Various types of stochastic integrals with respect to fractional Brownian sheet and their applications

In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let {BH z , z ∈ [0, 1]2} be a fractional Brownian sheet with Hurst parameters H = (H1,H2), and ([0, 1]2,B([0, 1]2), μ) a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in [0, 1]2, and four types of stochastic surface integrals: ϕ(s) dB γ i (s), i = 1, 2, α(a) dBH a , β(a,b)dBH a dBH b , β(a,b)dμ(a)dBH b , β(a,b)dBH a dμ(b). As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H1,H2 ∈ (1/4, 1). Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process. © 2007 Elsevier Inc. All rights reserved