No abstract

We extend to the multidimensional case a Wong-Zakai-type theorem proved by Hu and Øksendal in [7] for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô's SDEs, we consider for any finite partition of the time interval [0, T ] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally lead to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong-Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker-Planck equation and that the sequence converges to the solution of the Itô equation, as the mesh of the partition tends to zero.

We investigate some probabilistic aspects of the unique global strong solution of a two dimensional system of stochastic differential equations describing a preypredator model perturbed by Gaussian noise. We first establish, for any fixed t > 0, almost sure upper and lower bounds for the components X(t) and Y (t) of the solution vector: these explicit estimates emphasize the interplay between the various parameters of the model and agree with the asymptotic results found in the literature. Then, standing on the aforementioned bounds, we derive upper and lower estimates for the joint moments and distribution function of (X(t), Y (t)). Our analysis is based on a careful use of comparison theorems for stochastic differential equations and exploits several peculiar features of the noise driving the equation.

We consider the (unique) mild solution u(t, x) of a 1-dimensional stochastic heat equation on [0, T ] × R driven by time-homogeneous white noise in the Wick-Skorokhod sense. The main result of this paper is the computation of the spatial derivative of u(t, x), denoted by ∂xu(t, x), and its representation as a Feynman-Kac type closed form. The chaos expansion of ∂xu(t, x) makes it possible to find its (optimal) Hölder regularity especially in space.

In a recent article Lanconelli and Scorolli (2021) extended to the multidimensional case a Wong-Zakai-type approximation for Itô stochastic differential equations proposed by Øksendal and Hu (1996). The aim of the current paper is to extend the latter result to system of stochastic differential equations of Itô type driven by fractional Brownian motion like those considered by Hu (2018).The covariance structure of the fractional Brownian motion (fBm) precludes us from using the same approach as that used by Lanconelli and Scorolli and instead we employ a truncated Cameron-Martin expansion as the approximation for the fBm. We are naturally led to the investigation of a semilinear hyperbolic system of evolution equations in several space variables that we utilize for constructing a solution of the Wong-Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker-Planck equation and that the sequence converges to the solution of the Itô equation, as the number of terms in the expansion goes to infinite.