Stochastic calculus of variations for jump processes

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This monograph serves as a concise introduction to the stochastic calculus of variations, or Malliavin calculus, specifically for processes with jumps, including both pure jump processes and jump-diffusions. Aimed at researchers and graduate students, it presents many self-contained results, including those related to stochastic differential equations (SDEs) with jumps. The text explores applications of this calculus to control theory and mathematical finance, offering asymptotic expansions for functionals connected to financial assets of jump-diffusions, derived from the theory of asymptotic expansion on the Wiener–Poisson space. It also discusses the Hamilton–Jacobi–Bellman (HJB) equation of integro-differential type, linking it to the classical Merton problem and Ramsey theory. The field of jump processes is extensive, encompassing Lévy processes and SDEs with jumps, and recent advancements in stochastic analysis have allowed for a more compact expression of various results. This monograph fills a gap, as these topics have seldom been covered in a single volume. The contents include a preface, introduction, sections on Lévy processes and Itô calculus, perturbations, properties of probability laws, analysis of Wiener–Poisson functionals, applications, an appendix, bibliography, list of symbols, and an index.