Mean-Reverting Jump Diffusion Processes: Drift Adjustment to Preserve a Fixed Long-Term Mean

This note addresses the properties of mean-reverting stochastic processes of the Black-Karasinski type with additional stochastic jumps. For these processes, which are well suited for many financial applications such as the modelling of commodity prices and credit spreads, one would usually like to ensure a fixed long-term mean around which the process paths evolve. This paper shows the impact of jumps on the long-term asymptotic behaviour of the Black-Karasinski process and proposes a drift adjustment that ensures the convergence of the process expectation to a fixed long-term mean.