Numerical Solution of Stochastic Differential Equations with Jumps in Finance

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In financial and actuarial modeling, stochastic differential equations with jumps are used to describe the dynamics of various state variables. Solving these equations numerically is more complex than those driven solely by Wiener processes. This monograph builds on previous work and introduces stochastic differential equations with jumps, focusing on the necessary numerical methods for their solution. It presents new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor-corrector, extrapolation, Markov chain, and variance reduction methods, highlighting their numerical stability. Additionally, it covers exact simulation, estimation, and filtering. Serving as a foundational text on quantitative methods, it also provides access to numerous research problems in a rapidly expanding field. The focus on finance is due to recent research challenges in quantitative finance driving advancements in stochastic numerical methods. The volume introduces a modern benchmark approach for modeling in finance and insurance, extending beyond the standard risk-neutral framework. It requires an undergraduate background in mathematical or quantitative methods, making it accessible to a wide audience, including those seeking numerical recipes. Exercises included help readers deepen their understanding of the underlying mathematics.