## The Annals of Applied Probability

### Multilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adaptive Euler schemes

#### Abstract

In this article, we consider multilevel Monte Carlo for the numerical computation of expectations for stochastic differential equations driven by Lévy processes. The underlying numerical schemes are based on jump-adapted Euler schemes. We prove stable convergence of an idealised scheme. Further, we deduce limit theorems for certain classes of functionals depending on the whole trajectory of the process. In particular, we allow dependence on marginals, integral averages and the supremum of the process. The idealised scheme is related to two practically implementable schemes and corresponding central limit theorems are given. In all cases, we obtain errors of order $N^{-1/2}(\operatorname{log}N)^{1/2}$ in the computational time $N$ which is the same order as obtained in the classical set-up analysed by Giles [Oper. Res. 56 (2008) 607–617]. Finally, we use the central limit theorems to optimise the parameters of the multilevel scheme.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 1 (2016), 136-185.

Dates
Revised: November 2014
First available in Project Euclid: 5 January 2016

https://projecteuclid.org/euclid.aoap/1452003236

Digital Object Identifier
doi:10.1214/14-AAP1087

Mathematical Reviews number (MathSciNet)
MR3449315

Zentralblatt MATH identifier
1338.65004

#### Citation

Dereich, Steffen; Li, Sangmeng. Multilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adaptive Euler schemes. Ann. Appl. Probab. 26 (2016), no. 1, 136--185. doi:10.1214/14-AAP1087. https://projecteuclid.org/euclid.aoap/1452003236

#### References

• [1] Ankirchner, S., Dereich, S. and Imkeller, P. (2007). Enlargement of filtrations and continuous Girsanov-type embeddings. In Séminaire de Probabilités XL. Lecture Notes in Math. 1899 389–410. Springer, Berlin.
• [2] Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge Univ. Press, Cambridge.
• [3] Asmussen, S. and Rosiński, J. (2001). Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38 482–493.
• [4] Ben Alaya, M. and Kebaier, A. (2015). Central limit theorem for the multilevel Monte Carlo Euler method. Ann. Appl. Probab. 25 211–234.
• [5] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
• [6] Bruti-Liberati, N., Nikitopoulos-Sklibosios, C. and Platen, E. (2006). First order strong approximations of jump diffusions. Monte Carlo Methods Appl. 12 191–209.
• [7] Creutzig, J., Dereich, S., Müller-Gronbach, T. and Ritter, K. (2009). Infinite-dimensional quadrature and approximation of distributions. Found. Comput. Math. 9 391–429.
• [8] Dereich, S. (2008). The coding complexity of diffusion processes under supremum norm distortion. Stochastic Process. Appl. 118 917–937.
• [9] Dereich, S. (2011). Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction. Ann. Appl. Probab. 21 283–311.
• [10] Dereich, S. and Heidenreich, F. (2011). A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations. Stochastic Process. Appl. 121 1565–1587.
• [11] Dereich, S. and Li, S. (2015). Multilevel Monte Carlo implementation for SDEs driven by truncated stable processes. Preprint.
• [12] Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 607–617.
• [13] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering: Stochastic Modelling and Applied Probability. Applications of Mathematics (New York) 53. Springer, New York.
• [14] Heinrich, S. (2001). Multilevel Monte Carlo methods. In Large-Scale Scientific Computing. Lecture Notes in Comput. Sci. 2179 58–67. Springer, Berlin.
• [15] Jacod, J. (1997). On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de Probabilités, XXXI. Lecture Notes in Math. 1655 232–246. Springer, Berlin.
• [16] Jacod, J. and Mémin, J. (1981). Weak and strong solutions of stochastic differential equations: Existence and stability. In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Math. 851 169–212. Springer, Berlin.
• [17] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
• [18] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
• [19] Jeulin, T. and Yor, M., eds. (1985). Grossissements de Filtrations: Exemples et Applications. Lecture Notes in Math. 1118. Springer, Berlin.
• [20] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). Springer, New York.
• [21] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
• [22] Kohatsu-Higa, A. and Tankov, P. (2010). Jump-adapted discretization schemes for Lévy-driven SDEs. Stochastic Process. Appl. 120 2258–2285.
• [23] Kurtz, T. G. and Protter, P. E. (1996). Weak convergence of stochastic integrals and differential equations. In Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Math. 1627 1–41. Springer, Berlin.
• [24] Li, S. (2015). Multilevel Monte Carlo for Lévy-driven SDEs. Ph.D. thesis, in preparation.
• [25] Maghsoodi, Y. (1996). Mean square efficient numerical solution of jump-diffusion stochastic differential equations. Sankhyā Ser. A 58 25–47.
• [26] Mordecki, E., Szepessy, A., Tempone, R. and Zouraris, G. E. (2008). Adaptive weak approximation of diffusions with jumps. SIAM J. Numer. Anal. 46 1732–1768.
• [27] Platen, E. (1982). An approximation method for a class of Itô processes with jump component. Litovsk. Mat. Sb. 22 124–136.
• [28] Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 21. Springer, Berlin.
• [29] Rényi, A. (1963). On stable sequences of events. Sankhyā Ser. A 25 293–302.
• [30] Rosiński, J. (2001). Series representations of Lévy processes from the perspective of point processes. In Lévy Processes 401–415. Birkhäuser, Boston, MA.
• [31] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.