Dependence on the boundary condition for linear stochastic differential equations in the plane
The following linear stochastic differential equation is considered: \[ X(z,f)=f(s,0)+f(0,t)-f(0,0)+a\int_{(0,z]}X(z_ 1,f) W(dz_ 1), \] where \(z=(s,t)\in R^ 2_+\), \(a\in R\), W is a two-parameter Wiener process, and f is a real-valued continuous function on \(\partial R^ 2_+\). The strong solution of this equation is represented as a sum of series of multiple Ito-Wiener integrals of the function f. The main result of the paper is the following: If f(0,\(\cdot)\) and f(\(\cdot,0)\) are functions of bounded variation on every finite interval then there exists a version of the strong solution X(z,f) such that for any \(z_ 0\in R^ 2_+\) the mapping \(f(\cdot)\to X(z_ 0+\cdot,f)\) is linear and continuous with respect to the family of total variation seminorms as well as to families of \(\alpha\)-Hölder seminorms for \(\alpha\in (0,1/2).As\) a corollary, the existence of such a version of X(z,f) that the mapping (z,f)\(\to X(z,f)\) is jointly continuous is proven.
scientific article; zbMATH DE number 4135146