论文信息 - A martingale method for the convergence of a sequence of processes to a jump-diffusion process

A martingale method for the convergence of a sequence of processes to a jump-diffusion process

SummaryA convenient method for proving weak convergence of a sequence of non-Markovian processesxε(·) to a jump-diffusion process is proved. Basically, it is shown that the limit solves the martingale problem of Strook and Varadhan. The proofs are relatively simple, and the conditions apparently weaker than required by other current methods (in particular, for limit theorems for a sequence of ordinary differential equations with random right hand sides). In order to illustrate the relative ease of applicability in many cases, a simpler proof of a known result on averaging is given.

H. Kushner

[1] R. Has’minskiĭ. On Stochastic Processes Defined by Differential Equations with a Small Parameter , 1966 .

[2] R. Khas'minskii. A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides , 1966 .

[3] S. Varadhan,et al. Diffusion processes with continuous coefficients, I , 1969 .

[4] Thomas G. Kurtz,et al. Extensions of Trotter's operator semigroup approximation theorems , 1969 .

[5] T. Kurtz. A general theorem on the convergence of operator semigroups , 1970 .

[6] S. Varadhan,et al. Diffusion processes with boundary conditions , 1971 .

[7] Thomas G. Kurtz,et al. Semigroups of Conditioned Shifts and Approximation of Markov Processes , 1975 .

[8] A. Borodin,et al. A limit theorem for solutions of di erential equations with random right-hand side , 1978 .

[9] G. Papanicolaou,et al. Stability and Control of Stochastic Systems with Wide-band Noise Disturbances. I , 1978 .

[10] Harold J. Kushner,et al. Jump-Diffusion Approximations for Ordinary Differential Equations with Wide-Band Random Right Hand Sides, , 1979 .

[11] Harold J. Kushner,et al. Approximation of solutions to differential equations with random inputs by diffusion processes , 1979 .