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By using the white noise theory for fractional Brownian sheet, we prove the stochastic Green’s theorem with respect to fractional Brownian sheet with Hurst parameters H 1 , H 2 ∈ ( 0 , 1 ) . As an application, we derive an Ito formula for fractional Brownian sheet with H 1 , H 2 ∈ ( 0 , 1 ) .

By 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. Annals of Probability, 37(6), 2231–2261].

비대체와 회귀대체는 조사변수의 모형과 조사변수와 보조변수의 관계에 의존하며 모형이 성립되지 않는 경우 이들 대체법을 이용한 추정량의 불편성은 보장되지 않는다. 본 연구에서는 모형이 성립되지 않는 경우에도 추정량의 근사적 불편성이 성립되는 로버스트 대체법을 개발한다. 대체법 개발시 보조변수의 모수 정보를 이용하여 추정량의 효율 증대를 가져오게 한다. 모의실험을 실시하여 본 연구에 대한 이론적 결과의 타당성을 보인다.

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 Ito 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.

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 Ito formula for fractional Brownian sheet with Hurst parameters H 1 , H 2 ∈ ( 1 / 4 , 1 ) . Our proof is based on the repeated applications of Ito formula for one-parameter Gaussian process.

By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the fractional Brownian sheet with arbitrary Hurst parameters H 1 , H 2 ∈ ( 0 , 1 ) H_{1}, H_{2}\in (0,1) .

We study the asymptotic expansion in small time of the mean distance of Brownian motion on Riemannian manifolds. We compute the first four terms of the asymptotic expansion of the mean distance by using the decomposition of Laplacian into homogeneous components. This expansion can be expressed in terms of the scalar valued curvature invariants of order 2, 4, 6.

We present an extension of the Wong-Zakai type approximation theorem for a multiple stochastic integral. Using a piecewise linear approximation w (n) of a Wiener process w, we prove that the multiple integral process where f is a given symmetric function in the space 𝒞([0, T] m ), converge to the multiple Stratonovich integral of f in the uniform L 2-sense.

We present an extension of the Wong–Zakai approximation theorem for a stochastic differential equation on the plane driven by a two-parameter Wiener process. For an approximation of the two-parameter Wiener process, we use a two-parameter version of the one-parameter piecewise linear approximation. By our approximation to the two-parameter Wiener process we show that the solution of an ordinary differential equation converges, in the uniform L 2-sense, to that of a stochastic differential equation obtained by using Stratonovich integral. *This research was supported (in part) by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.

We prove that the logarithm of the formal power series, obtained from a stochastic differential equation, is an element in the closure of the Lie algebra generated by vector fields being coefficients of equations. By using this result, we obtain a representation of the solution of stochastic differential equations in terms of Lie brackets and iterated Stratonovich integrals in the algebra of formal power series.