Diffusion approximation and optimal stochastic control
type: Item
instance of: scholarly article
zbMATH Open document ID: 0974.60066
DOI: 10.1137/S0040585X97977860
author: Wolfgang J. Runggaldier
author: R. Liptser
author: Michael Taksar
publication date: 2001-05-02T00:00:00Z
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 93E20
zbMATH DE Number: 1593213
zbMATH Keywords: optimal stochastic control
zbMATH Keywords: diffusion processes
zbMATH Keywords: weak convergence
zbMATH Keywords: asymptotic optimality
zbMATH Keywords: Lipschitz feedback
MaRDI profile type: Publication
full work available at URL: s0040585x97977860
OpenAlex ID: W2017198824
title: Diffusion approximation and optimal stochastic control
published in: Theory of Probability and its Applications
review text: By the same goal with the previous paper of these authors [SIAM J. Control Optimization 34, No. 1, 161-178 (1996; Zbl 0867.93085)] but considering the case of a stochastic control model admitted a diffusion approximation, they show in the present paper that an optimal Lipschitz feedback control of the limit model NEWLINE\[NEWLINEdX_t = [A_0 (t,X_t)+ a_1 (t,X_t)u_t] dt + B(t, X_t) dW_tNEWLINE\]NEWLINE with the cost \( J(u)= E \{ \text{int}_0^T [p(t,X_t) +q(t,u_t)] dt +r(X_T) \}\) remains asymptotically optimal feedback for a family of controlled stochastic processes (prelimit models), indexed by a small parameter and associated with the same cost, whose dynamics are perturbed by additive stationary elements (wideband noises and martingales), a fast-fluctuating external process that contaminates some of the coefficients, as well as noises inducing singularly perturbed processes.NEWLINENEWLINENEWLINEHere, \( A_0\) and \(B^2\) depend on the invariant measure of the external process and the asymptotic intensity of the quadratic variation of the martingale; asymptotical optimality means \( \lim_{ \varepsilon \to 0} J^{\varepsilon}(u) = \lim_{ \varepsilon \to 0} \{ J_0^{\varepsilon} =\inf_{u^{\varepsilon}} J^{\varepsilon} (u^{\varepsilon}) \}\). Moreover, the statement holds for the case of \( \delta\)-optimality in the sense that \( J(u^{\delta}) \leq \{J_0 = \inf_u J(u) \} + {\delta}\) implies \(\limsup_{\varepsilon \to 0} ( J^{\varepsilon} (u^{\delta}) - J_0^{\varepsilon})\leq \delta \). The statement also remains true in the case of Markov feedback controls. All mentioned results and the estimation \(\liminf_{\varepsilon \to 0} J_0^{\varepsilon} \geq J_0 \) are obtained by using the technique of weak convergence of stochastic processes. Properties involving the prelimit model are described in Section 3 (relative compactness) and Section 5 (Lipschitz feedback control). Section 4 is devoted to problems of the limit model: feedback controls, existence of an optimal feedba…
description: scientific article