Risk-sensitive control and differential games in infinite dimensions
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.
scientific article; zbMATH DE number 1788027
scientific article