On irregular functionals of SDEs and the Euler scheme

Finance and Stochastics

Volumn 13, Issue 3, 2009, Pages 381-401

  Avikainen, Rainer ^a  

^a UNIVERSITY OF JYVÄSKYLÄ   (Finland)

Author keywords

Approximation; Euler scheme; Rate of convergence; Stochastic differential equations

Indexed keywords

EID: 70350666664     PISSN: 09492984     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00780-009-0099-7     Document Type: Article

References (27)

1. Avikainen, R.: Convergence rates for approximations of functionals of SDEs. arXiv:0712.3635v1 [math.PR] (2007).
2. 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, 43-60 (1996).
3. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, New York (1988).
4. Bouleau, N., Lépingle, D.: Numerical Methods for Stochastic Processes. Wiley, New York (1994).
5. Caballero, M.E., Fernández, B., Nualart, D.: Estimation of densities and applications. J. Theor. Probab. 11, 831-851 (1998).
6. Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, New York (1964).
7. p-variation of BSDEs with non-Lipschitz terminal condition (2008, in preparation).
8. Giles, M.B.: Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Keller, A., Heinrich, S., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 343-358. Springer, Berlin (2008).
9. Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56, 607-617 (2008).
10. Giles, M.B., Higham, D.J., Mao, X.: Analyzing multi-level Monte Carlo for options with non-globally Lipschitz payoff. Finance Stoch. 13, 403-414 (2009).
11. Guyon, J.: Euler scheme and tempered distributions. Stoch. Process. Appl. 116, 877-904 (2006).
12. Gyöngy, I.: A note on Euler's approximations. Potential Anal. 8, 205-216 (1998).
13. Gyöngy, I., Krylov, N.: Existence of strong solutions for Itô's stochastic equations via approximations. Probab. Theory Relat. Fields 105, 143-158 (1996).
14. Heinrich, S.: Multilevel Monte Carlo methods. In: Margenov, S., et al. (eds.) Large-Scale Scientific Computing, Third International Conference. Lecture Notes in Computer Science, vol. 2179, pp. 58-67. Springer, Berlin (2001).
15. Higham, D.J., Mao, X., Stuart, A.M.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041-1063 (2002).
16. Hofmann, N., Müller-Gronbach, T.: On the global error of Itô-Taylor schemes for strong approximation of scalar stochastic differential equations. J. Complex. 20, 732-752 (2004).
17. Hofmann, N., Müller-Gronbach, T., Ritter, K.: The optimal discretization of stochastic differential equations. J. Complex. 17, 117-153 (2001).
18. Hofmann, N., Müller-Gronbach, T., Ritter, K.: Linear vs. standard information for scalar stochastic differential equations. J. Complex. 18, 394-414 (2002).
19. Jacod, J., Protter, P.: Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26, 267-307 (1998).
20. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991).
21. Kebaier, A.: Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15, 2681-2705 (2005).
22. Kloeden, P.E., Platen, E.: Numerical Solutions of Stochastic Differential Equations. Springer, Berlin (1992).
23. Müller-Gronbach, T.: The optimal uniform approximation of systems of stochastic differential equations. Ann. Appl. Probab. 12, 664-690 (2002).
24. Müller-Gronbach, T.: Strong approximation of systems of stochastic differential equations. Habilitation thesis, TU Darmstadt (2002).
25. Müller-Gronbach, T.: Optimal pointwise approximation of SDEs based on Brownian motion at discrete points. Ann. Appl. Probab. 14, 1605-1642 (2004).
26. Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1974).
27. Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8, 483-509 (1990).