Ultrafilter convergence in stochastic analysis and mathematical finance

Abstract

Given a non-principal ultrafilter, we define and prove properties of ultralimits of measure spaces (including σ-algebras, filtrations and measures), random variables and discrete-time stochastic processes. Among other things, considering Brownian motion as the ultralimit of random walks, we define the stochastic integral as the ultralimit of sums involving the random walk and we show that solutions to stochastic differential equations can be written as the ultralimit of solutions to difference equations. We also show that the ultralimit of the Cox-Ross-Rubinstein model is the Black Scholes model.