A PDE approach to stochastic invariance
type: Item
instance of: scholarly article
title: A PDE approach to stochastic invariance
zbMATH Open document ID: 1013.60043
author: P. Loreti
author: Maria Elisabetta Tessitore
author: Hitoshi Ishii
published in: Discrete and Continuous Dynamical Systems
publication date: 2000-08-16T00:00:00Z
review text: The authors study an invariance property for the controlled stochastic differential equation of the Itô form \[ dX_t= \sigma(X_t, \alpha_t) dB_t+b (X_t,\alpha_t) dt,\quad X_0=x, \tag{1} \] where \(x\in \mathbb{R}^n\), \(\alpha_t\) is a control taking values in a given set \(A,B_z\) is an \(m\)-dimensional Brownian motion, and \(\sigma\) and \(b\) are given functions on \(\mathbb{R}^n\times A\) with values in \(\mathbb{R}^n\), respectively. Let \(K\) be a closed subset of \(\mathbb{R}^n\). The invariance property for (1) and \(K\) which we consider here is the following property: (IP) If \(x\in X\) and \(\alpha_t\) is any control, then \(X_t\in K\) for all \(t\geq 0\) almost surely. In optimal control the value function associated with (1) is \[ V(x)= \inf_\alpha E\int^\infty_0 \exp\left\{-\int^\infty_0 c(X_s,\alpha_s) ds\right\} f(X_t,\alpha_t) dt,\tag{2} \] where \(c\) and \(f\) are given real bounded functions on \(\mathbb{R}^n\times A\), \(f\) and \(c\), respectively, represent the running cost and the discount rate, \(c\) is assumed to satisfy \(\inf_\mathbb{R} n_{x\times a}c>0\), and \(E\) denotes the expectation with respect to the associated probability measure. The value function \(V\) satisfies the corresponding Hamilton-Jacobi-Bellman equation \[ F\bigl(x,V(x), DV(x), D^2 V(x) \bigr)=0 \quad \text{in }\mathbb{R}^n \tag{3} \] in the viscosity sense, where \(DV\) and \(D^2V\) denote the gradient of \(V\) and the Hessian of \(V\), respectively, and \(F\) is a real function on \(\mathbb{R}^n\times \mathbb{R}^n\times SL^n\), with \(SL^n\) denoting the space of real \(n\times n\) symmetric matrices, defined by \[ F(x,r,p, X)= \sup\Bigl\{- \frac 12\text{tr} \sigma(x,a) \sigma(x,a^T)X-\bigl(b(x,a), p \bigr)+c(x,a) r-f(x,a)\Bigr\}, \tag{4} \] with \(M\) denoting the trace of \(M\in SL^n\). The authors establish a few characterizations of the property (IP) in terms of Hamilton-Jacobi-Bellman equation (3). One of the properties proved to be equivalent to the invarian…
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 1492656
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: second-order Hamilton-Jacobi-Bellman equation
zbMATH Keywords: viscosity solutions
zbMATH Keywords: invariance property
MaRDI profile type: Publication
full work available at URL: dcds.2000.6.651
OpenAlex ID: W2011268550
DOI: 10.3934/DCDS.2000.6.651
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description: scientific article; zbMATH DE number 1492656