The Annals of Applied Probability

Iterative multilevel particle approximation for McKean–Vlasov SDEs

Lukasz Szpruch, Shuren Tan, and Alvin Tse

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

The mean field limits of systems of interacting diffusions (also called stochastic interacting particle systems (SIPS)) have been intensively studied since McKean (Proc. Natl. Acad. Sci. USA 56 (1966) 1907–1911) as they pave a way to probabilistic representations for many important nonlinear/nonlocal PDEs. The fact that particles are not independent render classical variance reduction techniques not directly applicable, and consequently make simulations of interacting diffusions prohibitive.

In this article, we provide an alternative iterative particle representation, inspired by the fixed-point argument by Sznitman (In École D’Été de Probabilités de Saint-Flour XIX—1989 (1991) 165–251, Springer). The representation enjoys suitable conditional independence property that is leveraged in our analysis. We establish weak convergence of iterative particle system to the McKean–Vlasov SDEs (McKV–SDEs). One of the immediate advantages of the iterative particle system is that it can be combined with the Multilevel Monte Carlo (MLMC) approach for the simulation of McKV–SDEs. We proved that the MLMC approach reduces the computational complexity of calculating expectations by an order of magnitude. Another perspective on this work is that we analyse the error of nested Multilevel Monte Carlo estimators, which is of independent interest. Furthermore, we work with state dependent functionals, unlike scalar outputs which are common in literature on MLMC. The error analysis is carried out in uniform, and what seems to be new, weighted norms.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 4 (2019), 2230-2265.

Dates
Received: July 2017
Revised: October 2018
First available in Project Euclid: 23 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1563869042

Digital Object Identifier
doi:10.1214/18-AAP1452

Mathematical Reviews number (MathSciNet)
MR3983338

Zentralblatt MATH identifier
07120708

Subjects
Primary: 65C30: Stochastic differential and integral equations 60H35: Computational methods for stochastic equations [See also 65C30]
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
McKean–Vlasov SDEs stochastic interacting particle systems nonlinear Fokker–Planck equations probabilistic numerical analysis

Citation

Szpruch, Lukasz; Tan, Shuren; Tse, Alvin. Iterative multilevel particle approximation for McKean–Vlasov SDEs. Ann. Appl. Probab. 29 (2019), no. 4, 2230--2265. doi:10.1214/18-AAP1452. https://projecteuclid.org/euclid.aoap/1563869042


Export citation

References

  • [1] Ali, A. L. H. (2012). Pedestrian flow in the mean field limit. Ph.D. thesis, King Abdullah Univ. Science and Technology (KAUST).
  • [2] Antonelli, F. and Kohatsu-Higa, A. (2002). Rate of convergence of a particle method to the solution of the McKean–Vlasov equation. Ann. Appl. Probab. 12 423–476.
  • [3] Bossy, M. (2004). Optimal rate of convergence of a stochastic particle method to solutions of 1D viscous scalar conservation laws. Math. Comp. 73 777–812.
  • [4] Bossy, M., Fezoui, L. and Piperno, S. (1997). Comparison of a stochastic particle method and a finite volume deterministic method applied to Burgers equation. Monte Carlo Methods Appl. 3 113–140.
  • [5] Bossy, M. and Jourdain, B. (2002). Rate of convergence of a particle method for the solution of a 1D viscous scalar conservation law in a bounded interval. Ann. Probab. 30 1797–1832.
  • [6] Bossy, M. and Talay, D. (1996). Convergence rate for the approximation of the limit law of weakly interacting particles: Application to the Burgers equation. Ann. Appl. Probab. 6 818–861.
  • [7] Bossy, M. and Talay, D. (1997). A stochastic particle method for the McKean–Vlasov and the Burgers equation. Math. Comp. 66 157–192.
  • [8] Buckdahn, R., Li, J., Peng, S. and Rainer, C. (2017). Mean-field stochastic differential equations and associated PDEs. Ann. Probab. 45 824–878.
  • [9] Bujok, K., Hambly, B. M. and Reisinger, C. (2015). Multilevel simulation of functionals of Bernoulli random variables with application to basket credit derivatives. Methodol. Comput. Appl. Probab. 17 579–604.
  • [10] Carmona, R., Delarue, F. and Lachapelle, A. (2013). Control of McKean–Vlasov dynamics versus mean field games. Math. Financ. Econ. 7 131–166.
  • [11] Chassagneux, J.-F., Crisan, D. and Delarue, F. (2014). A probabilistic approach to classical solutions of the master equation for large population equilibria. Available at arXiv:1411.3009.
  • [12] Delarue, F., Inglis, J., Rubenthaler, S. and Tanré, E. (2015). Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Ann. Appl. Probab. 25 2096–2133.
  • [13] Delarue, F., Inglis, J., Rubenthaler, S. and Tanré, E. (2015). Particle systems with a singular mean-field self-excitation. Application to neuronal networks. Stochastic Process. Appl. 125 2451–2492.
  • [14] E, W., Hutzenthaler, M., Jentzen, A. and Kruse, T. (2016). On full history recursive multilevel Picard approximations and numerical approximations of high-dimensional nonlinear parabolic partial differential equations. Available at arXiv:1607.03295.
  • [15] Friedman, A. (2006). Stochastic Differential Equations and Applications. Courier Corporation.
  • [16] Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 607–617.
  • [17] Giles, M. B., Nagapetyan, T. and Ritter, K. (2015). Multilevel Monte Carlo approximation of distribution functions and densities. SIAM/ASA J. Uncertain. Quantificat. 3 267–295.
  • [18] Haji-Ali, A.-L. and Tempone, R. (2018). Multilevel and multi-index Monte Carlo methods for the McKean–Vlasov equation. Stat. Comput. 28 923–935.
  • [19] Heinrich, S. (2001). Multilevel Monte Carlo methods. In Large-Scale Scientific Computing 58–67. Springer.
  • [20] Kebaier, A. (2005). Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15 2681–2705.
  • [21] Krylov, N. V. (1980). Controlled Diffusion Processes. Applications of Mathematics 14. Springer, New York. Translated from the Russian by A. B. Aries.
  • [22] Krylov, N. V. (2002). Introduction to the Theory of Random Processes. Graduate Studies in Mathematics 43. Amer. Math. Soc., Providence, RI.
  • [23] Lemaire, V. and Pagès, G. (2017). Multilevel Richardson–Romberg extrapolation. Bernoulli 23 2643–2692.
  • [24] Lions, P. L. (2014). Cours au collège de france: Théorie des jeux à champs moyens.
  • [25] McKean, H. P. Jr. (1966). A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56 1907–1911.
  • [26] Méléard, S. (1996). Asymptotic behaviour of some interacting particle systems; McKean–Vlasov and Boltzmann models. In Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Math. 1627 42–95. Springer, Berlin.
  • [27] Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Probability and Mathematical Statistics 3. Academic Press, New York.
  • [28] Pope, S. B. (2000). Turbulent Flows. Cambridge Univ. Press, Cambridge.
  • [29] Ricketson, L. F. (2015). A multilevel Monte Carlo method for a class of McKean–Vlasov processes. Available at arXiv:1508.02299.
  • [30] Sznitman, A.-S. (1991). Topics in propagation of chaos. In École D’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464 165–251. Springer, Berlin.
  • [31] Talay, D. and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 483–509 (1991).