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Various types of stochastic integrals with respect to fractional Brownian sheet and their applications |
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| Authors: | Yoon Tae Kim Jong Woo Jeon |
| Institution: | a Department of Statistics, Hallym University, Chuncheon, Gangwon-Do 200-702, South Korea b Department of Statistics, Seoul National University, Sillim-Dong, Gwanak-Gu, Seoul 151-742, South Korea c Centre for Mathematics and its Applications, Australian National University, ACT 0200, Australia d Statistical Research Center for Complex Systems, Seoul National University, Sillim-Dong, Gwanak-Gu, Seoul 151-742, South Korea |
| 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  be a fractional Brownian sheet with Hurst parameters H=(H1,H2), and (20,1],B(20,1]),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 20,1], and four types of stochastic surface integrals:  , i=1,2,  ,  ,  ,  . 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. |
| Keywords: | Skorohod integrals Itô formula Fractional Brownian sheet Malliavin derivative Stochastic line integrals |
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