Uniform Integrability of a Single Jump Local Martingale with State-Dependent Characteristics

We investigate a deterministic criterion to determine whether a diffusive local martingale with a single jump and state-dependent characteristics is a uniformly integrable martingale. We allow the diffusion coefficient, the jump hazard rate and the relative jump size to depend on the state and prove that the process is a uniformly integrable martingale if and only if the relative jump size is bounded away from one and the hazard rate is large enough compared to the diffusion component. The result helps to classify seemingly explosive behaviour in diffusive local martingales compensated by the existence of a jump. Moreover, processes of this type can be used to model financial bubbles in stock prices as deviation from the fundamental value. We present a simple framework to illustrate this application.