Section: New Results

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

Participants : Joachim Lebovits, Jacques Lévy Véhel.

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 $({𝕊}^{\text{'}}\left(ℝ\right),𝔹\left({𝕊}^{\text{'}}\left(ℝ\right)\right),\mu )$ where $\mu$ is probability measure given by Böchner 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 $({𝕊}^{\text{'}}\left(ℝ\right),𝔹\left({𝕊}^{\text{'}}\left(ℝ\right)\right),\mu )$ which admit a second order moment. The mBm $B^{\left(h\right)}$ has the following Wiener-Itô chaos decomposition in $\left(L^2\right)$:

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 $𝕊\left(ℝ\right)$ to $L^2\left(ℝ\right)$ 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 $ℝ$ to $\left({𝒮}^{*}\right)$, which is the derivative of $B^{\left(h\right)}$ in sense of $\left({𝒮}^{*}\right)$ by

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:

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

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.

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

where $I_t\left(B^h\right):\stackrel{𝑎.𝑠}{=}{e}^{\theta t}{B}_t^h-\theta {\int }_0^t{e}^{\theta s}{B}_s^{h}ds$.

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.