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On the martingale property of exponential local martingales: Criteria and applications to finance (Mikhail Urusov)

Abstract

The stochastic exponential Z of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale and for the process Z to be a uniformly integrable martingale in the case where  Mt = R t 0 b(Yu)dWu, the process Y is a one-dimensional diffusion, and the process W is a Brownian motion. These conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of Y. Such questions arise in stochastic analysis and its applications whenever one needs to perform a (locally) absolutely continuous measure change. For instance, one can characterize several notions of no-arbitrage and examine relations between them, or characterize one-dimensional diffusion models with bubbles using our results. This is a joint work with Aleksandar Mijatovic.