Numerical Solution of Stochastic Differential Equations with Jumps in Finance

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.

List of contents

Stochastic Differential Equations with Jumps.- Exact Simulation of Solutions of SDEs.- Benchmark Approach to Finance and Insurance.- Stochastic Expansions.- to Scenario Simulation.- Regular Strong Taylor Approximations with Jumps.- Regular Strong Itô Approximations.- Jump-Adapted Strong Approximations.- Estimating Discretely Observed Diffusions.- Filtering.- Monte Carlo Simulation of SDEs.- Regular Weak Taylor Approximations.- Jump-Adapted Weak Approximations.- Numerical Stability.- Martingale Representations and Hedge Ratios.- Variance Reduction Techniques.- Trees and Markov Chain Approximations.- Solutions for Exercises.

About the author

Professor Eckhard Platen is a joint appointment between the School of Finance and Economics and the Department of Mathematical Sciences to the 1997 created Chair in Quantitative Finance at the University of Technology Sydney. Prior to this appointment he was Founding Head of the Centre for Financial Mathematics at the Institute of Advanced Studies at the Australian National University in Canberra. He completed a PhD in Mathematics at the Technical University in Dresden in 1975 and obtained in 1985 his Dr. sc. from the Academy of Sciences in Berlin, where he headed at the Weierstrass Institute the Sector of Stochastics. He is co-author of two successful books on Numerical Methods for Stochastic Differential Equations, published by Springer Verlag, and has authored more than 100 research papers in quantitative finance and mathematics.

Summary

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.

Report

cid and clear writing style of the exposition in combination with many interesting examples from mathematical finance." (H. M. Mai, Zentralblatt MATH, Vol. 1225, 2012)

Product details

• Authors: Nicola Bruti-Liberati, Eckhar Platen, Eckhard Platen
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 23.08.2010
• EAN: 9783642120572
• ISBN: 978-3-642-12057-2
• No. of pages: 856
• Dimensions: 162 mm x 38 mm x 241 mm
• Weight: 1350 g
• Illustrations: XXVIII, 856 p.
• Series: Stochastic Modelling and Applied Probability; Stochastic Modelling and Applied Probability
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics