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

AbstractIn this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let {BzH,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)dBiγ(s), i=1,2, ∫α(a)dBaH, ∫∫β(a,b)dBaHdBbH, ∫∫β(a,b)dμ(a)dBbH, ∫∫β(a,b)dBaHdμ(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