Martingale property of generalized stochastic exponentials

TL;DR

This paper investigates the conditions under which a generalized stochastic exponential, associated with a diffusion process driven by Brownian motion, is a local, true, or uniformly integrable martingale, extending classical results.

Contribution

It introduces a generalized stochastic exponential for arbitrary Borel functions and provides deterministic criteria for its martingale properties.

Findings

Provides necessary and sufficient conditions for Z to be a local martingale.

Establishes criteria for Z to be a true martingale.

Determines when Z is uniformly integrable.

Abstract

For a real Borel measurable function b, which satisfies certain integrability conditions, it is possible to define a stochastic integral of the process b(Y) with respect to a Brownian motion W, where Y is a diffusion driven by W. It is well know that the stochastic exponential of this stochastic integral is a local martingale. In this paper we consider the case of an arbitrary Borel measurable function b where it may not be possible to define the stochastic integral of b(Y) directly. However the notion of the stochastic exponential can be generalized. We define a non-negative process Z, called generalized stochastic exponential, which is not necessarily a local martingale. Our main result gives deterministic necessary and sufficient conditions for Z to be a local, true or uniformly integrable martingale.