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

The existence of admissible strategies (, ) minimizing a function E₀¹ c (t, x, (t, x) ) dt is studied, with x = x (, ) a continuous stochastic process, and admissible strategies defined as those utilizing a specified pattern of information about x, here described by -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.