"Jump diffusion processes provide a mathematical framework for systems that evolve continuously yet undergo sudden, discontinuous shifts. Originating from early work on Lévy processes and extensions of classical diffusion theory, they have developed into a mature area of stochastic analysis equipped to handle jumps, regime switching, and intricate boundary interactions.

Recent advances have significantly strengthened the theoretical foundations of jump diffusions, including new results on stability, ergodicity, stochastic control, mean‑field models, martingale formulations, and numerical approximation. Together, these developments broaden the range of models that can be rigorously analyzed and effectively simulated.

Across applications, jump diffusions have become indispensable in fields where abrupt transitions are intrinsic, including finance, insurance, engineering, filtering, and population genetics. They support tractable analysis of first passage problems, optimal stopping, occupation times, and inference under jump‑correlated observations. This collection highlights the continued growth and relevance of jump diffusion theory in both mathematical development and real‑world modeling."

Collection created by Professor Linan Chen (McGill University)