Existence of an Optimal Markovian Filter for the Control under Partial Observations

This paper concerns the control of diffusions under partial observations. Part I studies the control of the signal process dXₜ = b(t,Xₜ ,Uₜ)dt + σ (t,Xₜ ,Uₜ )dBₜ, when the observation is dYₜ = h(t,Xₜ )dt + dWₜ, and when the objective is to maximize a reward function E\ ∫ ᵣT k(s,Xₛ, Uₛ )ds + g(XT )\. The existence of an optimal relaxed control is proved. Part II studies the separated problem and proves the existence of an optimal Markovian filter. Then, the authors compare the two problems and prove, under mild conditions, that the value functions for the two problems are equal.