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Exact simulation for multivariate Itô diffusions

Part of: Combinatorics, graph theory, probability theory, statistics Functional-differential and differential-difference equations Probabilistic methods, simulation and stochastic differential equations Equilibrium statistical mechanics

Published online by Cambridge University Press:  03 December 2020

Jose Blanchet  and
Fan Zhang
Jose Blanchet*
Affiliation:
Stanford University
Fan Zhang*
Affiliation:
Stanford University
*
*Postal address: Huang Engineering Center, 475 Via Ortega, Stanford, CA 94305, United States.
*Postal address: Huang Engineering Center, 475 Via Ortega, Stanford, CA 94305, United States.
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Abstract

We provide the first generic exact simulation algorithm for multivariate diffusions. Current exact sampling algorithms for diffusions require the existence of a transformation which can be used to reduce the sampling problem to the case of a constant diffusion matrix and a drift which is the gradient of some function. Such a transformation, called the Lamperti transformation, can be applied in general only in one dimension. So, completely different ideas are required for the exact sampling of generic multivariate diffusions. The development of these ideas is the main contribution of this paper. Our strategy combines techniques borrowed from the theory of rough paths, on the one hand, and multilevel Monte Carlo on the other.

Keywords

Exact simulationstochastic differential equationBrownian motionMonte Carlo method

MSC classification

Primary: 34K50: Stochastic functional-differential equations 65C05: Monte Carlo methods 82B80: Numerical methods (Monte Carlo, series resummation, etc.)
Secondary: 97K60: Distributions and stochastic processes
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 4 , December 2020 , pp. 1003 - 1034
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Copyright
© Applied Probability Trust 2020

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References

Ait-Sahalia, Y. (2008). Closed-form likelihood expansions for multivariate diffusions. Ann. Statist. 36, 906–937.CrossRefGoogle Scholar
Aronson, D. (1967). Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73, 890–896.CrossRefGoogle Scholar
Bally, V. (2006). Lower bounds for the density of locally elliptic Itô processes. Ann. Prob. 34, 2406–2440.CrossRefGoogle Scholar
Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12, 1077–1098.CrossRefGoogle Scholar
Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Prob. 15, 2422–2444.CrossRefGoogle Scholar
Blanchet, J., Chen, X. and Dong, J. (2017). $\epsilon$-strong simulation for multidimensional stochastic differential equations via rough path analysis. Ann. Appl. Prob. 27, 275–336.CrossRefGoogle Scholar
Chen, N. and Huang, Z. (2013). Localization and exact simulation of Brownian motion-driven stochastic differential equations. Math. Operat. Res. 38, 591–616.CrossRefGoogle Scholar
Fearnhead, P., Papaspiliopoulos, O., Roberts, G. O. and Stuart, A. (2010). Random-weight particle filtering of continuous time processes. J. R. Statist. Soc. B [Statist. Methodology] 72, 497–512.CrossRefGoogle Scholar
Fleming, W. H. and Rishel, R. W. (2012). Deterministic and Stochastic Optimal Control. Springer Science & Business Media, New York.Google Scholar
Friedman, A. (2013). Partial Differential Equations of Parabolic Type. Courier Corporation, Chelmsford, MA.Google Scholar
Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Operat. Res. 56, 607–617.CrossRefGoogle Scholar
Giles, M. B. and Szpruch, L. (2014). Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation. Ann. Appl. Prob. 24, 1585–1620.CrossRefGoogle Scholar
Henry-Labordère, P., Tan, X., Touzi, N. (2017). Unbiased simulation of stochastic differential equations. Ann. Appl. Prob. 27, 3305–3341.CrossRefGoogle Scholar
Huber, M. (2016). Nearly optimal Bernoulli factories for linear functions. Combinatorics Prob. Comput. 25, 577–591.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. (2012). Brownian Motion and Stochastic Calculus. Springer Science & Business Media, New York.Google Scholar
Kusuoka, S. and Stroock, D. (1987). Applications of the Malliavin calculus, part 3. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34, 397–442.Google Scholar
Łatuszyńki, K., Kosmidis, I., Papaspiliopoulos, O. and Roberts, G. O. (2011). Simulating events of unknown probabilities via reverse time martingales. Random Structures Algorithms 38, 441–452.CrossRefGoogle Scholar
McLeish, D. (2011). A general method for debiasing a Monte Carlo estimator. Monte Carlo Meth. Appl. 17, 301–315.CrossRefGoogle Scholar
Nacu, Ş. and Peres, Y. (2005). Fast simulation of new coins from old. Ann. Appl. Prob. 15, 93–115.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (2013). Continuous Martingales and Brownian Motion. Springer Science & Business Media, New York.Google Scholar
Rhee, C.-H. and Glynn, P. W. (2012). A new approach to unbiased estimation for SDE’s. In WSC ’12: Proceedings of the Winter Simulation Conference, Berlin, pp. 1–7.CrossRefGoogle Scholar
Rhee, C.-H. and Glynn, P. W. (2015). Unbiased estimation with square root convergence for SDE models. Operat. Res. 63, 1026–1043.CrossRefGoogle Scholar
Sheu, S.-J. (1991). Some estimates of the transition density of a nondegenerate diffusion Markov process. Ann. Prob. 19, 538–561.CrossRefGoogle Scholar