On the martingale property of certain local martingales: criteria and applications to finance

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 $M_t=\int_0^t b(Y_u)\,dW_u$, 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 $Y$. Furthermore, we give a deterministic necessary and sufficient condition in the one-dimensional setting for a discounted stock price to be a true martingale under the risk-neutral measure. This is relevant for ascertaining existence of financial bubbles in diffusion-based models. We give further applications of our results for characterising several notions of no-arbitrage and examining how they relate to each other. This is a joint work with A. Mijatovic.