Risk-sensitive control of an ergodic diffusion over an infinite horizon
This paper deals with the ergodic control of diffusion processes with small risk. Consider the stochastic differential equation in \(\mathbb{R}^n\), \[ dx_t= f(x_t)dt+ h(x_t, v_t) dt+ \sigma(x_t) dw_t, \] where \((v_t)\) is a control process with a given compact control region \(U\). To study ergodic risk sensitive control, the authors exploit the discrete-time technique as follows. Let \(\gamma\) and \(\Gamma\) be concentric spheres with radii \(r< R\), respectively. Put \(T_B=\) hitting time to \(B\), \(\Theta_t=\) time shift operator and \(\tau_1= T_\Gamma+ T_\gamma\circ\Theta_{T_\Gamma}\), \(\tau_{i+1}= \tau_i+ \tau_1\circ \Theta_{\tau_i}\). Putting \(v_s= u_i(x_s)\), on \(s\in [\tau_i, \tau_{i+1})\), choose \(\{x_s, s\leq \tau_i\}\) measurable \(U\)-valued functions \(u_i\), \(i= 0,1,\dots\), successively. The problem is to minimize the following risk-sensitive cost functional, by suitable \(u_i\), \(i= 0,1,\dots\); \[ {1\over\delta}\limsup_{n\to\infty} n^{-1}\log E_x\Biggl[\exp \int^{\tau_n}_0 \delta c(x_s, u_{n-1}(x_s)) ds\Biggr], \] where \(c\) is a given cost function and \(\delta\) is small and positive. As \(\delta\to 0\), applying the normalization of the optimal value of the risk-sensitive cost functional, the authors prove that its limit is independent of \((\gamma,\Gamma)\) and corresponds to the optimal value of the risk neutral average cost per unit time; \[ \limsup_{t\to\infty} {1\over t} E_x \int^t_0 c(x_s,u(x_s)) ds. \]
scientific article; zbMATH DE number 1756287