Existence of Optimal Strategies Based on Specified Information, for a Class of Stochastic Decision Problems

The existence of admissible strategies γ ( · , · ) minimizing a function E∫₀¹ c(t,x,γ (t,x))dt is studied, with x = x( · ,ω ) a continuous stochastic process, and admissible strategies defined as those utilizing a specified pattern of information about x, here described by σ-algebras Gₜ included in the “past” \ x(s),s t \. Conclusion: Moment conditions on x, growth conditions on c, and continuity of c in its third variable (the control) ensure that an optimal admissible strategy exists. The method of proof depends on properties of conditional expectations, and on a variant of a general Filippov (or implicit functions) lemma due to McShane and Warfield.