Interior estimates for second-order differences of solutions of finite-difference elliptic Bellman's equations

We establish interior estimates for the second-order finite differences of solutions of finite-difference approximations for uniformly elliptic Bellman’s equations. This article is a continuation of [15], where interior estimates for the first-order differences of solutions of finite-difference elliptic Bellman’s equations are obtained. However, the present article can be read independently since we only use a few elementary facts from [15] and for that reason we repeat part of the introduction from [15]. We are dealing with the second-order differences of solutions of finitedifference equations of the type (0.1) sup α∈A [ d1 ∑ |k|=1 [ ak (x)Δh,lkv(x) + b α k (x)δh,lkv(x) ] − c(x)v(x) + f(x) ] = 0, where x = (x, ..., x) ∈ R, Δh,lkv and δh,lkv are finite-difference approximations of the pure second-order derivative in the direction lk and the first-order derivative in the direction lk, respectively (see (1.1)), lk, k = ±1, ...,±d1, are fixed vectors in R, h > 0 is the step size of finite differences, d1 ≥ d is a fixed integer, A is a set, and ak (x), b α k (x), c (x), and f(x) are given functions on A×R, the detailed assumptions on which will be given later. For the reader more familiar with the theory of controlled Markov chains than with the theory of elliptic equations it is worth pointing out that under Assumption 1.1 the sum in (0.1) is rewritten as ∑ l∈Λ P h (x, l)[v(x+ hl)− v(x)], where Λ = {lk : |k| = 1, ..., d1} and P h (x, l) = h 2ak (x) + h bk (x) if l = lk. In addition, P α h > 0 under Assumption 1.2. Equation (0.1) is a finite-difference approximation for the following Bellman’s equation arising, for instance, in the theory of controlled diffusion processes (see, Received by the editor March 29, 2011 and, in revised form, November 25, 2011. 2010 Mathematics Subject Classification. Primary 35J60, 39A14.