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Jump-Adapted Strong Approximations

  • Eckhard PlatenEmail author
  • Nicola Bruti-Liberati
Chapter
  • 4.7k Downloads
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 64)

Abstract

This chapter describes jump-adapted strong schemes. The term jump-adapted refers to the time discretizations used to construct these schemes. These discretizations include all jump times generated by the Poisson jump measure. The form of the resulting schemes is much simpler than that of the regular schemes presented in Chaps. 6 and 7 which are based on regular time discretizations. The idea of jump-adapted time discretization goes back to Platen (1982a). It appeared later in various literature, for instance, Maghsoodi (1996). Jump-adapted schemes are not very efficient for SDEs driven by a Poisson measure with a high total intensity. In this case, regular schemes would usually be preferred. Some of the results of this chapter can be found in Bruti-Liberati et al. (2006) and in Bruti-Liberati & Platen (2007b). Results presented already in Kloeden & Platen (1999) and Chap. 5 are employed in the following when approximating the diffusion part of the solution of an SDE.

Keywords

Strong Convergence Time Step Size Discretization Error Euler Scheme Jump Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Bruti-Liberati, N., Nikitopoulos-Sklibosios, C. & Platen, E. (2006). First order strong approximations of jump diffusions, Monte Carlo Methods Appl. 12(3-4): 191–209. zbMATHCrossRefMathSciNetGoogle Scholar
  2. Bruti-Liberati, N. & Platen, E. (2007b). Strong approximations of stochastic differential equations with jumps, J. Comput. Appl. Math. 205(2): 982–1001. zbMATHCrossRefMathSciNetGoogle Scholar
  3. Kloeden, P. E. & Platen, E. (1999). Numerical Solution of Stochastic Differential Equations, Vol. 23 of Appl. Math., Springer. Third printing, (first edition (1992)). Google Scholar
  4. Maghsoodi, Y. (1996). Mean-square efficient numerical solution of jump-diffusion stochastic differential equations, SANKHYA A 58(1): 25–47. zbMATHMathSciNetGoogle Scholar
  5. Platen, E. (1982a). An approximation method for a class of Itô processes with jump component, Liet. Mat. Rink. 22(2): 124–136. zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.School of Finance and Economics, Department of Mathematical SciencesUniversity of Technology, SydneyBroadwayAustralia

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