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

The existence of admissible strategies $\gamma ( \cdot , \cdot )$ minimizing a function $E\int_0^1 {c(t,x,\gamma (t,x))dt} $ is studied, with $x = x( \cdot ,\omega )$ a continuous stochastic process, and admissible strategies defined as those utilizing a specified pattern of information about x, here described by $\sigma $-algebras $G_t $ included in the “past” $\{ {x(s),s \leqq t} \}$. 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.