Risk-sensitive control and differential games in infinite dimensions
type: Item
instance of: scholarly article
title: Risk-sensitive control and differential games in infinite dimensions
zbMATH Open document ID: 1030.93053
DOI: 10.1016/S0362-546X(01)00757-X
author: Andrzej Świȩch
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: The author provides a rigorous mathematical treatment of a stochastic risk-sensitive control problem in an infinite-dimensional Hilbert space \(H\). The problem is the following. For an admissible control \(u\), the system \(X_t\) (\(=X^u_t\)) is described by the stochastic differential equation in \(H\): \[ dX_s= f(X_s, u_s) ds+ \sqrt{\varepsilon}\sigma(X_s) dW_s, \] where \(W\) is an \(H\)-valued Brownian motion and \(\sigma\) is a linear bounded self-adjoint operator, \(\tau^\varepsilon\) (\(=\tau^{\varepsilon, u}\)) means the exit time of the system \(X\) from the closure of a given bounded convex open set, and the value function \(v_\varepsilon(x)\) is defined by \(v_\varepsilon(x)= \inf_u E_x(\exp(-\theta\tau^\varepsilon))\). Denoting its logarithmic transformation by \(w_\varepsilon\), he proves the following facts, which are infinite-dimensional versions of theorems of finite dimension. 1) \(v_\varepsilon\) and \(w_\varepsilon\) are unique viscosity solutions of the boundary value problems of the associated Hamilton-Jacobi-Bellman equations. 2. As \(\varepsilon\to 0\), \(w_\varepsilon\) converges uniformly. 3. Its limit is also a unique solution to a boundary value problem and turns out to be the upper value function of an associated differential game.
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 93C25
Mathematics Subject Classification ID: 91A23
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 1788027
zbMATH Keywords: stochastic risk-sensitive control
zbMATH Keywords: infinite-dimensional Hilbert space
zbMATH Keywords: exit time
zbMATH Keywords: viscosity solutions
zbMATH Keywords: boundary value problems
zbMATH Keywords: Hamilton-Jacobi-Bellman equations
zbMATH Keywords: differential game
reviewed by: Makiko Nisio
MaRDI profile type: Publication
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full work available at URL: s0362-546x(01)00757-x
OpenAlex ID: W2024453252
description: scientific article; zbMATH DE number 1788027