Abstract
We investigate the problem of computing a nested expectation of the form P[E[X Y ]≥ 0] = E[H(E[X Y ])] where H is the Heaviside function. This nested expectation appears, for example, when estimating the probability of a large loss from a financial portfolio. We present a method that combines the idea of using Multilevel Monte Carlo (MLMC) for nested expectations with the idea of adaptively selecting the number of samples in the approximation of the inner expectation, as proposed by [M. Broadie, Y. Du, and C. C. Moallemi, Manag. Sci., 57 (2011), pp. 1172 1194]. We propose and analyze an algorithm that adaptively selects the number of inner samples on each MLMC level and prove that the resulting MLMC method with adaptive sampling has an O (ϵ ^{2} logϵ ^{2}) complexity to achieve a root meansquared error ϵ . The theoretical analysis is verified by numerical experiments on a simple model problem. We also present a stochastic rootfinding algorithm that, combined with our adaptive methods, can be used to compute other risk measures such as ValueatRisk (VaR) and Conditional ValueatRisk (CVaR), the latter being achieved with O (ϵ ^{2}) complexity.
Original language  English 

Pages (fromto)  497–525 
Number of pages  29 
Journal  SIAM/ASA Journal on Uncertainty Quantification 
Volume  7 
Issue number  2 
Early online date  2 May 2019 
DOIs  
Publication status  Published  2019 
Keywords
 Monte Carlo
 Multilevel Monte Carlo
 Nested simulation
 Risk estimation
ASJC Scopus subject areas
 Statistics and Probability
 Modelling and Simulation
 Statistics, Probability and Uncertainty
 Discrete Mathematics and Combinatorics
 Applied Mathematics
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AbdulLateef HajiAli
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics  Associate Professor
Person: Academic (Research & Teaching)