On the Martingale Property of Certain Local Martingales

TL;DR

This paper establishes necessary and sufficient deterministic conditions for the stochastic exponential of certain local martingales to be a true or uniformly integrable martingale, with applications to bubble absence in financial models.

Contribution

It provides explicit deterministic criteria for the martingale property of stochastic exponentials associated with diffusions, extending previous results and applications.

Findings

Criteria for Z to be a true martingale

Criteria for Z to be uniformly integrable

Application to bubble absence in financial models

Abstract

The stochastic exponential $Z_t=\exp\{M_t-M_0-(1/2) <M,M>_t\}$ 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 in the case where $M_t=\int_0^t b(Y_u)\,dW_u$ and $Y$ is a one-dimensional diffusion driven by a Brownian motion $W$. Furthermore, we provide a necessary and sufficient condition for $Z$ to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function $b$ and the drift and diffusion coefficients of $Y$. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.