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 H z , z ∈ [0, 1] 2 } be a fractional Brownian sheet with Hurst parameters H = (H 1 , H 2 ), 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) dB H a , β(a, b) dB H a dB H b , β(a, b) dμ(a) dB H b , β(a, b) dB H a dμ (b). As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H 1 , H 2 ∈ (1/4, 1). Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.

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a b s t r a c tBy applying the techniques of Malliavin calculus, we study a central limit theorem of cross-variation related to the standard Brownian sheet. Moreover, we provide the exact Berry-Esseen bound on the Kolmogorov distance by using a recent result proven by Nourdin and Peccati [Nourdin, I., & Peccati, G. (2009b). Stein's method and exact Berry-Esseen asymptotics for functionals of Gaussian fields.

We introduce two types of the Stratonovich stochastic integrals for two-parameter processes, and investigate the relationship of these Stratonovich integrals and various types of Skorohod integrals with respect to a fractional Brownian sheet. By using this relationship, we derive a differentiation formula in the Stratonovich sense for fractional Brownian sheet through Itô formula. Also the relationship between the two types of the Stratonovich integrals will be obtained and used to derive a differentiation formula in the Stratonovich sense. In this case, our proof is based on the repeated applications of differentiation formulas in the Stratonovich form for one-parameter Gaussian processes.

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