Rough limit theorems on large deviations for Markov stochastic processes.~II

A. D. Wentzell

Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 3, pp. 512-526

Résumé

The first part of this paper was published in vol. XXI, No. 2 (1976) of this journal. In the second part we deal not with Markov processes but rather with their characteristics. The results of § 5 are used to establish properties of the standardized action functional and the relation between it and the action functional. In § 7 we verify, in some natural cases, a somewhat intricate continuity condition imposed on the characteristics of the Markov processes under consideration. In § 6 we introduce two classes (discrete-time and continuous-time) of families of Markov processes, and we prove our main results. Markov processes $\xi^{\alpha}(t)$ of these families have jumps becoming smaller and smaller as $\alpha$ increases but more and more frequent (and the diffusion parts of the processes are changing in accordance with the jumps). The most close to our results are those of [2] (concerning processes with independent increments) and [3] (concerning diffusion processes). Our results are valid for a wider class of families of Markov processes which includes both families of processes with independent increments and of diffusion processes with small diffusion. The results of the present paper are analogous to limit theorems on large deviations for sums of independent random variables concerning «very large» deviations of order $\sqrt n$ (for precise results for sums of independent random variables see [1], Theorem 6). The case, analogous to that of «not very large» deviations (of order $o(\sqrt n)$ see [1], Theorem 1), will be considered in another paper.