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 {B_z^H, 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) d B_i^γ (s), i = 1, 2, ∫ α (a) d B_a^H, ∫ ∫ β (a, b) d B_a^H d B_b^H, ∫ ∫ β (a, b) d μ (a) d B_b^H, ∫ ∫ β (a, b) d B_a^H d μ (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.