ben Said

9.2 Bilateral grants with industry

Chair Ecole Polytechnique-Ecole des Ponts ParisTech-Sorbonne Université-Société Générale "Financial Risks" of the Risk fondation.

Participants: Aurélien Alfonsi, Benjamin Jourdain.

Postdoctoral grant : G.Szulda
Chair Ecole des Ponts ParisTech - Université Paris-Cité - BNP Paribas "Futures of Quantitative Finance"

Participants: Julien Guyon.
Institut Europlace de Finance Louis Bachelier and Labex Louis Bachelier grant : "Multi-Agent Reinforcement Learning in Large Financial Networks with Heterogeneous Interactions" from November 2023.

Participants: Agnès Sulem, Hamed Amini.

10 Partnerships and cooperations

Participants: Mathrisk Team.

10.1 International research visitors

10.1.1 Visits of international scientists

Hamed Amini, Associate Professor, University of Florida, Research stay, April - July; 15-30 November 2024. Research stay on risk processes on stochastic networks and Stochastic Graphon games with A. Sulem.
Arturo Kohatsu Higa, Professor, Ritsumeikan University, 03/01 - 04/01 2024 : Research visit on the topic of approximations to the invariant measure of McKean-Vlasov equations (in collaboration with A. Alfonsi and V. Bally)
Antonino Zanette, Professor, University of Udine, Research stays in connection with Premia, August 2 - September 4th 2024.
Paul Hager : 4-11 August 2024, Technische Universität Berlin, Research stay in connection with Premia
Michael Samet, 28/10-06/11 2024, Aachen University, PhD student, Research stay in connection with Premia
Alvaro Leitao, 4-6 June, University of A Coruña, Research stay in connection with Premia
Fabio Baschetti (January-May 2024), Scuola Normale Superiore di Pisa, 4th year PhD student

10.1.2 Visits to international teams

Research stays abroad

A. Zanette visited Prof.Lucia Caramellino, Department of Mathematics ,University of Roma Tor Vergata to work on pricing issues in the Sabr model.
K. Shao visited Prof Mathieu Laurière, NYU Shangai, Project: Learning Extended Graphon Mean-Field Games May 13 - May 17, 2024; December 2 - December 20, 2024.

10.2 National initiatives

FMSP (Fondation Sciences Mathématiques de Paris) PhD grants :

Cofund MathInParis program: K. Shao (2021 - Present)(INRIA)
Labex Bezout

11 Dissemination

11.1 Promoting scientific activities

A. Alfonsi

Co-organizer of the Mathrisk seminar “Méthodes stochastiques et finance”

Co-organizer of the Bachelier (Mathematical Finance) seminar IHP, Paris.
V. Bally

Organizer of the seminar of the LAMA laboratory, Université Gustave Eiffel.
A. Sulem

Co-organizer of the seminar INRIA-MathRisk /Université Paris Diderot LPSM “Numerical probability and mathematical finance”

11.1.1 Scientific events: organisation

J. Guyon co-organized the minisymposium "Latest advances in volatility modeling" during the 2024 Bachelier Congress.
A. Sulem and A. Zanette organized the Premia meeting for the delivery of the 26th release of the software to the Consortium. Talks by A. Zanette (Univ Udine), 5th June 2024, INRIA Paris.

11.1.2 Journal

Member of the editorial boards

A. Alfonsi

Member of the editorial board of the Book Series "Mathématiques et Applications" of Springer.
J. Guyon

Associate editor of
- Finance and Stochastics
- Quantitative Finance
- SIAM Journal on Financial Mathematics
- Journal of Dynamics and Games
B. Jourdain

Associate editor of
- ESAIM : Proceedings and Surveys
- Stochastic Processes and their Applications (SPA)
- Stochastic and Partial Differential Equations : Analysis and Computations
D. Lamberton

Associate editor of
- Mathematical Finance,
- ESAIM Probability & Statistics
A. Sulem

Associate editor of
- Journal of Mathematical Analysis and Applications (JMAA)
- SIAM Journal on Financial Mathematics (SIFIN)

Reviewer - reviewing activities

J. Guyon : Reviewer for Finance and Stochastics, Quantitative Finance, SIAM Journal on Financial Mathematics, Mathematical Finance International Journal of Theoretical and Applied Finance, Frontiers of Mathematical Finance.
B. Jourdain : Reviewer for Mathematical Reviews
A. Sulem: Reviewer for Mathematical Reviews

11.2 Invited talks

A. Alfonsi
- 06 02 2024: "How many inner simulations to compute conditional expectations with least-square Monte Carlo?", Quants' seminar of Société Générale.
- 14 03 2024: "Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation.", Math Finance Seminar at Imperial College, London.
- 04 04 2024: "Risk valuation of quanto derivatives on temperature and electricity", International Conference on Computational Finance 2024, Amsterdam.
- 29 11 2024: "A pure dual approach for hedging Bermudan options.", Séminaire FDD-FIME, IHP Paris.
E. Devey
- Young Researchers days at Domaine de La Tour (Calvados) - 12/06 to 14/06. McKean-Vlasov optimal control problem and its application to electricity management
J. Guyon
- Research In Options 2024, Fundação Getulio Vargas, Rio de Janeiro, December 2024.
- QuantMinds 2024, London, November 2024.
- 12th World Congress of the Bachelier Finance Society, Rio de Janeiro, July 2024.
- 4th Italian Meeting on Probability and Mathematical Statistics, Rome, June 2024.
- Workshop "Fairness in sports", Ghent, June 2024.
- Workshop "The memory of volatility", Padova, March 2024.
- Institut Henri Poincaré, Bachelier seminar, November 2024.
- Bank of America, New York, Invited seminar, October 2024.
- CACIB, Paris, Seminar of the LPSM working group on math finance, May 2024.
- Columbia University, Columbia-NYU Financial Engineering Colloquium, May 2024.
- Bloomberg, New York, Keynote speaker at BBQ (Bloomberg Quant Seminar), March 2024.
- Societe Generale, Quantitative finance and volatility seminar, La Clusaz, March 2024.
- Imperial College London, London Mathematical Finance Seminar, February 2024.
- Bayes Business School, London, Financial Engineering Workshop, January 2024.
B. Jourdain
- Stochastics in Mathematical Finance and Physics conference, Hammamet, 21-25 october 2024 : Convexity propagation and convex ordering for one-dimensional stochastic differential equations
- Berlin stochastic analysis and stochastic finance seminar, 27 june 2024 : Convexity propagation and convex ordering for one-dimensional stochastic differential equations
- Berlin Probability Colloquium of the IRTG, 26 june 2024 : Central limit theorem for nonlinear functionals of empirical measures and fluctuations of mean-field interacting particle systems
- Seminar of the probability team at CMAP Ecole Poilytechnique, 30 april 2024 : Central limit theorem for nonlinear functionals of empirical measures and fluctuations of mean-field interacting particle systems
- Workshop Stochastic and Deterministic Analysis for Irregular Models, CIRM Marseille, 8-12 january 2024 : 4.5h course on Numerical methods for SDEs with singular coefficients
K. Shao
- Groupe de Travail Méthodes Stochastiques et Finance, Champs-sur-Marne, France. (January 16, 2024)
- Séminaire Bachelier Doctorants, IHP, Paris, France (Feburary 2, 2024)
- Ceremade PhD students seminar, Paris, France. (March 26, 2024)
- Bachelier World Congress 2024, Rio de Janeiro, Brazil. (July 8-12, 2024)
A. Sulem
- Conference A Backward stochastic excursion with Ying Hu, Rennes, June 2024

11.3 Scientific expertise

A. Alfonsi
- Member of the council of the Bachelier Finance Society
A. Sulem
- Reviewer of a proposal for an International Research Training Group entitled “Stochastic Analysis in Interaction” Technische Universität, Weiersrtass Institute Berlin in cooperation with University of Oxford (Berlin, 11-12 Mars 2024)
- Member of the Nominating Committee of the Bachelier Finance Society

11.4 Research administration

A. Alfonsi
- Deputy director of the CERMICS.
- In charge of the Master “Finance and Data” at Ecole des Ponts.
V. Bally
- Responsible of the Master 2, mathèmatiques de la Finance et des données. , Université Gustave Eiffel
- Member of the LAMA committee, UGE.
J. Guyon
- Head of the Applied probability team at CERMICS, ENPC since September 2024.
B. Jourdain
- Deputy head of the Labex Bézout.
A. Sulem
- Member of the Scientific Committee of AMIES( Agence pour les Mathématiques en Interaction avec l'Entreprise et la Société)
- Member of the Committee for INRIA international Chairs

11.5 Teaching - Supervision - Juries

11.5.1 Teaching

A. Alfonsi
- “Probabilités”, first year course at the Ecole des Ponts.
- “Données Haute Fréquence en finance”, lecture for the Master at UPEMLV.
- “Mesures de risque”, Master course of UPEMLV and Sorbonne Université.
- Professeur chargé de cours at Ecole Polytechnique.
V. Bally
- Course "Taux d'Intêret" M2 Finance.
- Course "Calcul de Malliavin et applications en finance" M2 Finance
- Course "Analyse du risque" M2 Actuariat,
- Course "Calcul Stochastiques" M2 Recherche
- Course "Probabilités approfondies" M1
E. Devey
- Lecturer at CPES (Cycle Pluridisciplinaire d'Études Supérieures) for second-year undergraduate students. Course: Introduction to probability ; Teaching hours : 30 ; 09/09/2024-10/01/2025.
J. Guyon
- course "Probability Theory", 1st year ENPC
- course "Volatility Modeling", Master Mathematics for Finance and Data (MFD), 3rd year ENPC - UGE
- course "Advanced calibration methods and VIX derivatives", joint lecture of the BNP Paribas chair Futures of Quantitative Finance, Master Probabilités et Finance, Master M2MO, Master MFD (Sorbonne Université, Université Paris Cité, and ENPC-UGE)
- J. Guyon, B. Liang : course "Nonlinear Option Pricing", Master MAFN, Columbia University
- J. Guyon : course "Volatility Modeling", Master of Science in Financial Engineering, NYU
- minicourse: QuantMinds International, London, Nov 2024
- minicourse: BNP Paribas, Paris, Seminar of the BNP Paribas chair "Futures of Quantitative Finance", May and June 2024.
B. Jourdain
- course "Mathematical finance", 2nd year ENPC
- "Monte-Carlo methods", 3rd year ENPC and Research Master Mathématiques et Application, Université Gustave Eiffel
- course "Monte-Carlo Markov chain methods and particle algorithms", Research Master Probabilités et Modèles Aléatoires, Sorbonne Université
- course "Machine Learning 1", MSC Data Science for Business, X-HEC
- course "Randomness", 1st year Ecole Polytechnique
D. Lamberton
- "Arbitrage, volatilité et gestion de portefeuille", Master 2 course, Université Gustave Eiffel.
- "Intégration et probabilités", L3 course, Université Gustave Eiffel.
- "Théorie des distributions et équations aux dérivées partielles", Master 1 course, Université Gustave Eiffel.
A. Sulem
- Master of Mathematics, Université du Luxembourg, Responsible of the course on "Numerical Methods in Finance", and lectures (22 hours)

11.5.2 Supervision

Postdoral fellows
- Guido Gazzani (May 2023 - April 2024); ENPC, Advisor: J. Guyon
- Guillaume Szulda, ENPC, Advisor: A. Alfonsi (From January 2023 to December 2024)
PhD defended
- Nerea Vadillo Fernandez (CIFRE AXA Climate), “Risk valuation for weather derivatives in index-based insurance”, defended on January 11 2024, supervised by A. Alfonsi
- Hervé Andres (started in June 2021) "Modeling and validation of real-world economic scenarios in insurance : how to take into account pathwise properties?", supervised by B. Jourdain, defended on December 16th 2024, ENPC.
PhD in progress
- Elise Devey (started October 2023), "Graphon Mean-Field Games and Renewable Energy Systems", Supervisor: Agnès Sulem, INRIA doctoral grant
- Hervé Andrès (started in June 2021) "Dependence modelling in economic scenario generation for insurance", supervised by B. Jourdain
- Faten Ben Said (CIFRE EDF, co-advisor: Julien Reygner), “Caractérisation et prise en compte des dépendances statistiques dans le cadre d'applications de dynamique sédimentaire”, started in March 2023, supervised by A. Alfonsi
- François Escolan, “Mean-field limit of stochastic particle systems on manifolds”, started in November 2024,ERC HighLEAP, supervised by A. Alfonsi (co-advisors: Virginie Ehrlacher and Julien Reygner)
- Kexin Shao (started in October 2021) "Martingale optimal transport and financial applications", supervised by B. Jourdain and A. Sulem
- Edoardo Lombardo, “High order numerical approximation for some singular stochastic processes and related PDEs”, started in November 2020, International PhD, advisors: Aurélien Alfonsi and Lucia Caramellino (Tor Vegata Roma University)
- Thibault Jeannin (started in november 2024) "Calibration of pure path-dependent volatility models", supervised by J. Guyon and B. Jourdain
- Arthur Bourdon (started in November 2024) "Approximation and explication of insurance valuation computations by Artificial Intelligence", supervised by B. Jourdain
Internship
- Adi Aldirani (ENSTA), Neural SDEs, June-October2024, supervision: L. Goudenege.
- Mariam Maatoug (Ecole polytechnique de Tunis) 01/03/2024 - 31/07/2024 on the Calibration by neural networks of rough path stochastic volatity models
- Augustin Chenevois (ENSIMAG) : 27/O5 to 26/07 2024 on : An unsupervised deep learning approach in solving partial integro-differential equations
- Thibault Jeannin (Master M2MO, May-October 2024), Calibration of pure path-dependent volatility models (co-supervision by J. Guyon and with B. Jourdain)
Project supervision
- J. Guyon, F. Meunier: Project of the ENPC course TDLOG (2023-24): Live probability calculator for the draw of the Round of 16 of European football cups
- J. Guyon: Génie Industriel Department, Scientific project (Spring 2024): Algorithms for the calculation of the probabilities of the draw of the Round of 16 of European football cups
- J. Guyon: IMI Department Project (2023-24): Drawing and Scheduling Matchups in the New UEFA Champions League Format
- J. Guyon: Project of the ENPC course TDLOG (2024-25): Draw simulator, league phase of the New UEFA Champions League Format
- J. Guyon: IMI Department Project (2024-25): Simulation of the New UEFA Champions League Format

11.5.3 Juries

A. Alfonsi
- Referee of the PhD thesis of Leila Bassou “Optimal control methods for systemic risk”
- Referee of the PhD thesis of Cecilia Aubrun “Unraveling Financial Market Quakes: Exploring Endogenous Volatility Dynamics in Interconnected Markets”.
- Referee of the PhD thesis of Victor Le Coz “Microscopic modeling of the yield curve”.
- Referee of the PhD thesis of Houssem Dahbi “Parametric estimation for a class of multidimensional affine processes”.
- Member of the jury of the PhD thesis of Hervé Andres “Modelling and validation of economic scenarios in insurance: taking into account taking into account trajectory properties”. 16/12/2024
- Member of the recruiting commitee for Professor positions in Mathematical Finance at Ecole Polytechnique.
J. Guyon
- Reviewer of the PhD thesis examination of Shaun Li (Université Paris 1 Panthéon-Sorbonne, October 2024): The Quintic Volatility Model for SPX and VIX: a 360 Tour
B. Jourdain
- PhD of Guillaume Boutoille, defended on December 3, Sorbonne University, B. Jourdain
- PhD of Anh Dung LÍ, defended on December 13, Toulouse School of Economics, B. Jourdain
A. Sulem
- President of the jury Prix Marc Yor
- Reviewer of the habilitation thesis of Daniel Bartl, "Optimal transport, stochastic processes, and high dimensional statistics", University of Vienna, August 2024.
- President of the jury of the PhD thesis of Hervé Andres “Modelling and validation of economic scenarios in insurance: taking into account trajectories properties.
- Habilitation of Céline Labart, Some contributions to numerical schemes for Backward Stochastic Differential Equations, Université de Savoie Mont Blanc, 26 June 2024
- PhD of Nerea Vadillo, Evaluation de risque des dérivés climatiques liés au marché de l’énergie, ENPC, 11 January 2024
- Member of the recruiting commitee for a Professor position in "Probability, Applications and Interactions", Université Evry

11.6 Popularization

Julien Guyon

11.6.1 Productions (articles, videos, podcasts, serious games, ...)

Quant of the Year , Risk, November 2024
Live interviews on the French TV network La chaîne L'Équipe on August 29, 2024 and December 12, 2024
Interview in the French sports daily L'Equipe December 13, 2024

12 Scientific production

12.1 Major publications

1 articleA.Anis Al Gerbi, B.Benjamin Jourdain and E.Emmanuelle Clément. Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators.Monte Carlo Method and Applications223https://arxiv.org/abs/1508.06492July 2016, 197-228HAL
2 bookA.Aurélien Alfonsi. Affine Diffusions and Related Processes: Simulation, Theory and Applications.2015HAL DOI
3 articleA.Aurélien Alfonsi and V.Vlad Bally. A generic construction for high order approximation schemes of semigroups using random grids.Numerische Mathematik2021HAL DOI
4 articleA.Aurélien Alfonsi and P.Pierre Blanc. Dynamic optimal execution in a mixed-market-impact Hawkes price model.Finance and Stochasticshttps://arxiv.org/abs/1404.0648January 2016HAL DOI back to text
5 articleA.Aurélien Alfonsi, A.Adel Cherchali and J. A.Jose Arturo Infante Acevedo. A full and synthetic model for Asset-Liability Management in life insurance, and analysis of the SCR with the standard formula.European Actuarial Journal2020HAL DOI
6 articleA.Aurélien Alfonsi, J.Jacopo Corbetta and B.Benjamin Jourdain. Sampling of probability measures in the convex order by Wasserstein projection.Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques5632020, 1706-1729HAL DOI back to text
7 articleA.Aurélien Alfonsi, B.Benjamin Jourdain and A.Arturo Kohatsu-Higa. Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme.Electronic Journal of Probabilityhttps://arxiv.org/abs/1405.70072015HAL
8 articleH.Hamed Amini, A.Andreea Minca and A.Agnès Sulem. A dynamic contagion risk model with recovery features.Mathematics of Operations ResearchNovember 2021HAL DOI back to text
9 articleH.Hamed Amini, A.Andreea Minca and A.Agnès Sulem. Control of interbank contagion under partial information.SIAM Journal on Financial Mathematics61December 2015, 24HAL back to text back to text
10 articleV.Vlad Bally and L.Lucia Caramellino. Convergence and regularity of probability laws by using an interpolation method.Annals of Probability4522017, 1110--1159HAL
11 articleA.Aych Bouselmi and D.Damien Lamberton. The critical price of the American put near maturity in the jump diffusion model.SIAM Journal on Financial Mathematics71https://arxiv.org/abs/1406.6615May 2016, 236--272HAL DOI
12 articleR.Roxana Dumitrescu, M.-C.Marie-Claire Quenez and A.Agnès Sulem. A Weak Dynamic Programming Principle for Combined Optimal Stopping/Stochastic Control with $E^{f}$ -Expectations.SIAM Journal on Control and Optimization5442016, 2090-2115HAL DOI back to text
13 articleR.Roxana Dumitrescu, M.-C.Marie-Claire Quenez and A.Agnès Sulem. Game Options in an Imperfect Market with Default.SIAM Journal on Financial Mathematics81January 2017, 532 - 559HAL DOI back to text
14 articleM.Miryana Grigorova, M.-C.Marie-Claire Quenez and A.Agnès Sulem. European options in a non-linear incomplete market model with default.SIAM Journal on Financial Mathematics113September 2020, 849–880HAL DOI back to text
15 bookB.Benjamin Jourdain. Probabilités et statistique.seconde éditionEllipses2016HAL
16 bookB.Bernt Øksendal and A.Agnès Sulem. Applied Stochastic Control of Jump Diffusions.3rd editionSpringer, Universitext2019, 436HAL DOI back to text

12.2 Publications of the year

International journals

17 articleA.Aurélien Alfonsi and N.Nerea Vadillo. A stochastic volatility model for the valuation of temperature derivatives.IMA Journal of Management Mathematics354October 2024, 737-785HAL DOI back to text
18 articleH.Hamed Amini, Z.Zhongyuan Cao and A.Agnès Sulem. Graphon Mean-Field Backward Stochastic Differential Equations With Jumps and Associated Dynamic Risk Measures.Finance and Stochastics2025. In press. HAL DOI back to text
19 articleH.Hervé Andrès, A.Alexandre Boumezoued and B.Benjamin Jourdain. Signature-based validation of real-world economic scenarios.ASTIN Bulletin542April 2024, 410-440HAL DOI back to text
20 articleR.Roberta Flenghi and B.Benjamin Jourdain. Central limit theorem over non-linear functionals of empirical measures: beyond the iid setting.Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques2024. In press. HAL back to text
21 articleR.Roberta Flenghi and B.Benjamin Jourdain. Convergence to the uniform distribution of vectors of partial sumsmodulo one with a common factor.Journal of Theoretical Probability2024HAL back to text
22 articleL.Ludovic Goudenège, A.Andrea Molent and A.Antonino Zanette. Backward hedging for American options with transaction costs.Decisions in Economics and FinanceAugust 2024HAL DOI back to text
23 articleBest paperJ.Julien Guyon and F.Florian Bourgey. Fast Exact Joint S&P 500/VIX Smile Calibration in Discrete and Continuous Time.RiskFebruary 2024HAL DOI back to text
24 articleB.Benjamin Jourdain and S.Stéphane Menozzi. Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q − L ρ Drift Coefficient and Additive Noise.The Annals of Applied Probability341BFebruary 2024HAL DOI back to text
25 articleB.Benjamin Jourdain and G.Gilles Pagès. Convex ordering for stochastic Volterra equations and their Euler schemes.Finance and Stochastics292025, 1-62In press. HAL back to text
26 articleB.Benjamin Jourdain and G.Gudmund Pammer. An extension of martingale transport and stability in robust finance.Electronic Journal of Probability29572024HAL back to text
27 articleB.Benjamin Jourdain and K.Kexin Shao. Maximal Martingale Wasserstein Inequality.Electronic Communications in Probability29262024HAL back to text
28 articleB.Benjamin Jourdain and K.Kexin Shao. Non-decreasing martingale couplings.ESAIM: Probability and StatisticsJanuary 2025HAL DOI back to text

Doctoral dissertations and habilitation theses

29 thesisH.Hervé Andrès. Modelling and validation of real-world economic scenarios in insurance: taking pathwise properties into account.Ecole nationale des ponts et chausséesDecember 2024HAL back to text
30 thesisN.Nerea Vadillo Fernandez. Risk valuation for weather derivatives related to the energy market.Ecole des Ponts; MATHRISKJanuary 2024HAL back to text

Reports & preprints

31 miscA.Aurélien Alfonsi, V.Vlad Bally and L.Lucia Caramellino. Stochastic sewing lemma on Wasserstein space.December 2024HAL back to text
32 miscA.Aurélien Alfonsi, A.Ahmed Kebaier and J.Jérôme Lelong. A pure dual approach for hedging Bermudan options.April 2024HAL back to text
33 miscA.Aurélien Alfonsi and E.Edoardo Lombardo. High order approximations and simulation schemes for the log-Heston process.July 2024HAL back to text
34 miscA.Aurélien Alfonsi and G.Guillaume Szulda. On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients.December 2024HAL back to text
35 miscH.Hamed Amini, Z.Zhongyuan Cao, A.Andreea Minca and A.Agnes Sulem. Ruin Probabilities for Risk Processes in Stochastic Networks.February 2024HAL DOI back to text
36 miscH.Hamed Amini, Z.Zhongyuan Cao and A.Agnes Sulem. Central Limit Theorems for Price-Mediated Contagion in Stochastic Financial Networks.2024HAL DOI back to text
37 miscH.Hamed Amini, Z.Zhongyuan Cao and A.Agnes Sulem. Stochastic Graphon Mean Field Games with Jumps and Approximate Nash Equilibria.2024HAL DOI back to text
38 miscH.Hervé Andrès, A.Alexandre Boumezoued and B.Benjamin Jourdain. Implied volatility (also) is path-dependent.June 2024HAL back to text
39 miscH.Hervé Andrès and B.Benjamin Jourdain. Existence, uniqueness and positivity of solutions to the Guyon-Lekeufack path-dependent volatility model with general kernels.August 2024HAL back to text
40 miscT.Tiziano de Angelis and D.Damien Lamberton. A probabilistic approach to continuous differentiability of optimal stopping boundaries.May 2024HAL back to text back to text
41 miscM.Mathis Fitoussi, B.Benjamin Jourdain and S.Stéphane Menozzi. Weak well-posedness and weak discretization error for stable-driven SDEs with Lebesgue drift.May 2024HAL back to text
42 miscG.Guido Gazzani and J.Julien Guyon. Pricing and Calibration in the 4-Factor Path-Dependent Volatility Model.2024HAL DOI back to text
43 miscB.Benjamin Jourdain and G.Gilles Pagès. Convex comparison of Gaussian mixtures.October 2024HAL back to text

12.3 Cited publications

44 articleA.Abdelkoddousse Ahdida, A.Aurélien Alfonsi and E.Ernesto Palidda. Smile with the Gaussian term structure model.The Journal of Computational Finance2112017HAL DOI back to text
45 articleA.Aurélien Alfonsi and V.Vlad Bally. Construction of Boltzmann and McKean Vlasov type flows (the sewing lemma approach).The Annals of Applied Probability335October 2023HAL DOI back to text
46 articleA.Aurélien Alfonsi and P.Pierre Blanc. Extension and calibration of a Hawkes-based optimal execution model.Market microstructure and liquidityAugust 2016HAL DOI back to text
47 articleA.Aurélien Alfonsi, A.Adel Cherchali and J. A.Jose Arturo Infante Acevedo. A full and synthetic model for Asset-Liability Management in life insurance, and analysis of the SCR with the standard formula.European Actuarial Journal2020HAL DOI back to text
48 articleA.Aurélien Alfonsi, A.Adel Cherchali and J. A.José Arturo Infante Acevedo. Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests.Insurance: Mathematics and Economics2021HAL DOI back to text back to text
49 articleA.Aurélien Alfonsi, R.Rafaël Coyaud and V.Virginie Ehrlacher. Constrained overdamped Langevin dynamics for symmetric multimarginal optimal transportation.Mathematical Models and Methods in Applied Sciences2021HAL back to text
50 articleA.Aurélien Alfonsi, R.Rafaël Coyaud, V.Virginie Ehrlacher and D.Damiano Lombardi. Approximation of Optimal Transport problems with marginal moments constraints.Mathematics of Computation2020HAL DOI back to text
51 articleA.Aurélien Alfonsi and B.Benjamin Jourdain. Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability.ESAIM: Probability and Statistics242020, 703-717HAL DOI back to text
52 articleA.Aurélien Alfonsi, B.Bernard Lapeyre and J.Jérôme Lelong. How many inner simulations to compute conditional expectations with least-square Monte Carlo?Methodology and Computing in Applied Probability253June 2023, 71HAL DOI back to text
53 articleA.Aurélien Alfonsi and E.Edoardo Lombardo. High order approximations of the Cox-Ingersoll-Ross process semigroup using random grids.IMA Journal of Numerical AnalysisAugust 2023HAL DOI back to text
54 articleA.Aurélien Alfonsi. Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation..Stochastic Processes and their Applications181February 2023, 104535HAL DOI back to text
55 articleA.Aurélien Alfonsi, A.Alexander Schied and F.Florian Klöck. Multivariate transient price impact and matrix-valued positive definite functions.Mathematics of Operations ResearchMarch 2016HAL DOI back to text
56 unpublishedH.Hamed Amini, Z.Zhongyuan Cao and A.Agnès Sulem. Graphon Mean-Field Backward Stochastic Differential Equations With Jumps and Associated Dynamic Risk Measures.October 2022, working paper or preprintHAL DOI back to text
57 articleH.Hamed Amini, A.Andreea Minca and A.Agnès Sulem. Optimal equity infusions in interbank networks.Journal of Financial Stability31August 2017, 1-17HAL DOI back to text
58 articleV.Vlad Bally, L.Lucia Caramellino and A.Arturo Kohatsu-Higa. Using moment approximations to study the density of jump driven SDEs.Electronic Journal of Probability27January 2022HAL DOI back to text
59 articleV.Vlad Bally, L.Lucia Caramellino and P.Paolo Pigato. Tube estimates for diffusions under a local strong Hörmander condition.Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques5542019, 2320--2369HAL DOI back to text
60 articleV.Vlad Bally, L.Lucia Caramellino and G.Guillaume Poly. Non universality for the variance of the number of real roots of random trigonometric polynomials.Probability Theory and Related Fields1743-42019, 887-927HAL DOI back to text
61 articleV.Vlad Bally, L.Lucia Caramellino and G.Guillaume Poly. Regularization lemmas and convergence in total variation.Electronic Journal of Probability250January 2020, paper no. 74, 20 ppHAL DOI back to text back to text
62 articleV.Vlad Bally and L.Lucia Caramellino. Total variation distance between stochastic polynomials and invariance principles.Annals of Probability472019, 3762 - 3811HAL DOI back to text
63 articleV.Vlad Bally and L.Lucia Caramellino. Transfer of regularity for Markov semigroups.Journal of Stochastic Analysis 232021, Article 13HAL back to text
64 articleV.Vlad Bally, D.Dan Goreac and V.Victor Rabiet. Regularity and Stability for the Semigroup of Jump Diffusions with State-Dependent Intensity.The Annals of Applied Probability285August 2018, 3028 - 3074HAL DOI back to text back to text
65 articleV.Vlad Bally and Y.Yifeng Qin. Total variation distance between a jump-equation and its Gaussian approximation.Stochastics and Partial Differential Equations: Analysis and ComputationsAugust 2022HAL DOI back to text back to text
66 articleV.Vlad Bally. Upper bounds for the function solution of the homogenuous 2D Boltzmann equation with hard potential.The Annals of Applied Probability2019HAL back to text back to text
67 articleM.Mathias Beiglböck, B.Benjamin Jourdain, W.William Margheriti and G.Gudmund Pammer. Approximation of martingale couplings on the line in the weak adapted topology.Probability Theory and Related Fields1831-237 pages, 2 figures2022, 359--413HAL DOI back to text
68 articleM.Mathias Beiglböck, B.Benjamin Jourdain, W.William Margheriti and G.Gudmund Pammer. Stability of the Weak Martingale Optimal Transport Problem.The Annals of Applied Probability336BDecember 2023HAL DOI back to text
69 articleM.Mathias Beiglböck, P.-H.Pierre-Henry Labordère and F.Friedrich. Penkner. Model-independent bounds for option prices - a mass transport approach.Finance Stoch.1732013, 477-501back to text
70 articleO.Oumaima Bencheikh and B.Benjamin Jourdain. Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures.Journal of Approximation Theory27410568428 pages2022HAL DOI back to text
71 articleO.Oumaima Bencheikh and B.Benjamin Jourdain. Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean.ESAIM: Proceedings and Surveys6514 pagesApril 2019, 219-235HAL DOI back to text
72 articleO.Oumaima Bencheikh and B.Benjamin Jourdain. Weak and strong error analysis for mean-field rank based particle approximations of one dimensional viscous scalar conservation law.The Annals of Applied Probability3262022, 4143--4185HAL DOI back to text
73 thesisZ.Zhongyuan Cao. Systemic risk, complex financial networks and graphon mean field interacting systems.Université Paris sciences et lettresSeptember 2023HAL back to text
74 phdthesisR.Rui Chen. Dynamic optimal control for distress large financial networks and Mean field systems with jumps.Université Paris-DauphineJuly 2019HAL back to text
75 articleR.Rui Chen, A.Andreea Minca and A.Agnès Sulem. Optimal connectivity for a large financial network.ESAIM: Proceedings and Surveys59Editors : B. Bouchard, E. Gobet and B. Jourdain2017, 43 - 55HAL back to text
76 incollectionR.Roxana Dumitrescu, M.Miryana Grigorova, M.-C.Marie-Claire Quenez and A.Agnès Sulem. BSDEs with default jump.Computation and Combinatorics in Dynamics, Stochastics and Control - The Abel Symposium, Rosendal, Norway August 201613The Abel Symposia book seriesSpringer2018HAL DOI back to text back to text
77 articleR.Roxana Dumitrescu, M.-C.Marie-Claire Quenez and A.Agnès Sulem. Mixed generalized Dynkin game and stochastic control in a Markovian framework.Stochastics: An International Journal of Probability and Stochastic Processes8912017, 400-429HAL DOI back to text
78 articleR.Roxana Dumitrescu, M.-C.Marie-Claire Quenez and A.Agnès Sulem. American Options in an Imperfect Complete Market with Default.ESAIM: Proceedings and Surveys2018, 93--110HAL DOI back to text back to text
79 articleR.Roxana Dumitrescu, M.-C.Marie-Claire Quenez and A.Agnès Sulem. Generalized Dynkin games and doubly reflected BSDEs with jumps.Electronic Journal of Probability2016HAL DOI back to text
80 articleR.Roxana Dumitrescu, M.-C.Marie-Claire Quenez and A.Agnès Sulem. Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems.Journal of Optimization Theory and Applications16712015, 23HAL DOI back to text
81 articleC.Claudio Fontana, B.Bernt \O{}}ksendal and A.Agn{ès Sulem. Market viability and martingale measures under partial information.Methodol Comput Appl Probab1792015, 15-39DOI back to text
82 articleL.Ludovic Goudenège, A.Andrea Molent and A.Antonino Zanette. Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models.Quantitative Finance204April 2020, 573-591HAL DOI back to text
83 articleL.Ludovic Goudenège, A.Andrea Molent and A.Antonino Zanette. Moving average options: Machine learning and Gauss-Hermite quadrature for a double non-Markovian problem.European Journal of Operational Research3032December 2022, 958-974HAL DOI back to text
84 incollectionL.Ludovic Goudenège, A.Andrea Molent and A.Antonino Zanette. Variance Reduction Applied to Machine Learning for Pricing Bermudan/American Options in High Dimension.Applications of Lévy ProcessesNova Science PublishersAugust 2021HAL back to text
85 articleM.Miryana Grigorova, M.-C.Marie-Claire Quenez and A.Agnès Sulem. American options in a non-linear incomplete market model with default.Stochastic Processes and their Applications1422021HAL DOI back to text
86 article J.Julien Guyon and M.Mehdi El Amrani. Does the Term-Structure of the At-the-Money Skew Really Follow a Power Law? Risk August 2023 HAL back to text back to text back to text back to text
87 articleJ.Julien Guyon. Inversion of convex ordering in the VIX market.Quantitative Finance20102020, 1597-1623URL: https://doi.org/10.1080/14697688.2020.1753885DOI back to text
88 articleJ.Julien Guyon and J.Jordan Lekeufack. Volatility is (mostly) path-dependent.Quantitative Finance239July 2023, 1221-1258HAL DOI back to text back to text back to text back to text back to text
89 articleBest paperJ.Julien Guyon and S.Scander Mustapha. Neural Joint S&P 500/VIX Smile Calibration.Risk MagazineDecember 2023HAL DOI back to text back to text back to text back to text
90 articleY.Yaozhong Hu, B.Bernt \O{}}ksendal and A.Agn{ès Sulem. Singular mean-field control games.Stochastic Analysis and Applications355June 2017, 823-851HAL DOI back to text
91 articleB.Benjamin Jourdain and A.Ahmed Kebaier. Non-asymptotic error bounds for The Multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient.Electronic Journal of Probability24122019, 1-34HAL DOI back to text
92 articleB.Benjamin Jourdain and W.William Margheriti. A new family of one dimensional martingale couplings.Electronic Journal of Probability251362020, 1-50HAL DOI back to text
93 articleB.Benjamin Jourdain and W.William Margheriti. Martingale Wasserstein inequality for probability measures in the convex order.Bernoulli2822022, 830-858HAL DOI back to text
94 articleB.Benjamin Jourdain and W.William Margheriti. One dimensional martingale rearrangement couplings.ESAIM: Probability and Statistics2639 pages2022, 495-527HAL DOI back to text
95 articleB.Benjamin Jourdain, W.William Margheriti and G.Gudmund Pammer. Lipschitz continuity of the Wasserstein projections in the convex order on the line.Electronic Communications in Probability28noneJanuary 2023HAL DOI back to text
96 articleB.Benjamin Jourdain and G.Gilles Pagès. Convex order, quantization and monotone approximations of ARCH models.Journal of Theoretical Probability3542022, 2480-2517HAL DOI back to text
97 articleB.Benjamin Jourdain and G.Gilles Pagès. Convex ordering for stochastic Volterra equations and their Euler schemes.Finance and Stochastics2023HAL back to text
98 articleB.Benjamin Jourdain and G.Gilles Pagès. Optimal dual quantizers of 1D log-concave distributions: uniqueness and Lloyd like algorithm.Journal of Approximation Theory2671055812021HAL back to text
99 articleB.Benjamin Jourdain and G.Gilles Pagès. Quantization and martingale couplings.ALEA : Latin American Journal of Probability and Mathematical Statistics192022HAL DOI back to text
100 articleB.Benjamin Jourdain and A.Alvin Tse. Central limit theorem over non-linear functionals of empirical measures with applications to the mean-field fluctuation of interacting diffusions.Electronic Journal of Probability261542021HAL DOI back to text
101 articleB.Benjamin Jourdain and A.Alexandre Zhou. Existence of a calibrated Regime Switching Local Volatility model.Mathematical Finance302April 2020, 501-546HAL DOI back to text
102 articleD.Damien Lamberton. On the binomial approximation of the American put.Applied Mathematics and Optimization2018HAL back to text
103 unpublishedD.Damien Lamberton and G.Giulia Terenzi. Properties of the American price function in the Heston-type models.April 2019, working paper or preprintHAL back to text
104 articleD.Damien Lamberton and G.Giulia Terenzi. Variational formulation of American option prices in the Heston Model.SIAM Journal on Financial Mathematics101April 2019, 261-368HAL DOI back to text
105 articleB.Bernard Lapeyre and J.Jérôme Lelong. Neural network regression for Bermudan option pricing.Monte Carlo Methods and Applications273September 2021, 227-247HAL DOI back to text
106 articleA.Andreea Minca and A.Agnès Sulem. Optimal Control of Interbank Contagion Under Complete Information.Statistics & Risk Modeling with Applications in Finance and Insurance3112014, 1001-1026HAL DOI back to text
107 articleB.Bernt Oksendal and A.Agnès Sulem. Forward--Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty.Journal of Optimization Theory and Applications1611April 2014, 22-55HAL DOI back to text
108 articleM.-C.Marie-Claire Quenez and A.Agnès Sulem. Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps.Stochastic Processes and their Applications1249September 2014, 23HAL back to text
109 articleA.Alexandre Richard, X.Xiaolu Tan and F.Fan Yang. Discrete-time simulation of stochastic Volterra equations.Stochastic Process. Appl.1412021, 109--138URL: https://doi.org/10.1016/j.spa.2021.07.003DOI back to text
110 articleX.Xicheng Zhang. Euler schemes and large deviations for stochastic Volterra equations with singular kernels.J. Differential Equations24492008, 2226--2250URL: https://doi.org/10.1016/j.jde.2008.02.019DOI back to text
111 articleB.Bernt \O{}}ksendal and A.Agn{ès Sulem. Dynamic Robust Duality in Utility Maximization.Applied Mathematics and Optimization2016, 1-31HAL back to text
112 incollectionB.Bernt \O{}}ksendal and A.Agn{ès Sulem. Optimal control of predictive mean-field equations and applications to finance.Springer Proceedings in Mathematics & Statistics138Stochastic of Environmental and Financial EconomicsSpringer Verlag2016, 319HAL DOI back to text back to text

MATHRISK - 2024

MATHRISK - 2024

2024Activity reportProject-TeamMATHRISK

Keywords

Computer Science and Digital Science

Other Research Topics and Application Domains

1 Team members, visitors, external collaborators

Research Scientist

Faculty Members

Post-Doctoral Fellows

PhD Students

Interns and Apprentices

Administrative Assistants

Visiting Scientists

External Collaborators

2 Overall objectives

3 Research program

3.1 Systemic risk in financial networks

3.2 Stochastic Control, optimal stopping and non-linear backward stochastic differential equations (BSDEs) with jumps

Option pricing in incomplete and nonlinear financial market models with default.

Optimal stopping.

Stochastic control with jumps.

3.3 Volatility Modeling

3.4 Insurance modeling

Asset Liability Management.

3.5 (Martingale) Optimal Transport and Mean-field systems

3.5.1 Numerical methods for Optimal transport

3.5.2 Mean-field systems

Mean-field systems and optimal transport.

3.5.3 Martingale Optimal Transport

Quantization.

3.5.4 Martingale Schrödinger problems

3.6 Deep learning for large dimensional financial problems

Neural networks and Machine Learning techniques for high dimensional American options.

3.7 Advanced numerical probability methods and Computational finance

3.7.1 Approximation of stochastic differential equations

High order schemes.

Stochastic Volterra Equations.

Abstract Malliavin calculus and convergence in total variation.

Invariance principles.

Analysis of jump type SDEs.

3.7.2 Monte-Carlo and Multi-level Monte-Carlo methods

Error bounds of MLMC.

3.8 Remarks

4 Application domains

4.1 Financial Mathematics, Insurance

5 Social and environmental responsibility

6 Highlights of the year

6.1 Awards

7 New software, platforms, open data

7.1 New software

7.1.1 PREMIA

7.2 New platforms

8 New results

8.1 Risk processes in stochastic networks

Central Limit Theorems for Price-Mediated Contagion in Stochastic Financial Networks

Ruin Probabilities for Risk Processes in Stochastic Networks

8.2 Graphon Mean-field games

8.2.1 Stochastic Graphon Mean-field Games and approximate Nash Equilibria

8.2.2 Reinforcement Learning for Graphon Mean-Field Games

8.3 Decentraized control problems in energy networks

8.4 Optimal stopping and American options

8.5 Martingale Optimal transport

8.6 Convex order

8.7 Volatility modeling in Insurance and Finance

8.8 Pricing and calibration of path-dependent volatility models

8.9 Numerical probability

8.9.1 Approximations of Stochastic Differential Equations (SDEs)

High order schemes for the weak error for the CIR and Heston processes.

Weak error analysis

Approximation of Stochastic Volterra Equations (SVE).

8.9.2 Abstract Malliavin calculus and convergence in total variation

8.9.3 Sewing Lemma

8.10 Deep learning for large dimensional financial problems

8.11 Quantum Computing in Finance

9 Bilateral contracts and grants with industry

9.1 Bilateral contracts with industry

9.2 Bilateral grants with industry

10 Partnerships and cooperations

10.1 International research visitors