Unbiased Estimators and Multilevel Monte Carlo

Published Online:https://doi.org/10.1287/opre.2017.1670

References

  • Agapiou S, Roberts GO, Vollmer SJ (2014) Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models. Bernoulli Forthcoming.Google Scholar
  • Alaya MB, Kebaier A (2015) Central limit theorem for the multilevel Monte Carlo Euler method. Ann. Appl. Probab. 25(1):211–234.CrossrefGoogle Scholar
  • Beskos A, Roberts GO (2005) Exact simulation of diffusions. Ann. Appl. Probab. 15(4):2422–2444.CrossrefGoogle Scholar
  • Borkar VS (2008) Stochastic Approximation: A Dynamical Systems Viewpoint (Cambridge University Press, Cambridge, UK).CrossrefGoogle Scholar
  • Collier N, Haji-Ali AL, Nobile F, Von Schwerin E, Tempone R (2015) A continuation multilevel Monte Carlo algorithm. BIT 55(2):399–432.CrossrefGoogle Scholar
  • Delyon B, Lavielle M, Moulines E (1999) Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27(1):94–128.Google Scholar
  • Dereich S, Mueller-Gronbach T (2015) General multilevel adaptations for stochastic approximation algorithms. Preprint arXiv:1506.05482.Google Scholar
  • Dick J, Kuo FY, Sloan IH (2013) High-dimensional integration: The quasi-Monte Carlo way. Acta Numer. 22:133–288.CrossrefGoogle Scholar
  • Douc R, Cappé O, Moulines E (2005) Comparison of resampling schemes for particle filtering. Proc. 4th Internat. Sympos. Image and Signal Processing and Analysis, ISPA ’05 (IEEE, Piscataway, NJ), 64–69.Google Scholar
  • Feller W (1946) A limit theoerm for random variables with infinite moments. Amer. J. Math. 68(2):257–262.CrossrefGoogle Scholar
  • Giles MB (2008) Multilevel Monte Carlo path simulation. Oper. Res. 56(3):607–617.LinkGoogle Scholar
  • Giles MB (2015) Multilevel Monte Carlo methods. Acta Numer. 24:259–328.CrossrefGoogle Scholar
  • Giles MB, Szpruch L (2014) Antithetic multilevel Monte Carlo estimation for multi-dimensional sdes without Lévy area simulation. Ann. Appl. Probab. 24(4):1585–1620.CrossrefGoogle Scholar
  • Giles MB, Waterhouse BJ (2009) Multilevel quasi-Monte Carlo path simulation. Albrecher H, Runggaldier WJ, Schachermayer W, eds. Advanced Financial Modelling (De Gruyter, Berlin),165–181.Google Scholar
  • Glasserman P (2003) Monte Carlo Methods in Financial Engineering (Springer, New York).CrossrefGoogle Scholar
  • Glynn P (1983) Randomized estimators for time integrals. Technical report, Mathematical Research Center, University of Wisconsin, Madison.Google Scholar
  • Glynn PW, Rhee CH (2014) Exact estimation for Markov chain equilibrium expectations. J. Appl. Probab. 51A:377–389.CrossrefGoogle Scholar
  • Glynn PW, Whitt W (1992a) The asymptotic efficiency of simulation estimators. Oper. Res. 40(3):505–520.LinkGoogle Scholar
  • Glynn PW, Whitt W (1992b) The asymptotic validity of sequential stopping rules for stochastic simulations. Ann. Appl. Probab. 2(1):180–198.CrossrefGoogle Scholar
  • Haji-Ali AL, Nobile F, Von Schwerin E, Tempone R (2016) Optimization of mesh hierarchies in multilevel Monte Carlo samplers. Stoch. Partial Differ. Equ. Anal. Comput. 4(1):76–112.CrossrefGoogle Scholar
  • Hansen MH, Hurwitz WN, Madow WG (1953) Sample Survey Methods and Theory, Methods and Applications, Vol. 1 (John Wiley & Sons, New York).Google Scholar
  • Heinrich S (2001) Multilevel Monte Carlo methods. Margenov S, Waśniewski J, Yalamov P, eds. Large-Scale Scientific Computing (Springer, Berlin), 58–67.CrossrefGoogle Scholar
  • Hoel H, Von Schwerin E, Szepessy A, Tempone R (2012) Adaptive multilevel Monte Carlo simulation. Engquist B, Runborg O, Tsai YH, eds. Numerical Analysis of Multiscale Computations (Springer, Berlin), 217–234.CrossrefGoogle Scholar
  • Huber ML (2015) Perfect Simulation (Chapman Hall/CRC, Boca Raton, FL).Google Scholar
  • Jacob PE, Thiery AH (2015) On nonnegative unbiased estimators. Ann. Statist. 43(2):769–784.CrossrefGoogle Scholar
  • Jentzen A, Kloeden PE, Neuenkirch A (2009) Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global lipschitz coefficients. Numer. Math. 112(1):41–64.CrossrefGoogle Scholar
  • Kahl C, Jäckel P (2006) Fast stong approximation Monte Carlo schemes for stochastic volatility models. Quant. Finance 6(6):513–536.CrossrefGoogle Scholar
  • Kebaier A, Lelong J (2015) Coupling importance sampling and multilevel Monte Carlo using sample average approximation. J. Methodol. Comput. Appl. Probab. Forthcoming.Google Scholar
  • Kloeden PE, Platen E (1992) Numerical Solution of Stochastic Differential Equations (Springer, Berlin).CrossrefGoogle Scholar
  • Kushner HJ, Yin GG (2003) Stochastic Approximation and Recursive Algorithms and Applications (Springer, New York).Google Scholar
  • Lyne AM, Girolami M, Atchade Y, Strathmann H, Simpson D (2015) On Russian roulette estimates for Bayesian inference with doubly-intractable likelihoods. Statist. Sci. 30(4):443–467.CrossrefGoogle Scholar
  • McLeish D (2011) A general method for debiasing a Monte Carlo estimator. Monte Carlo Methods Appl. 17(4):301–315.CrossrefGoogle Scholar
  • Propp JG, Wilson DB (1996) Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9(1–2):223–252.CrossrefGoogle Scholar
  • Rhee CH, Glynn PW (2012) A new approach to unbiased estimation for SDE’s. Proc. Winter Simulation Conf. (IEEE, Piscataway, NJ).Google Scholar
  • Rhee CH, Glynn PW (2015) Unbiased estimation with square root convergence for SDE models. Oper. Res. 63(5):1026–1043.LinkGoogle Scholar
  • Robbins H, Monro S (1951) A stochastic approximation method. Ann. Math. Statist. 22(3):400–407.CrossrefGoogle Scholar
  • Rychlik T (1990) Unbiased nonparametric estimation of the derivative of the mean. Statist. Probab. Lett. 10(4):329–333.CrossrefGoogle Scholar
  • Rychlik T (1995) A class of unbiased kernel estimates of a probability density function. Appl. Math. (Warsaw) 22(4):485–497.CrossrefGoogle Scholar
  • Strathmann H, Sejdinovic D, Girolami M (2015) Unbiased Bayes for big data: Paths of partial posteriors. Preprint arXiv:1501.03326.Google Scholar
  • Vihola M (2015) Unbiased estimators and multilevel Monte Carlo. Preprint arXiv:1512.01022v2.Google Scholar
  • Vihola M, Helske J, Franks J (2016) Importance sampling type estimators based on approximate marginal Markov chain Monte Carlo. Preprint arXiv:1609.02541.Google Scholar
  • Walter C (2017) Point process-based Monte Carlo estimation. Statist. Comput. 27(1):219–236.CrossrefGoogle Scholar
  • Zheng Z, Glynn PW (2017) A CLT for infinitely stratified estimators, with applications to debiased MLMC. ESAIM: Proc. Surv. 59:104–114.CrossrefGoogle Scholar
  • Zheng Z, Blanchet J, Glynn PW (2017) Rates of convergence and CLTs for subcanonical debiased MLMC. Manuscript.Google Scholar
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