On the rate of convergence of the finite-difference approximations for parabolic Bellman equations with constant coefficients.

摘要

The error bounds of order h + tau½ for two types of finite-difference approximation schemes of parabolic Bellman equations with constant coefficients are obtained, where h is x-mesh size and tau is t-mesh size. The key methods employed are the maximum principles for the Bellman equation and the approximation schemes.; The difference of two finite-difference approximation schemes lies in the step sizes in t. In one scheme, the step sizes are fixed and always equal tau. In the other scheme, starting from the left end point T of the interval [0, T], the first step size is T-t for t ∈ [T - tau, T] and step size is always tau after that. We have that the solution v˜h,,tau of the latter is Lipschitz in t while the Lipschitz continuity in t of the solution vh,tau of the former is unknown.; The main ideas of the proof of the convergence rate are based on earlier work by Dong and Krylov. The same convergence rate was proved by Dong and Krylov for elliptic Bellman equation. Herein we extend the result to the case of parabolic Bellman equation. As far as we know, the analysis of the solution v˜h,,tau is new. By using an induction approach, we also give new proof of the smoothness of solutions v h,,tau (v˜h, ,tau) and the comparison principle of the approximation schemes.