MLMC for Nested Expectations

  • Michael B. GilesEmail author


This paper discusses progress and future research possibilities in applying MLMC ideas to nested expectations of the form \({\mathbb {E}}[\, g({\mathbb {E}}[\,f(X,Y) | X]) \,]\), with an outer expectation with respect to one random variable X, and an inner conditional expectation with respect to a second random variable Y . The difficulty in treating such applications is shown to depend on whether the function g is (1) smooth, (2) continuous and piecewise smooth, or (3) discontinuous.



The author is very grateful to a number of people for discussions and collaborations: Ian Sloan and Frances Kuo on alternatives to the Multi-Index Monte Carlo method; Takashi Goda and Howard Thom on EVPPI estimation; Wenhui Gou, Abdul-Lateef Haji-Ali, Ralf Korn and Klaus Ritter on Value-at-Risk estimation.


  1. 1.
    Ades, A., Lu, G., Claxton, K.: Expected value of sample information calculations in medical decision modeling. Med. Decis. Mak. 24(2), 207–227 (2004)CrossRefGoogle Scholar
  2. 2.
    Bierig, C., Chernov, A.: Approximation of probability density functions by the multilevel Monte Carlo maximum entropy method. J. Comput. Phys. 314, 661–681 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bratvold, R., Bickel, J., Lohne, H.: Value of information in the oil and gas industry: past, present, and future. SPE Reserv. Eval. Eng. 12, 630–638 (2009)CrossRefGoogle Scholar
  4. 4.
    Brennan, A., Kharroubi, S., O’Hagan, A., Chilcott, J.: Calculating partial expected value of perfect information via Monte Carlo sampling algorithms. Med. Decis. Mak. 27, 448–470 (2007)CrossRefGoogle Scholar
  5. 5.
    Broadie, M., Du, Y., Moallemi, C.: Efficient risk estimation via nested sequential simulation. Manag. Sci. 57(6), 1172–1194 (2011)CrossRefGoogle Scholar
  6. 6.
    Bujok, K., Hambly, B., Reisinger, C.: Multilevel simulation of functionals of Bernoulli random variables with application to basket credit derivatives. Methodol. Comput. Appl. Probab. 17(3), 579–604 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, N., Liu, Y.: Estimating expectations of functionals of conditional expected via multilevel nested simulation. In: Presentation at conference on Monte Carlo and Quasi-Monte Carlo Methods, Sydney (2012)Google Scholar
  9. 9.
    Fang, W., Giles, M.: Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift: Part I, finite time interval (2016). ArXiv preprint: 1609.08101Google Scholar
  10. 10.
    Giles, M.: Multilevel Monte Carlo methods. Acta Numer. 24, 259–328 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Giles, M., Goda, T.: Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI (2017). ArXiv preprint: 1708.05531Google Scholar
  12. 12.
    Giles, M., Ramanan, K.: MLMC with adaptive timestepping for reflected Brownian diffusions (2018, in preparation)Google Scholar
  13. 13.
    Giles, M., Reisinger, C.: Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance. SIAM J. Financ. Math. 3(1), 572–592 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Giles, M., Szpruch, L.: Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation. Ann. Appl. Probab. 24(4), 1585–1620 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Giles, M., Lester, C., Whittle, J.: Non-nested adaptive timesteps in multilevel Monte Carlo computations. In: Cools, R., Nuyens, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2014. Springer, Basel (2016)zbMATHGoogle Scholar
  16. 16.
    Glasserman, P., Heidelberger, P., Shahabuddin, P.: Variance reduction techniques for estimating value-at-risk. Manag. Sci. 46, 1349–1364 (2000)CrossRefGoogle Scholar
  17. 17.
    Glasserman, P., Heidelberger, P., Shahabuddin, P.: Portfolio value-at-risk with heavy-tailed risk factors. Math. Finance 12, 239–269 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gordy, M., Juneja, S.: Nested simulation in portfolio risk measurement. Manag. Sci. 56(10), 1833–1848 (2010)CrossRefGoogle Scholar
  19. 19.
    Gou, W.: Estimating value-at-risk using multilevel Monte Carlo maximum entropy method. MSc Thesis, University of Oxford (2016)Google Scholar
  20. 20.
    Haji-Ali, A.L.: Pedestrian flow in the mean-field limit. MSc Thesis, KAUST (2012)Google Scholar
  21. 21.
    Haji-Ali, A.L., Nobile, F., Tempone, R.: Multi index Monte Carlo: when sparsity meets sampling. Numer. Math. 132(4), 767–806 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Korn, R., Korn, E., Kronstadt, G.: Monte Carlo Methods and Models in Finance and Insurance. Chapman and Hall/CRC Financial Mathematics. CRC Press, Boca Raton (2010)CrossRefGoogle Scholar
  23. 23.
    Korn, R., Pupashenko, M.: A new variance reduction technique for estimating value-at-risk. Appl. Math. Finance 22(1), 83–98 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nakayasu, M., Goda, T., Tanaka, K., Sato, K.: Evaluating the value of single-point data in heterogeneous reservoirs with the expectation maximization. SPE Econ. Manag. 8, 1–10 (2016)CrossRefGoogle Scholar
  25. 25.
    Rhee, C.H., Glynn, P.: Unbiased estimation with square root convergence for SDE models. Oper. Res. 63(5), 1026–1043 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rosin, M., Ricketson, L., Dimits, A., Caflisch, R., Cohen, B.: Multilevel Monte Carlo simulation of Coulomb collisions. J. Comput. Phys. 247, 140–157 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations