Nested Multilevel Monte Carlo with Biased and Antithetic Sampling
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We consider the problem of estimating a nested structure of two expectations taking the form U_0 = E[max{U_1(Y), π(Y)}], where U_1(Y) = E[X | Y]. Terms of this form arise in financial risk estimation and option pricing. When U_1(Y) requires approximation, but exact samples of X and Y are available, an antithetic multilevel Monte Carlo (MLMC) approach has been well-studied in the literature. Under general conditions, the antithetic MLMC estimator obtains a root mean squared error ε with order ε^-2 cost. If, additionally, X and Y require approximate sampling, careful balancing of the various aspects of approximation is required to avoid a significant computational burden. Under strong convergence criteria on approximations to X and Y, randomised multilevel Monte Carlo techniques can be used to construct unbiased Monte Carlo estimates of U_1, which can be paired with an antithetic MLMC estimate of U_0 to recover order ε^-2 computational cost. In this work, we instead consider biased multilevel approximations of U_1(Y), which require less strict assumptions on the approximate samples of X. Extensions to the method consider an approximate and antithetic sampling of Y. Analysis shows the resulting estimator has order ε^-2 asymptotic cost under the conditions required by randomised MLMC and order ε^-2|logε|^3 cost under more general assumptions.
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