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Affine Diffusions and Related Processes: Simulation, Theory and Applications pp 37–66Cite as

An Introduction to Simulation Schemes for SDEs

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Part of the book series: Bocconi & Springer Series ((BS,volume 6))

Abstract

Let us start this chapter by a general motivation for having simulation schemes. To fix the ideas, we consider a continuous process (X t , t ∈ [0, T]) that takes values in \(\mathbb{R}^{d}\) and a function \(F: \mathcal{C}([0,T], \mathbb{R}^{d}) \rightarrow \mathbb{R}\) such that \(\mathbb{E}[\vert F(X_{t},t \in [0,T])\vert ] < \infty \).

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References

  1. Alfonsi, A.: High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comput. 79(269), 209–237 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alfonsi, A., Jourdain, B., Kohatsu-Higa, A.: Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme. Ann. Appl. Probab. 24(3), 1049–1080 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Relat. Fields 104(1), 43–60 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Friz, P.K., Victoir, N.B.: Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge Studies in Advanced Mathematics, vol. 120. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  5. Fujiwara, T.: Sixth order methods of Kusuoka approximation (2006)

    Google Scholar 

  6. Gaines, J.G., Lyons, T.J.: Random generation of stochastic area integrals. SIAM J. Appl. Math. 54(4), 1132–1146 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  7. Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Guyon, J.: Euler scheme and tempered distributions. Stoch. Process. Appl. 116(6), 877–904 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kanagawa, S.: On the rate of convergence for Maruyama’s approximate solutions of stochastic differential equations. Yokohama Math. J. 36(1), 79–86 (1988)

    MathSciNet  Google Scholar 

  10. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)

    Google Scholar 

  11. Kebaier, A.: Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15(4), 2681–2705 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York), vol. 23. Springer, Berlin (1992)

    Google Scholar 

  13. Kurtz, T.G., Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19(3), 1035–1070 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kurtz, T.G., Protter, P.: Wong-Zakai corrections, random evolutions, and simulation schemes for SDEs. In: Stochastic Analysis, pp. 331–346. Academic, Boston, MA (1991)

    Google Scholar 

  15. Lyons, T., Victoir, N.: Cubature on Wiener space. Proc. R. Soc. Lond. Ser. A 460(2041), 169–198 (2004). Stochastic analysis with applications to mathematical finance

    Google Scholar 

  16. Lyons, T.J., Caruana, M., Lévy, T.: Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics, vol. 1908. Springer, Berlin (2007). Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, With an introduction concerning the Summer School by Jean Picard

    Google Scholar 

  17. Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Mathematics and its Applications, vol. 313. Kluwer Academic Publishers Group, Dordrecht (1995). Translated and revised from the 1988 Russian original

    Google Scholar 

  18. Ninomiya, S., Victoir, N.: Weak approximation of stochastic differential equations and application to derivative pricing. Appl. Math. Finance 15(1–2), 107–121 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Oshima, K., Teichmann, J., Velušček, D.: A new extrapolation method for weak approximation schemes with applications. Ann. Appl. Probab. 22(3), 1008–1045 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  21. Talay, D.: Discrétisation d’une équation différentielle stochastique et calcul approché d’espérances de fonctionnelles de la solution. RAIRO Modél. Math. Anal. Numér. 20(1), 141–179 (1986)

    MATH  MathSciNet  Google Scholar 

  22. Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8(4), 483–509 (1990, 1991)

    Google Scholar 

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  1. CERMICS, Ecole Nationale des Ponts et Chaussées, Champs-sur-Marne, France

    Aurélien Alfonsi

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  1. Aurélien Alfonsi
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Alfonsi, A. (2015). An Introduction to Simulation Schemes for SDEs. In: Affine Diffusions and Related Processes: Simulation, Theory and Applications. Bocconi & Springer Series, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-05221-2_2

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