EXISTENCE THEORY FOR STOCHASTIC OPTIMAL CONTROL SYSTEMS

The aim of the present paper is to present certain existence theorems for stochastic control systems whose state variables, χ(t;ω), are continuous functions from the set R+ = {t;t ≥ 0} into the space L2(Ω, A, μ). That is, for each t R+, χ(t;ω) is a vector‐valued random variable whose second absolute moment exists. U = μ(t), the admissible controls, are taken as measurable functions of t only. It is assumed that the initial time is fixed but allow the terminal time tf(ω) to vary with ω∈Ω. The usual space constraints and boundary conditions are also allowed to vary with ω∈Ω. The cost functional is taken to be a continuous functional over a suitable class of continuous functions.