Mathematical Sciences: Mathematical Logic and Applications

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A major objective of this project is to use nonstandard analysis to give a general metatheorem which reduces existence problems for continuous-time stochastic processes to corresponding statements about discrete-time processes. A step towards the objective was taken in the forthcoming paper, "From Discrete to Continuous Time". In that paper, a concept of forcing was introduced, and it was proved that a wide class of statements about continuous-time stochastic processes are true if and only if forced by a sequence of discrete-time stochastic processes. This forcing theorem can be used to prove a variety of existence theorems. For example, the method gives proofs of the existence of solutions of many stochastic differential equations. The discrete-time statement is often easy to prove, and the forcing theorem does most of the work required to get from there to the continuous-time statement. The forcing theorem is presently formulated in a way which is natural from the point of view of mathematical logic but is often more difficult to apply than one would hope. One reason for this is that one must first show that the statement to be proved is expressible in terms of "liftable functions", which are defined in a technical way using nonstandard analysis. The investigator is working toward a more general and more convenient forcing theorem which would use nonstandard notions in its proof but not in its application. To complement this effort, the scope of the method will be tested by developing a wider variety of applications to particular problems in probability theory. It is surprising that deep methods from mathematical logic can be used to obtain alternative and often simpler proofs in the theory of probability. However, the investigator has been very successful in demonstrating this by example over a period of many years.