EN FR
EN FR

## Section: New Results

### White Noise-based Stochastic Calculus with respect to Multifractional Brownian Motion

The purpose of this work is to build a stochastic calculus with respect to (mBm) with a view to applications in finance and particularly to stochastic volatility models. We use an approach based on white noise theory.

#### White Noise-based Stochastic Calculus with respect to multifractional Brownian motion

The following results may be found in [28] . Integration with respect to mBm requires stochastic spaces in which we can differentiate or integrate stochastic processes. Considering the probability space $\left({𝕊}^{\text{'}}\left(ℝ\right),𝔹\left({𝕊}^{\text{'}}\left(ℝ\right)\right),\mu \right)$ where $\mu$ is probability measure given by BÃ¶chner Minlos theorem, one can build to spaces, noted $\left(𝒮\right)$ and $\left({𝒮}^{*}\right)$ which will play an analogous role to the spaces $𝕊\left(ℝ\right)$ and ${𝕊}^{\text{'}}\left(ℝ\right)$ for tempered distributions. We recall that $𝕊\left(ℝ\right)$ is the Schwartz space of rapidly decreasing functions which are infinitely differentiable and ${𝕊}^{\text{'}}\left(ℝ\right)$ is the space of tempered distributions. Let us moreover note $\left({L}^{2}\right)$ the space of random variables defined on the probability space $\left({𝕊}^{\text{'}}\left(ℝ\right),𝔹\left({𝕊}^{\text{'}}\left(ℝ\right)\right),\mu \right)$ which admit a second order moment. The mBm ${B}^{\left(h\right)}$ has the following Wiener-ItÃ´ chaos decomposition in $\left({L}^{2}\right)$:

 ${B}^{\left(h\right)}\left(t\right)=\sum _{k=0}^{+\infty }{<{\mathbb{1}}_{\left[0;t\right]},{M}_{h\left(t\right)}\left({e}_{k}\right)>}_{{L}^{2}\left(ℝ\right)}\phantom{\rule{-4.26773pt}{0ex}}<.,\underset{={e}_{k}}{\underbrace{{M}_{h\left(t\right)}\left({d}_{k}^{\left(t\right)}\right)}}>=\sum _{k=0}^{+\infty }\left({\int }_{0}^{t}{M}_{h\left(t\right)}\left({e}_{k}\right)\left(s\right)ds\right)<.,{e}_{k}>$ (6)

where ${\left({e}_{k}\right)}_{k\in ℕ}$ denotes the family of Hermite functions, defined for every integer $k$ in $ℕ$, by ${e}_{k}\left(x\right):={\pi }^{-1/4}{\left({2}^{k}k!\right)}^{-1/2}{e}^{-{x}^{2}/2}{h}_{k}\left(x\right)$ and where ${\left({h}_{k}\right)}_{k\in ℕ}$ is the family of Hermite polynomial, defined for every integer $k$ in $ℕ$, by ${h}_{k}\left(x\right):={\left(-1\right)}^{k}{e}^{{x}^{2}}\frac{{d}^{k}}{d{x}^{k}}\left({e}^{-{x}^{2}}\right)$. Note moreover that ${M}_{H}$ is an operator from $𝕊\left(ℝ\right)$ to ${L}^{2}\left(ℝ\right)$ for every real $H$ in $\left(0,1\right)$ and $<.,{e}_{k}>$ is a centered random Gaussian variable with variance equal to 1 for all $k$ in $ℕ$. We can now define a process, noted ${W}^{\left(h\right)}$, from $ℝ$ to $\left({𝒮}^{*}\right)$, which is the derivative of ${B}^{\left(h\right)}$ in sense of $\left({𝒮}^{*}\right)$ by

 ${W}^{\left(h\right)}\left(t\right)=\sum _{k=0}^{+\infty }\left[\frac{d}{dt}\left({\int }_{0}^{t}\phantom{\rule{2.84544pt}{0ex}}{M}_{h\left(t\right)}\left({e}_{k}\right)\left(s\right)\phantom{\rule{2.84544pt}{0ex}}ds\right)\right]\phantom{\rule{2.84544pt}{0ex}}<.,{e}_{k}>.$ (7)

Hence we define integral with respect to mBm of any process $\Phi :ℝ\to \left({𝒮}^{*}\right)$ as being the element of $\left({𝒮}^{*}\right)$ given by:

 ${\int }_{ℝ}\Phi \left(s,\omega \right)d{B}^{\left(h\right)}\left(s\right)={\int }_{ℝ}\Phi \left(s\right)\diamond {W}^{\left(h\right)}\left(s\right)ds\phantom{\rule{2.84544pt}{0ex}}\left(\omega \right),$ (8)

where $\diamond$ denotes the Wick product on $\left({𝒮}^{*}\right)$. It is then possible to get ItÃ´ formulas and Tanaka formula such as

 $\begin{array}{cc}\hfill {\int }_{0}^{T}\phantom{\rule{2.84544pt}{0ex}}\frac{\partial f}{\partial x}\left(t,{B}^{\left(h\right)}\left(t\right)\right)\phantom{\rule{2.84544pt}{0ex}}d{B}^{\left(h\right)}\left(t\right)& =f\left(T,{B}^{\left(h\right)}\left(T\right)\right)-f\left(0,0\right)-{\int }_{0}^{T}\phantom{\rule{2.84544pt}{0ex}}\frac{\partial f}{\partial t}\left(t,{B}^{\left(h\right)}\left(t\right)\right)\phantom{\rule{2.84544pt}{0ex}}dt\hfill \\ & -\frac{1}{2}\phantom{\rule{2.84544pt}{0ex}}{\int }_{0}^{T}\phantom{\rule{2.84544pt}{0ex}}\left(\frac{d}{dt}\left[{R}_{h}\left(t,t\right)\right]\right)\phantom{\rule{2.84544pt}{0ex}}\frac{{\partial }^{2}f}{\partial {x}^{2}}\left(t,{B}^{\left(h\right)}\left(t\right)\right)\phantom{\rule{2.84544pt}{0ex}}dt.\hfill \end{array}$ (9)

for functions with sub exponential growth and where the last equality holds in ${L}^{2}$.

Once this stochastic calculus with respect to mBm is defined, we can solve differential equations arising in mathematical finance.

#### Multifractional stochastic volatility

Multifractional stochastic volatility

The results of this part may be found in [6] . We assume that, under the risk-neutral measure, the forward price of a risky asset is the solution of the S.D.E.

 $\left\{\begin{array}{c}d{F}_{t}={F}_{t}{\sigma }_{t}d{W}_{t},\hfill \\ dln\left({\sigma }_{t}\right)=\theta \left(\mu -ln\left({\sigma }_{t}\right)\right)dt+{\gamma }_{h}{d}^{\diamond }{B}_{t}^{h}+{\gamma }_{\sigma }d{W}_{t}^{\sigma },\phantom{\rule{1.em}{0ex}}{\sigma }_{0}>0,\phantom{\rule{4pt}{0ex}}\theta >0,\hfill \end{array}\right\$ (10)

where $W$ and ${W}^{\sigma }$ are two standard Brownian motions and ${B}^{h}$ is a multifractional Brownian motion independent of $W$ and ${W}^{\sigma }$ with functional parameter $h$, which is assumed to be continuously differentiable. We assume that $W$ is decomposed into $\rho d{W}_{t}^{\sigma }+\sqrt{1-{\rho }^{2}}d{W}_{t}^{F}$, where ${W}^{F}$ is a Brownian motion independent of ${W}^{\sigma }$. Note that ${d}^{\diamond }{B}_{t}^{h}$ denotes differentiation in the sense of white Noise theory. The solution of the volatility process ${\left({\sigma }_{t}\right)}_{t\in \left[0,T\right]}$ is

 ${\sigma }_{t}\stackrel{a.s.}{=}exp\left(ln\left({\sigma }_{0}\right){e}^{-\theta t}+\mu \left(1-{e}^{-\theta t}\right)+{\gamma }_{\sigma }{\int }_{0}^{t}{e}^{\theta \left(s-t\right)}d{W}_{s}^{\sigma }+{\gamma }_{h}\phantom{\rule{4pt}{0ex}}{e}^{-\theta t}{I}_{t}\left({B}^{h}\right)\right),$ (11)

where ${I}_{t}\left({B}^{h}\right):\stackrel{𝑎.𝑠}{=}{e}^{\theta t}{B}_{t}^{h}-\theta {\int }_{0}^{t}\phantom{\rule{4pt}{0ex}}{e}^{\theta s}\phantom{\rule{4pt}{0ex}}{B}_{s}^{h}\phantom{\rule{4pt}{0ex}}ds$.

Since the solution the previous S.D.E. is not explicit for ${\left({F}_{t}\right)}_{t\in \left[0,T\right]}$ we use preconditioning and then cubature methods in order to get an approximation of it. This model allows to take into account the well-known "smile" effect of volatility, as well as its evolution at various maturities.

#### Approximation of mBm by fBms

In [18] , we establish that a sequence of well-chosen lumped fractional Brownian motions converges in law to a multifractional Brownian motion. This allows to define stochastic integrals with respect to mBm by "transporting" corresponding stochastic integrals with respect to fBm.