Occupation densities of stratonovitch stochastic differential equations with boundary conditions

Abstract

To solve diffusion type stochastic differential equations in which the Stratonovitch integral describes the coupling to the driving Wiener process, and with boundary conditions at the endpoints of the parameter space, the unit interval, one can consider the “flow” of diffusions corresponding to the initial condition , and look for a random variable X 0 such that X 0 and satisfy tne given boundary condition. Then solves the equation. By using semimartingale inequalities, we prove that the corresponding “flow” L(x) of occupation densities of is continuous in x. This way we are able to identify occupation densities of solutions of the diffusion type equations with coupled boundary data, and give conditions under which they are continuous.