Optimisation of Measures on a Hyperfinite Adapted Probability Space

Abstract

The minimisation problem for a functional $P\mapsto u(\Gamma ,P,\tilde{g})$ is considered, where $\tilde{g}$ is an ℝⁿ-valued stochastic process, defined on some filtered probability space $\Gamma=(\Gamma,({\mathcal{G}}_{t})_{t\in[0,1]},\mathbb{P})$ , and P is an admissible probability measure in the sense that it obeys (1) some uniform equivalence condition with respect to the given measure ℙ on Γ, and (2) a finite number (possibly zero) of arbitrarily given other conditions that require the expectation (with respect to P) of some continuous bounded function φ of $(\tilde{g}_{t_{1}},\ldots,\tilde{g}_{t_{k}})$ , for t ₁,…,t _k ∈[0,1], to lie within some closed set. We assume that u can be formulated through finite compositions of conditional expectations and bounded continuous functions.

Under the assumption of |φ| being uniformly bounded from below and some condition on the dimension of $\phi(\tilde{g}_{t_{1}},\ldots ,\tilde{g}_{t_{k}})$ , the existence of a solution on hyperfinite adapted probability spaces, as well as its minimality among admissible measures on any other adapted probability space, is proven. Also, a coarseness result for the Loeb operation is established.

The main result of this paper, however, is a “standard result”: It does not include any reference to nonstandard analysis and can be perfectly understood without any familiarity with nonstandard analysis.