Stochastic calculus for jump processes

  • Andrea Pascucci
Part of the Bocconi & Springer Series book series (BS)


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.


Stochastic Volatility Quadratic Variation Jump Process Stochastic Integral Stochastic Volatility Model 
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Copyright information

© Springer-Verlag Italia 2011

Authors and Affiliations

  • Andrea Pascucci
    • 1
  1. 1.Department of MathematicsUniversity of BolognaBologna

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