ASYMPTOTIC EXPANSION FOR THE FUNCTIONAL OF MARKOVIAN EVOLUTION IN Rd IN THE CIRCUIT OF DIFFUSION APPROXIMATION

The problems of asymptotic expansion for solutions of PDE and PDE systems were studied by many authors. A lot of references could be found in [5]. As a rule, border problems are studied with the small parameter being denoted at the higher derivative by t. For example, in [8, page 155] the system of first-order equations is studied with the small parameter denoted by t and x that corresponds to the telegraph equation. In this paper we study asymptotic expansion for solution of singularly perturbed equation for functional of Markovian evolution in Rd. Let x ∈ Rd and ξ(s) is an ergodic Markovian process in the set E = {1, . . . ,N} with the intensity matrix Q = {qi j , i, j = 1,N}. The probability of being in the ith state longer than t is P{θi > t} = e−qit, where qi = ∑ j =i qi j . Let a(i) = (a1(i), . . . ,ad(i)) be a vector-function on E. We regard a vector-function as a corresponding vector-column. Put matrix A= {ak(i), k = 1,d, i= 1,N}. We study evolution