Stochastic calculus for jump processes
In this chapter we introduce the basics of stochastic calculus for jump processes. We follow the approaches proposed by Protter  for the general theory of stochastic integration and by Applebaum  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 ), Galerkin schemes (see, for instance, Platen and Bruti-Liberati ) and Monte Carlo methods (see, for instance, Glasserman ). 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.
KeywordsStochastic Volatility Quadratic Variation Jump Process Stochastic Integral Stochastic Volatility Model
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