Properties of solutions of stochastic differential equations with random coefficients, non-Lipschitzian diffusion, and Poisson measures

Abstract

Abstract. The existence and uniqueness of a solution of a stochastic differential equation with random coefficients, non-Lipschitzian diffusion, and with centered as well as with non-centered Poisson measures are proved. We estimate the probability that a solution eventually becomes negative. We find conditions for the existence of a nonnegative solution.

Rate of convergence in the Euler scheme for stochastic differential equations with non-Lipschitz diffusion and Poisson measure

“…Assume that the coefficients of the stochastic differential equation (1) are such that the following conditions for the existence and uniqueness of a nonnegative strong solution are satisfied [8]:…”

On a Class of Stochastic Differential Equations with Random and Hölder Continuous Coefficients Arising in Biological Modeling

“…Our idea is to embed the rather special system of SDE's of the model in a slightly more encompassing class, like the one in (3.9) below, in order to establish a general proof of strong existence and uniqueness. Our technique relies on the construction of an explicit approximating sequence of stochastic processes (inspired by the work of Zubchenko [13]) in such a way that all the relevant features of the solution appear to be directly constructed from scratch. In Section 3 we give a detailed proof of existence and uniqueness of the SDE (3.9).…”