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In this paper, we consider a class of backward doubly stochastic differential equations (BDSDEs for short) with general terminal value and general random generator. Those BDSDEs do not involve any forward diffusion processes. By using the techniques of Malliavin calculus, we are able to establish the L p -Hölder continuity of the solution pair. Then, an implicit numerical scheme for the BDSDE is proposed and the rate of convergence is obtained in the L p -sense. As a by-product, we obtain an explicit representation of the process Y in the solution pair to a linear BDSDE with random coefficients. [ABSTRACT FROM AUTHOR]

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: ∂ β + ν 2 (− Δ) α ∕ 2 u (t , x) = I t γ ρ (u (t , x)) W ̇ (t , x) , t > 0 , x ∈ R d , where W ̇ is the space–time white noise, α ∈ (0 , 2 ] , β ∈ (0 , 2) , γ ≥ 0 and ν > 0. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang's condition: d < 2 α + α β min (2 γ − 1 , 0). In some cases, the initial data can be measures. When β ∈ (0 , 1 ] , we prove the sample path regularity of the solution. [ABSTRACT FROM AUTHOR]

Abstract We present an explicit solution triplet (Y , Z , K) to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a compensated Poisson random measure. The process Y is expressed by an integral whose kernel is explicitly given. The processes Z and K are expressed by Hida–Malliavin derivatives involving Y. [ABSTRACT FROM AUTHOR]

This paper studies the weak and strong solutions to the stochastic differential equation d X ( t ) = − 1 2 W ̇ ( X ( t ) ) d t + d B ( t ) , where ( B ( t ) , t ≥ 0 ) is a standard Brownian motion and W ( x ) is a two sided Brownian motion, independent of B . It is shown that the Itô–McKean representation associated with any Brownian motion (independent of W ) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Itô calculus for the solution is developed. For dealing with the singularity of drift term ∫ 0 T W ̇ ( X ( t ) ) d t , the main idea is to use the concept of local time together with the polygonal approximation W π . Some new results on the local time of Brownian motion needed in our proof are established. [ABSTRACT FROM AUTHOR]

Abstract: We extend the classical Garsia–Rodemich–Rumsey inequality to the multiparameter situation. The new inequality is applied to obtain some joint Hölder continuity along the rectangles for fractional Brownian fields and for the solution of the stochastic heat equation with additive white noise. [Copyright &y& Elsevier]

Abstract: In this paper, we establish a version of the Feynman–Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman–Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogeneous Gaussian noise: first, an explicit expression for the Malliavin derivatives of the solutions is obtained. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the Hölder continuity of the solutions. [Copyright &y& Elsevier]

Abstract: We study the problem of parameter estimation for generalized Ornstein–Uhlenbeck processes driven by -stable noises, observed at discrete time instants. Least squares method is used to obtain an asymptotically consistent estimator. The strong consistency and the rate of convergence of the estimator have been studied. The estimator has a higher order of convergence in the general stable, non-Gaussian case than in the classical Gaussian case. [Copyright &y& Elsevier]

We prove a stochastic maximum principle for controlled processes X(t)=X(u)(t) of the formdX(t)=b(t,X(t),u(t)) dt+σ(t,X(t),u(t)) dB(H)(t),where B(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter H=(H1,…,Hm)∈(1/2,1)m. As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian motion. [Copyright &y& Elsevier]