Numerical Solution of Stochastic Differential Equations with Jumps in Finance

1. Front Matter

   Pages I-XXVIII

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2. Stochastic Differential Equations with Jumps

   • Eckhard Platen, Nicola Bruti-Liberati

   Pages 1-60

3. Exact Simulation of Solutions of SDEs

   • Eckhard Platen, Nicola Bruti-Liberati

   Pages 61-137

4. Benchmark Approach to Finance and Insurance

   • Eckhard Platen, Nicola Bruti-Liberati

   Pages 139-185

5. Stochastic Expansions

   • Eckhard Platen, Nicola Bruti-Liberati

   Pages 187-231

6. Introduction to Scenario Simulation

   • Eckhard Platen, Nicola Bruti-Liberati

   Pages 233-271

7. Regular Strong Taylor Approximations with Jumps

   • Eckhard Platen, Nicola Bruti-Liberati

   Pages 273-307

8. Regular Strong Itô Approximations

   • Eckhard Platen, Nicola Bruti-Liberati

   Pages 309-346

9. Jump-Adapted Strong Approximations

   • Eckhard Platen, Nicola Bruti-Liberati

   Pages 347-388

10. Estimating Discretely Observed Diffusions

    • Eckhard Platen, Nicola Bruti-Liberati

    Pages 389-417

11. Filtering

    • Eckhard Platen, Nicola Bruti-Liberati

    Pages 419-475

12. Monte Carlo Simulation of SDEs

    • Eckhard Platen, Nicola Bruti-Liberati

    Pages 477-505

13. Regular Weak Taylor Approximations

    • Eckhard Platen, Nicola Bruti-Liberati

    Pages 507-522

14. Jump-Adapted Weak Approximations

    • Eckhard Platen, Nicola Bruti-Liberati

    Pages 523-569

15. Numerical Stability

    • Eckhard Platen, Nicola Bruti-Liberati

    Pages 571-590

16. Martingale Representations and Hedge Ratios

    • Eckhard Platen, Nicola Bruti-Liberati

    Pages 591-635

17. Variance Reduction Techniques

    • Eckhard Platen, Nicola Bruti-Liberati

    Pages 637-695

18. Trees and Markov Chain Approximations

    • Eckhard Platen, Nicola Bruti-Liberati

    Pages 697-753

19. Solutions for Exercises

    • Eckhard Platen, Nicola Bruti-Liberati

    Pages 755-780

20. Back Matter

    Pages 781-856

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In financial and actuarial modeling and other areas of application, stochastic differential equations with jumps have been employed to describe the dynamics of various state variables. The numerical solution of such equations is more complex than that of those only driven by Wiener processes, described in Kloeden & Platen: Numerical Solution of Stochastic Differential Equations (1992). The present monograph builds on the above-mentioned work and provides an introduction to stochastic differential equations with jumps, in both theory and application, emphasizing the numerical methods needed to solve such equations. It presents many new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor corrector, extrapolation, Markov chain and variance reduction methods, stressing the importance of their numerical stability. Furthermore, it includes chapters on exact simulation, estimation and filtering. Besides serving as a basic text on quantitativemethods, it offers ready access to a large number of potential research problems in an area that is widely applicable and rapidly expanding. Finance is chosen as the area of application because much of the recent research on stochastic numerical methods has been driven by challenges in quantitative finance. Moreover, the volume introduces readers to the modern benchmark approach that provides a general framework for modeling in finance and insurance beyond the standard risk-neutral approach. It requires undergraduate background in mathematical or quantitative methods, is accessible to a broad readership, including those who are only seeking numerical recipes, and includes exercises that help the reader develop a deeper understanding of the underlying mathematics.

• Variance
• jump diffusions
• linear optimization
• numerical methods
• quantitative finance
• simulation
• stochastic differential equations

• Dept. Mathematical Sciences, School of Finance and Economics and Depa, University of Technology, Sydney, Sydney, Australia

  Eckhard Platen

• Dept. Mathematical Sciences, University of Technology, Sydney, Broadway, Australia

  Nicola Bruti-Liberati

Prof. Eckhard Platen holds the Chair in Quantitative Finance at the University of Technology, Sydney. Author of books on numerical methods for stochastic differential equations and recent book on benchmark approach at Springer Verlag. Has written more than 140 papers in finance, insurance and applied mathematics and serves on the editorial boards of five international journals including Mathematical Finance and Quantitative Finance

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