Almost sure optimality and optimality in probability for stochastic control problems over an infinite time horizon
A pathwise optimality criterion for stochastic control problems is proposed, which is aimed at reducing the risk connected with the fluctuation of cost: Consider \[ dx^u_t= f(x^u_t,u_t)d \tau+\sigma (x^u_t,u_t) dw_t, \] with cost function \[ J_T(u)= \int^T_0 c(x_t^u,u_t) dt, \] and look for a control \(u^*\) optimal in the sense that \[ \lim_{T\to \infty}g_T \bigl[J_T (u^*)-J_T(u) \bigr]^+ =0\text{ a.s. for any }u\in {\mathcal U_\infty}, \] with \(g\) positive, non-increasing, typically \(o(1/t)\). This may be of interest, e.g., in economical applications, when risky situations are particularly dangerous. The control problem is solved under a number of technical conditions, and two examples are discussed, in particular the linear Gaussian model with quadratic cost function.
scientific article; zbMATH DE number 1373565