Existence theorems in probability theory
The paper suggests new important notions and methods of solving existence problems in various fields of mathematics. Existence proofs often involve arguments which verify that a sequence of approximations converges to the desired solution. The most useful tools in existence proofs are families of compact sets. The goal of the paper is to go beyond the usual case of convergence in a compact set. The authors deal with probability spaces \((\Omega,P,B,G_t)_{t\in B}\), where \(P\) is a probability measure on a \(\sigma\)-algebra \(G\), \(B\) is the set of dyadic rationals in \([0,T)\), and \((G_t)\) is a filtration or a flow of \(\sigma\)-subalgebras of \(G\). Let \(L^0(\Omega,M)\) be a metric of all \(P\)-measurable functions from \(\Omega\) into a complete separable metric space \(M\), identifying functions which are equal \(P\) -- almost surely, with the metric of convergence in probability. The key new concept is that of neocompact subset of a space \(L^0(\Omega,M)\). The notion of a neocompact set is a generalization of the notion of a compact set. It is a modification of the notion of the family of neocompact sets introduced by the second author [Ann. Pure Appl. Logic 52, No. 1/2, 99-141 (1991; Zbl 0756.60036)]. The motivation for this notion comes from nonstandard analysis. The family of neocompact sets is much larger than the family of compact sets. It provides new opportunities for proving existence theorems by approximation. Nonstandard analysis' main contribution to probability theory is the introduction of ``very rich'' spaces. With neocompact sets it becomes possible to define the notion of a rich adapted probability space in conventional terms. A theory of neometric spaces is investigated. A neometric space is a metric space endowed with a collection of neocompact sets. Then the neometric theory of sets of stochastic processes and stochastic integrals is developed. It is shown how a whole new class of optimality problems can be solved in rich probability spaces. It is proved that for every continuous stochastic process \(x\) on a rich adapted space \(\Omega\) there exists a Brownian motion \(w\) on \(\Omega\) which best approximates \(x\) (in the metric of convergence in probability with the sup metric on paths). At last some nontrivial applications of the approximation theorem to the theory of existence of solutions of stochastic differential equations are considered.
scientific article; zbMATH DE number 915907