A PDE approach to stochastic invariance
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 invariance property for (1) and \(K\) is the restriction property for (3) and \(K\). This restriction is: (RP) If \(\Omega\) is a neighborhood of \(K,g\in C(\Omega)\), and \(a\in C(\Omega)\) is a solution of \[ F(x,u(x),Du(x), D^2u(x))=g(x) \text{ in }\Omega,\tag{5} \] then \(u\) is also a solution of (5) in \(K\). Another property is a monotonicity condition of \(F\) on the boundary of \(K\). Let \(Z\) denote the function on \(K\) defined by \(Z(x)=0\). The property is: (MP) If \((x,r,p,X) \in(\partial K)\times \mathbb{R}\times \mathbb{R} \times SL^n\) and \((qY)\in J^{2p}_k Z(x)\), then \(F(x,r,p+q,X+Y)\leq F(x,r,p,X)\). In the case where \(K=\Omega\) and \(\Omega\) is an open subset of \(\mathbb{R}^n\), the authors introduce the extension property of (3) and \(K\): (EP) If \(g\in C(\overline \Omega)\) and \(u\in C(\overline \Omega)\) is a solution of (5) in \(\Omega\), then \(u\) is a solution of (5) in \(\overline \Omega\). This note represents an extended version due to the second and third authors in which a weaker form of the fact that (MP) implies (EP) has been established. In the case of ordinary differential equations, condition (MP) as an equivalent condition for (IP) goes back to Nagumo (1942). Friedman (1975) has given sufficient conditions for uncontrolled stochastic differential equations to be invariant for closed sets. For other details see the authors' references.
second-order Hamilton-Jacobi-Bellman equation
scientific article; zbMATH DE number 1492656