The descriptive complexity of stochastic integration

Abstract

For most functionals for which pathwise stochastic integration with respect to Brownian motion is defined, sample Brownian paths for which the integral exists are very hard to construct. There exist on the unit interval, functions ω that can be uniformly approximated by sequences of continuous piece-linear functions (ωn) such that each ωn is encoded by a finite binary string of high Kolmogorov–Chaitin complexity. Such functions ω are called complex oscillations. Their set has Wiener measure 1 and they are fully characterised by infinite binary strings of high complexity. In this paper we study stochastic integration from the point of view of complex oscillations. We prove that, under some computability properties on integrands, pathwise stochastic integrals exist for any complex oscillation. We prove also that Itô’s lemma holds for each complex oscillation. Thus constructing a continuous function satisfying Itô’s lemma is reduced to constructing an infinite binary string of high Kolmogorov–Chaitin complexity.