Preprint 2003-011

On the Rate of Convergence of Approximation Schemes for Bellman Equations Associated with Optimal Stopping Time Problems

Abstract: We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes.These results extend (and slightly improve) resent results by Barles \& Jakobsen to the more difficult time dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles \& Jakobsen. As applications we consider so-called control schemes based on the dynamic programming principle and finite difference methods (though not in the most general case). In the optimal stopping case these methods are similar to the Brennan \& Schwartz scheme. A simple observation allow us to obtain the optimal rate $1/2$ for the finite difference methods, and this is an improvement over previous results by Krylov and Barles \& Jakobsen. Finally, we present an idea that allow us to improve all the above mentioned results in the linear case. In particular, we are able to handle finite difference methods with variable diffusion coefficients without the reduction of order of convergence observed by Krylov in the non-linear case.

Paper:

Available as PostScript (384 Kbytes) or gzipped PostScript (144 Kbytes; uncompress using gunzip).

Author(s):

Espen R. Jakobsen, <erj@math.ntnu.no>

Publishing information:

To appear in Math. Models Methods Appl. Sci. (M3AS)

Comments:

Submitted by:

<erj@math.ntnu.no> January 29 2003.