Numerical solutions 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. Their numerical solutions are more complex than those driven solely by Wiener processes, as detailed in Kloeden & Platen's earlier work. This monograph serves as an introduction to these equations, focusing on both theory and application, particularly the numerical methods necessary 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, with an emphasis on numerical stability. Additionally, it covers exact simulation, estimation, and filtering. This text not only serves as a foundational resource on quantitative methods but also highlights numerous potential research problems in a rapidly expanding field. Finance is chosen as the primary application area, reflecting recent research challenges in quantitative finance. The volume introduces a modern benchmark approach for modeling in finance and insurance, extending beyond the traditional risk-neutral framework. It requires an undergraduate background in mathematical or quantitative methods and is accessible to a wide audience, including those seeking numerical recipes. Exercises are included to enhance understanding of the underlying mathematics.