Some Questions Associated With Stochastic Calculus Of Fractional Martingales

In this article, we consider the fractional martingale (it is also calledα-martingale) defined bywhere M = {M_t,F_t} is a continuous local martingale andα∈(?) is the index of fractional martingale. This process arises from the study of fractional Brownian motion. It is the extension of the following special situation:where B = B_t, t≥0 is a standard Brownian motion. It is more and more popular since it has a lot of properties similar to martingale and fractional Brownian motion. However, there has been little systematic investigation on this process. In this article we mainly consider the fractional Gaussian martingale. That is, we consider the fractional martingale when M is a Gaussian martingale. We discuss some questions associated with stochastic calculus of this process.We first obtain the stochastic integral of a measurable process u = {u_t,t≥0}. with respect to any fractional martingale M^(α)and obtain an Ito formula. Next we consider the local time and Tanaka formula of fractional martingales. Finally, we investigate the weighted quadratic covariation (?) of f(M^(α)) and M^(α), defined bywhere the limit is uniform in probability. We prove the existence of (?) for some special fractional martingale M^(α). We construct a Banach space W_p of measurable functions such that the weighted quadratic covariation exists for f∈W_p. As an application, we give a generalized the Ito formula.