Time Reversal of Volterra Processes Driven Stochastic Differential Equations

We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past-dependent stochastic differential equations driven by a standard Brownian motion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning. 1. Introduction Fractional Brownian motion (fBm for short) of Hurst index is the Gaussian process which admits the following representation: for any , where is a one-dimensional Brownian motion and is a triangular kernel, that is, for , the definition of which is given in (46). Fractional Brownian motion is probably the first process which is not a semimartingale and for which it is still interesting to develop a stochastic calculus. That means we want to define a stochastic integral and solve stochastic differential equations driven by such a process. From the very beginning of this program, two approaches do exist. One approach is based on the H？lder continuity or the finite variation of the fBm sample paths. The other way to proceed relies on the gaussianity of fBm. The former is mainly deterministic and was initiated by Z？hle [1], Feyel and de la Pradelle [2], and Russo and Vallois [3, 4]. Then, came the notion of rough paths was introduced by Lyons [5], whose application to fBm relies on the work of Coutin and Qian [6]. These works have been extended in the subsequent works [7–17]. A new way of thinking came with the independent but related works of Feyel, de la Pradelle [18], and Gubinelli [19]. The integral with respect to fBm was shown to exist as the unique process satisfying some characterization (analytic in the case of [18], algebraic in [19]). As a byproduct, this showed that almost all the existing integrals throughout the literature were all the same as they all satisfy these two conditions. Behind each approach, but the last too, is a construction of an integral defined for a regularization of fBm, then the whole work is to show that, under some convenient hypothesis, the approximate integrals converge to a quantity which is called the stochastic integral with respect to fBm. The main tool to prove the convergence is either integration by parts in the sense of fractional deterministic calculus, or enrichment of the fBm by some iterated integrals proved to exist independently or by analytic continuation [20, 21]. In the probabilistic approach [22–30], the idea is also to define an approximate integral and then prove its convergence. It turns out that the key tool is here the integration by