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  . Integration with respect to mBm requires stochastic spaces in which we can differentiate or integrate stochastic processes. Considering the probability space where is probability measure given by BÃ¶chner Minlos theorem, one can build to spaces, noted and which will play an analogous role to the spaces and for tempered distributions. We recall that is the Schwartz space of rapidly decreasing functions which are infinitely differentiable and is the space of tempered distributions. Let us moreover note the space of random variables defined on the probability space which admit a second order moment. The mBm has the following Wiener-ItÃ´ chaos decomposition in :
where denotes the family of Hermite functions, defined for every integer in , by and where is the family of Hermite polynomial, defined for every integer in , by . Note moreover that is an operator from to for every real in and is a centered random Gaussian variable with variance equal to 1 for all in . We can now define a process, noted , from to , which is the derivative of in sense of by
Hence we define integral with respect to mBm of any process as being the element of given by:
where denotes the Wick product on . It is then possible to get ItÃ´ formulas and Tanaka formula such as
for functions with sub exponential growth and where the last equality holds in .
Once this stochastic calculus with respect to mBm is defined, we can solve differential equations arising in mathematical finance.
Multifractional stochastic volatilityMultifractional stochastic volatility
The results of this part may be found in  . We assume that, under the risk-neutral measure, the forward price of a risky asset is the solution of the S.D.E.
where and are two standard Brownian motions and is a multifractional Brownian motion independent of and with functional parameter , which is assumed to be continuously differentiable. We assume that is decomposed into , where is a Brownian motion independent of . Note that denotes differentiation in the sense of white Noise theory. The solution of the volatility process is
Since the solution the previous S.D.E. is not explicit for 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  , 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.