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

Abstract

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 = ( 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 ) 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 H 1 , H 2 ∈ ( 1 / 4 , 1 ) . Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.