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Section: New Results

Dependence modeling

With his PHD student J. Reygner, B. Jourdain has studied a mean-field version of rank-based models of equity markets, introduced by Fernholz in the framework of stochastic portfolio theory ([38] ). When the number of companies grows to infinity, they obtain an asymptotic description of the market in terms of a stochastic differential equation nonlinear in the sense of McKean. The diffusion and drift coefficients depend on the cumulative distribution function of the current marginal law of the capitalizations. Using results on the longtime behavior of such SDEs derived in [66] , they discuss the long-term capital distribution in this asymptotic model, as well as the performance of simple portfolio rules. In particular, they highlight the influence of the volatility structure of the model on the growth rates of portfolios.

Another approach to handle the question of stochastic modeling in a multidimensional framework consists in dealing with stochastic differential equations that are defined on matrices in order to model either the instantaneous covariance or the instantaneous correlation between the assets.

The research on the estimation of the parameters of a Wishart process has started this year together with the thesis of Clément Rey. A. Alfonsi, A. Kebaier and C. Rey are studying the Maximum Likelihood Estimator for the Wishart processes and in particular its convergence in the ergodic and the non ergodic case.

Correlation issues are crucial in the modeling of volatility. In his thesis, Ould Aly ( [77] ) proposes a revised version of Bergomi's model for the variance curve which proves to be very tractable for calibration and for the pricing of variance derivatives (see [23] ). He also obtains results on the monotonicity of option prices with respect to the correlation between the stock price and the volatility in the Heston model (see [78] ).

In [34] , [15] , L. Abbas-Turki and D. Lamberton study the monotonicity of option prices with respect to cross-asset correlations in a multidimensional Heston model.

Modeling the dependence is not only useful for the equity market. In credit risk, getting a model that describes the dynamic of the joint distribution of a basket of defaults is still a challenge. The Loss Intensity model proposed by Schönbucher allows to fit perfectly the marginal distributions of the number of defaults in a basket. Then, Stochastic Loss Intensity models extend this model and can also in principle fit the marginal distributions. However, these models appear as a non-linear differential equation with jumps. A Alfonsi, C. Labart and J. Lelong have shown that these models are well-defined by using a particles system ( [44] ). Besides, this particles system gives a very convenient way to run a Monte-Carlo algorithm and to compute expectations in this model. Interacting particle systems are studied by B. Jourdain and his PhD student Julien Reygner in [39] , [21] .

Application of optimal transport. A. Alfonsi and B. Jourdain study in [43] the Wasserstein distance between two probability measures in dimension n sharing the same copula C. The image of the probability measure dC by the vectors of pseudo-inverses of marginal distributions is a natural generalization of the coupling known to be optimal in dimension n=1. In dimension n>1, it turns out that for cost functions equal to the p-th power of the Lq norm, this coupling is optimal only when p=q i.e. when the cost function may be decomposed as the sum of coordinate-wise costs.