Multilevel Monte Carlo Approximation of Distribution Functions and Densities

We construct and analyze multilevel Monte Carlo methods for the approximation of distribution functions and densities of univariate random variables. Since, by assumption, the target distribution is not known explicitly, approximations have to be used. We provide a general analysis under suitable assumptions on the weak and strong convergence. We apply the results to smooth path-independent and path-dependent functionals and to stopped exit times of stochastic differential equations (SDEs).

  • 1.  A. Alfonsi B. Jourdain and  A. Kohatsu-Higa , Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme , Ann. Appl. Probab. , 24 ( 2014 ), pp. 1049 -- 1080 . CrossrefISIGoogle Scholar

  • 2.  M. Altmayer and  A. Neuenkirch , Multilevel Monte Carlo quadrature of discontinuous payoffs in the generalized Heston model using Malliavin integration by parts , SIAM J. Financial Math. , 6 ( 2015 ), pp. 22 -- 52 . LinkISIGoogle Scholar

  • 3.  R. Avikainen , On irregular functionals of SDEs and the Euler scheme , Finance Stoch. , 13 ( 2009 ), pp. 381 -- 401 . CrossrefISIGoogle Scholar

  • 4.  V. Bally and  D. Talay , The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function , Probab. Theory Related Fields , 104 ( 1996 ), pp. 43 -- 60 . CrossrefISIGoogle Scholar

  • 5.  V. Bally and  D. Talay , The law of the Euler scheme for stochastic differential equations: II. Convergence rate of the density , Monte Carlo Methods Appl. , 2 ( 1996 ), pp. 93 -- 128 . CrossrefGoogle Scholar

  • 6.  B. Bouchard S. Geiss and  E. Gobet , First Time to Exit of a Continuous Itô Process: General Moment Estimates and $L_1$-Convergence Rate for Discrete Time Approximations, preprint, arXiv:1307.4247 , 2013 . Google Scholar

  • 7.  C. de Boor , A Practical Guide to Splines , Appl. Math. Sci. 27 , Springer , Berlin , 1978 . Google Scholar

  • 8.  M. B. Giles , Improved multilevel Monte Carlo convergence using the Milstein scheme , in Monte Carlo and Quasi-Monte Carlo Methods 2006 , A. Keller, S. Heinrich, and H. Niederreiter, eds., Springer , Berlin, 2008, pp. 343 -- 358 . Google Scholar

  • 9.  M. B. Giles and  Multilevel Monte Carlo path simulation , Oper. Res. , 56 ( 2008 ), pp. 607 -- 617 . CrossrefISIGoogle Scholar

  • 10.  M. B. Giles K. Debrabant and  A. Rößler , Numerical Analysis of Multilevel Monte Carlo Path Simulation Using the Milstein Discretisation, preprint, arXiv:1302.4676 , 2013 . Google Scholar

  • 11.  M. B. Giles D. J. Higham and  X. Mao , Analysing multilevel Monte Carlo for options with non-globally Lipschitz payoff , Finance Stoch. , 13 ( 2009 ), pp. 403 -- 413 . CrossrefISIGoogle Scholar

  • 12.  E. Gobet and  C. Labart , Sharp estimates for the convergence of the density of the Euler scheme in small time , Elect. Comm. Probab. , 13 ( 2008 ), pp. 352 -- 363 . CrossrefISIGoogle Scholar

  • 13.  E. Gobet and  S. Menozzi , Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme , Stochastic Process. Appl. , 112 ( 2004 ), pp. 201 -- 223 . CrossrefISIGoogle Scholar

  • 14.  S. Heinrich Carlo complexity of global solution of integral equations , J. Complexity , 14 ( 1998 ), pp. 151 -- 175 . CrossrefISIGoogle Scholar

  • 15.  D. J. Higham X. Mao M. Roj Q. Song and  G. Yin , Mean exit times and the multi-level Monte Carlo method , SIAM/ASA J. Uncertainty Quantification , 1 ( 2013 ), pp. 2 -- 18 . LinkGoogle Scholar

  • 16.  J. Jacod and  P. Protter , Asymptotic error distributions for the Euler method for stochastic differential equations , Ann. Probab. , 26 ( 1998 ), pp. 267 -- 307 . CrossrefISIGoogle Scholar

  • 17.  A. Kebaier and  Statistical Romberg : A new variance reduction method and applications to options pricing , Ann. Appl. Probab. , 14 ( 2005 ), pp. 2681 -- 2705 . CrossrefISIGoogle Scholar

  • 18.  A. Kebaier and  A. Kohatsu-Higa , An optimal control variance reduction method for density estimation , Stochastic Process. Appl. , 118 ( 2008 ), pp. 2143 -- 2180 . CrossrefISIGoogle Scholar

  • 19.  P. Massart , The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality , Ann. Probab. , 18 ( 1990 ), pp. 1269 -- 1283 . CrossrefISIGoogle Scholar

  • 20.  G. N. Milstein J. Schoenmakers and  V. Spokoiny , Transition density estimation for stochastic differential equations via forward-reverse representations , Bernoulli , 10 ( 2004 ), pp. 281 -- 312 . CrossrefISIGoogle Scholar

  • 21.  T . Müller-Gronbach, The optimal uniform approximation of systems of stochastic differential equations , Ann. Appl. Probab. , 12 ( 2002 ), pp. 664 -- 690 . CrossrefISIGoogle Scholar

  • 22.  S. E. Shreve , Stochastic Calculus for Finance II: Continuous-Time Models , Springer-Verlag , New York , 2004 . Google Scholar

  • 23.  D. Talay and  Z. Zheng , Approximation of quantiles of components of diffusion processes , Stochastic Process. Appl. , 109 ( 2004 ), pp. 23 -- 46 . CrossrefISIGoogle Scholar

  • 24.  A. B. Tsybakov , Introduction to Nonparametric Estimation , Springer , Berlin , 2009 . Google Scholar