A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one
We will focus — in dimension one — on the SDEs of the type dX t = σ(X t )dB t + b(X t )dt where B is a fractional Brownian motion. Our principal aim is to describe a simple theory — from our point of view — allowing to study this SDE, and this for any H∈(
- PDF / 298,373 Bytes
- 17 Pages / 439.37 x 666.142 pts Page_size
- 66 Downloads / 147 Views
tic differential equation; fractional Brownian motion; Russo-Vallois integrals; Newton-Cotes functional; Approximation schemes; Doss-Sussmann transformation. MSC 2000: 60G18, 60H05, 60H20.
1 Introduction The fractional Brownian motion (fBm) B = {Bt , t ≥ 0} of Hurst index H ∈ (0, 1) is a centered and continuous Gaussian process verifying B0 = 0 a.s. and (1) E[(Bt − Bs )2 ] = |t − s|2H for all s, t ≥ 0. Observe that B 1/2 is nothing but standard Brownian motion. Equality (1) implies that the trajectories of B are (H − ε)-H¨older continuous, for any ε > 0 small enough. As the fBm is selfsimilar (of index H) and has stationary increments, it is used as a model in many fields (for example, in hydrology, economics, financial mathematics, etc.). In particular, the study
182
I. Nourdin
of stochastic differential equations (SDEs) driven by a fBm is important in view of the applications. But, before raising the question of existence and/or uniqueness for this type of SDEs, the first difficulty is to give a meaning to the integral with respect to a fBm. It is indeed well-known that B is not a semimartingale when H = 1/2. Thus, the Itˆ o or Stratonovich calculus does not apply to this case. There are several ways of building an integral with respect to the fBm and of obtaining a change of variables formula. Let us point out some of these contributions: 1. Regularization or discretization techniques. Since 1993, Russo and Vallois [31] have developed a regularization procedure, whose philosophy is similar to the discretization. They introduce forward (generalizing Itˆ o), backward, symmetric (generalizing Stratonovich, see Definition 3 below) stochastic integrals and a generalized quadratic variation. The regularization, or discretization technique, for fBm and related processes have been performed by [12, 17, 32, 36], in the case of zero quadratic variation (corresponding to H > 1/2). Note also that Young integrals [35], which are often used in this case, coincide with the forward integral (but also with the backward or symmetric ones, since covariation between integrand and integrator is always zero). When the integrator has paths with finite p-variation for p > 2, forward and backward integrals cannot be used. In this case, one can use some symmetric integrals introduced by Gradinaru et al. in [15] (see Section 2 below). We also refer to Errami and Russo [11] for the specific case where H ≥ 1/3. 2. Rough paths. An other approach was taken by Lyons [20]. His absolutely pathwise method based on L´evy stochastic areas considers integrators having p-variation for any p > 1, provided one can construct a canonical geometric rough path associated with the process. We refer to the survey article of Lejay [18] for more precise statements related to this theory. Note however that the case where the integrator is a fBm with index H > 1/4 has been studied by Coutin and Qian [7] (see also Feyel and de La Pradelle [13]). See also Nourdin and Simon [26] for a link between the regularization technique and the rough paths theory. 3. Malliavin calc
Data Loading...
