Jump-Diffusion Approximations for Ordinary Differential Equations with Wide-Band Random Right Hand Sides

Abstract

Let $y( \cdot )$ be a stationary mixing process and $J^\varepsilon ( \cdot )$ an approximation to a random impulsive process. Kurtz’s (1975) results on approximation of a general semigroup by a Markov semigroup are used to prove (weak and a similar type of) convergence of the solutions to (1.1) and (1.2) to jumping diffusions. Previous results are generalized in various ways. The case of unbounded $y( \cdot )$ is also treated as is the combined jump-diffusion case. Also, a limit theorem for an integral with respect to “approximate white noise” in terms of an Itôintegral is given. The method has the advantages of generality and relative ease of use.