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Exact Simulation of Solutions of SDEs

  • Eckhard PlatenEmail author
  • Nicola Bruti-Liberati
Chapter
  • 4.8k Downloads
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 64)

Abstract

Accurate scenario simulation methods for solutions of multi-dimensional stochastic differential equations find applications in the statistics of stochastic processes and many applied areas, in particular in finance. They play a crucial role when used in standard models in various areas. These models often dominate the communication and thinking in a particular field of application, even though they may be too simple for advanced tasks. Various simulation techniques have been developed over the years. However, the simulation of solutions of some stochastic differential equations can still be problematic. Therefore, it is valuable to identify multi-dimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems of those simulation methods that use discrete-time approximations. This chapter follows closely Platen & Rendek (2009a) and provides methods for the exact simulation of paths of multi-dimensional solutions of stochastic differential equations, including Ornstein-Uhlenbeck, square root, squared Bessel, Wishart and Lévy type processes. Other papers that could be considered to be related with exact simulation include Lewis & Shedler (1979), Beskos & Roberts (2005), Broadie & Kaya (2006), Kahl & Jäckel (2006), Smith (2007), Andersen (2008), Burq & Jones (2008) and Chen (2008).

Keywords

Wiener Process Transition Density Bessel Process Standard Wiener Process Heston Model 
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. Andersen, L. (2008). Simple and efficient simulation of the Heston stochastic volatility model, J. Comput. Finance 11(3): 1–42. Google Scholar
  2. Beskos, A. & Roberts, G. (2005). Exact simulation of diffusions, Ann. Appl. Probab. 15: 2422–2444. zbMATHCrossRefMathSciNetGoogle Scholar
  3. Broadie, M. & Kaya, O. (2006). Exact simulation of stochastic volatility and other affine jump diffusion processes, Oper. Res. 54: 217–231. zbMATHCrossRefMathSciNetGoogle Scholar
  4. Burq, Z. A. & Jones, O. D. (2008). Simulation of Brownian motion at first-passage times, Math. Comput. Simulation 77: 64–81. zbMATHCrossRefMathSciNetGoogle Scholar
  5. Chen, N. (2008). Exact simulation of stochastic differential equations. Chinese Univ. of Hong Kong (working paper). Google Scholar
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  7. Lewis, P. A. W. & Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning, Naval Research Logistics Quarterly 26: 403–413. zbMATHCrossRefMathSciNetGoogle Scholar
  8. Platen, E. & Rendek, R. (2009a). Exact scenario simulation for selected multi-dimensional stochastic processes, Commun. on Stoch. Analysis 3(3): 443–465. MathSciNetGoogle Scholar
  9. Smith, R. D. (2007). An almost exact simulation method for the Heston model, J. Comput. Finance 11(1): 115–125. Google 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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