## Section: New Results

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

Participants : Joachim Lebovits, Jacques LÃ©vy VÃ©hel.

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

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

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

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

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

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

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

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

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

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

Since the solution the previous S.D.E. is not explicit for ${\left({F}_{t}\right)}_{t\in [0,T]}$ 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.