Pathwise optimality in stochastic control (Q2706164)

This paper deals with the pathwise optimality for stochastic control problems over an infinite time horizon. The authors considered the following problems. For an admissible control \(u_t\) and its response \(x^u_t\), the running cost is given by \(J_T(u)=\int^T_0 c(x^u_t,u_t)dt\). For a given nonnegative nonincreasing function \(g\) with \(\lim_{T\to\infty} g(T)=0\), \(u^*\) is called a \(g\)-optimal a.s. whenever \(\lim_{T\to\infty} g(T)(J_T(u*)- J_T(u))^+ =0\), a.s. holds, for any admissible control \(u\). The authors give sufficient conditions for the existence of \(g\)-optimal a.s. controls and analyze both cases of diffusion processes and of processes with discrete state space.