The Annals of Applied Probability

Unbiased simulation of stochastic differential equations

Pierre Henry-Labordère, Xiaolu Tan, and Nizar Touzi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We propose an unbiased Monte 3 estimator for $\mathbb{E}[g(X_{t_{1}},\ldots,X_{t_{n}})]$, where $X$ is a diffusion process defined by a multidimensional stochastic differential equation (SDE). The main idea is to start instead from a well-chosen simulatable SDE whose coefficients are updated at independent exponential times. Such a simulatable process can be viewed as a regime-switching SDE, or as a branching diffusion process with one single living particle at all times. In order to compensate for the change of the coefficients of the SDE, our main representation result relies on the automatic differentiation technique induced by the Bismut–Elworthy–Li formula from Malliavin calculus, as exploited by Fournié et al. [Finance Stoch. 3 (1999) 391–412] for the simulation of the Greeks in financial applications. In particular, this algorithm can be considered as a variation of the (infinite variance) estimator obtained in Bally and Kohatsu-Higa [Ann. Appl. Probab. 25 (2015) 3095–3138, Section 6.1] as an application of the parametrix method.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3305-3341.

Dates
Received: March 2016
Revised: November 2016
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1513328702

Digital Object Identifier
doi:10.1214/17-AAP1281

Mathematical Reviews number (MathSciNet)
MR3737926

Zentralblatt MATH identifier
06848267

Subjects
Primary: 65C05: Monte Carlo methods 60J60: Diffusion processes [See also 58J65]
Secondary: 60J85: Applications of branching processes [See also 92Dxx] 35K10: Second-order parabolic equations

Keywords
Unbiased simulation of SDEs regime switching diffusion linear parabolic PDEs

Citation

Henry-Labordère, Pierre; Tan, Xiaolu; Touzi, Nizar. Unbiased simulation of stochastic differential equations. Ann. Appl. Probab. 27 (2017), no. 6, 3305--3341. doi:10.1214/17-AAP1281. https://projecteuclid.org/euclid.aoap/1513328702


Export citation

References

  • [1] Andersson, P. and Kohatsu-Higa, A. (2017). Unbiased simulation of stochastic differential equations using parametrix expansions. Bernoulli 23 2028–2057.
  • [2] Bally, V. and Kohatsu-Higa, A. (2015). A probabilistic interpretation of the parametrix method. Ann. Appl. Probab. 25 3095–3138.
  • [3] Ben Alaya, M. and Kebaier, A. (2015). Central limit theorem for the multilevel Monte Carlo Euler method. Ann. Appl. Probab. 25 211–234.
  • [4] Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12 1077–1098.
  • [5] Beskos, A., Papaspiliopoulos, O., Roberts, G. O. and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 333–382.
  • [6] Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Probab. 15 2422–2444.
  • [7] Blanchet, J., Chen, X. and Dong, J. (2017). $\varepsilon$-Strong simulation for multidimensional stochastic differential equations via rough path analysis. Ann. Appl. Probab. 27 275–336.
  • [8] Bompis, R. and Gobet, E. (2014). Stochastic approximation finite element method: Analytical formulas for multidimensional diffusion process. SIAM J. Numer. Anal. 52 3140–3164.
  • [9] Fahim, A., Touzi, N. and Warin, X. (2011). A probabilistic numerical method for fully nonlinear parabolic PDEs. Ann. Appl. Probab. 21 1322–1364.
  • [10] Fournié, E., Lasry, J.-M., Lebuchoux, J., Lions, P.-L. and Touzi, N. (1999). Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch. 3 391–412.
  • [11] Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 607–617.
  • [12] Giles, M. B. and Szpruch, L. (2014). Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation. Ann. Appl. Probab. 24 1585–1620.
  • [13] Graham, C. and Talay, D. (2013). Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation. Stochastic Modelling and Applied Probability 68. Springer, Heidelberg.
  • [14] Henry-Labordère, P. (2012). Counterparty risk valuation: A marked branching diffusion approach. Preprint. Available at arXiv:1203.2369.
  • [15] Henry-Labordère, P., Oudjane, N., Tan, X., Touzi, N. and Warin, X. (2016). Branching diffusion representation of semilinear PDEs and Monte Carlo approximation. Work in progress.
  • [16] Henry-Labordère, P., Tan, X. and Touzi, N. (2014). A numerical algorithm for a class of BSDEs via the branching process. Stochastic Process. Appl. 124 1112–1140.
  • [17] Jourdain, B. and Sbai, M. (2007). Exact retrospective Monte Carlo computation of arithmetic average Asian options. Monte Carlo Methods Appl. 13 135–171.
  • [18] Kebaier, A. (2005). Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15 2681–2705.
  • [19] Kloeden, P. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics 23. Springer, Berlin.
  • [20] Rhee, C.-H. and Glynn, P. W. (2015). Unbiased estimation with square root convergence for SDE models. Oper. Res. 63 1026–1043.
  • [21] Talay, D. and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 483–509.
  • [22] Wagner, W. (1988). Unbiased multi-step estimators for the Monte Carlo evaluation of certain functional integrals. J. Comput. Phys. 79 336–352.
  • [23] Wagner, W. (1989). Unbiased Monte Carlo estimators for functionals of weak solutions of stochastic differential equations. Stoch. Stoch. Rep. 28 1–20.