The generalized Bouleau-Yor identity for a sub-fractional Brownian motion

Let S (H) be a sub-fractional Brownian motion with index 0 < H < 1/2. In this paper we study the existence of the generalized quadratic covariation [f(S (H) ), S (H) ]((W)) defined by [f(S-H), S-H](t)((W)) = lim(epsilon down arrow 0) 1/(epsilon)2H integral(t)(0){f(S-s+epsilon(H)) - f(S-s(H))}(S-s+epsilon(H) - S-s(H))ds(2H), provided the limit exists in probability, where x -> f(x) is a measurable function. We construct a Banach space H of measurable functions such that the generalized quadratic covariation exists in L-2 provided f is an element of H. Moreover, the generalized Bouleau-Yor identity takes the form integral(R)f(x)L-H(dx,t) - (2 - 2(2H-1))[f(S-H), S-H](t)((W)) for all f is an element of H, where L-H (x, t) is the weighted local time of S-H . This allows us to write the generalized Ito's formula for absolutely continuous functions with derivative belonging to H.