New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

Abstract

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h 3/2 ) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h 2 ).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.

The efficient solution of the (quietly constrained) noisy, linear regulator problem

“…The algorithm given below is easily modified as in [4] to accommodate different boundary conditions. Our results will be sufficient to solve the noisy-quietly constrained problems we have considered which form the major interest for applications.…”

“…and The error ofŌ(h 7/2 ) comes from estimates of higher order iterated integrals [4]. Motivated by the deterministic case [5], [6], or [7], our algorithm is given in (5.6).…”

“…We have the following: The arguments in [6] and [8] (deterministic case) and [4] (stochastic case) lead to the desired pointwise, maximal, global error ofŌ(h) and, under weak additional conditions,Ō(h 3/2 ).…”