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

Abstract

Abstract. Elliptic Bellman equations with coefficients independent of the variable x are considered. Error bounds for certain types of finite-difference schemes are obtained. These estimates are sharper than the earlier results in Krylov's article of 1997. §1. IntroductionOur main purpose in this paper is to present some new estimates for the rate of convergence of finite-difference approximations in the problem of finding viscosity or probabilistic solutions of degenerate elliptic Bellman equations. Historically, the first estimates were obtained in [8] for the equations with "constant" coefficients. There, the convergence rate of order h 1/3 , where h is the mesh size, was established in the case where the "free" term is Lipschitz continuous and the finite-difference approximations are monotone, translation invariant, and apart from that, almost arbitrary with the order of consistency h. It was also shown in [8] that in the general framework the order of accuracy cannot be better than h 1/2 . After that, in [9] the results were extended to parabolic degenerate Bellman equations with variable coefficients. It was proved that if the data are Hölder 1/2 continuous in the time variable and Lipschitz continuous in the space variables, then, again, the approximation error for quite arbitrary finite-difference approximations admits an estimate of order of accuracy h 1/3 from one side and of order of accuracy h 1/21 from the other. Also in [9], some approximations similar to finitedifference ones were suggested with the order of accuracy not less than h 1/3 ; see, e.g., Theorem 5.7 in [9]. This theorem is close to Theorem 3.5 in [1], where in a particular elliptic situation the error bound of order of accuracy h 1/2 was obtained on the account of a special approximation. The authors of [1] and [2] did a very good job of surveying the literature related to finite-difference approximations for the Hamilton-Jacobi and Bellman equations; instead of copying their comments, we allow ourselves to refer the interested reader to [1] and [2] for that information.One of the purposes in the papers [9] and [10] (the latter is the basis for the former) was not only to give the rates of convergence, but also prove the convergence itself. In a subsequent paper, the second author intends to establish such convergence for general approximation schemes in the case where the limit function is not a viscosity solution of the corresponding Bellman equation.2000 Mathematics Subject Classification. Primary 65M15, 35J60, 93E20.

An unconditionally monotone numerical scheme for the two-factor uncertain volatility model

“…In general, no fixed point stencil finite difference scheme can produce a monotone scheme for arbitrary two factor diffusion tensors (Dong and Krylov, 2006). To ensure monotonicity for problems with non-constant diffusion tensors, first order wide stencil methods have been suggested.…”

On the Rate of Convergence of Difference Approximations for Uniformly Nondegenerate Elliptic Bellman’s Equations

“…We need two additional auxiliary results. Estimate (5.3) is quite similar to (4.6) of [6] the latter being a particular case of the former which occurs when N * 1 = 0 (in (4.6) of [6] it is assumed that u ∈ C 0,1 ). Lemma 5.1.…”

“…Next assume that a and c are independent of x as in [6]. Then we can mollify all terms in (5.10) and obtain that in Ω h+ε we have…”

“…In such cases the optimal rate h 1/2 was obtained in the joint work of Hongjie Dong and the author [7] for parabolic Bellman's equations with Lipschitz coefficients in domains. Both ideas of symmetry and "shaking the coefficients" is used in [7] as well as in [6]. In the paper by Hongjie Dong and the author [6] we consider among other things weakly nondegenerate Bellman's equations with constant "coefficients" in the whole space and obtain the rate of convergence h, where h is the mesh size.…”

“…Both ideas of symmetry and "shaking the coefficients" is used in [7] as well as in [6]. In the paper by Hongjie Dong and the author [6] we consider among other things weakly nondegenerate Bellman's equations with constant "coefficients" in the whole space and obtain the rate of convergence h, where h is the mesh size. It may be tempting to say that this result is an improvement of earlier results, however it is just a better rate under different conditions.…”