MARTINGALE PROPERTY OF GENERALIZED STOCHASTIC EXPONENTIALS

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 \textit{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.

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