Abstract
In this chapter we introduce the basics of stochastic calculus for jump processes. We follow the approaches proposed by Protter [287] for the general theory of stochastic integration and by Applebaum [11] for the presentation of Lévy-type stochastic integrals. We extend to this framework, the analysis performed in the previous chapters for continuous processes: in particular, we prove Itô formula and a Feynman-Kač type representation theorem for solutions to SDEs with jumps. For simplicity, most statements are given in the one-dimensional case. Then we show how to derive the integro-differential equation for a quite general exponential model driven by the solution of a SDE with jumps: these results open the way for the use of deterministic and probabilistic numerical methods, such as finite difference schemes (see, for instance, Cyganowski, Grüne and Kloeden [82]), Galerkin schemes (see, for instance, Platen and Bruti-Liberati [281]) and Monte Carlo methods (see, for instance, Glasserman [158]). In the last part of the chapter, we examine some stochastic volatility models with jumps: in particular, we present the Bates and the Barndorff-Nielsen and Shephard models.
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© 2011 Springer-Verlag Italia
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Pascucci, A. (2011). Stochastic calculus for jump processes. In: PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series. Springer, Milano. https://doi.org/10.1007/978-88-470-1781-8_14
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DOI: https://doi.org/10.1007/978-88-470-1781-8_14
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-1780-1
Online ISBN: 978-88-470-1781-8
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