We consider the problem of estimating a nested structure of two expectations
taking the form U0=E[max{U1(Y),π(Y)}], where U1(Y)=E[X∣Y].
Terms of this form arise in financial risk estimation and option pricing. When
U1(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 U1, which can be
paired with an antithetic MLMC estimate of U0 to recover order
ε−2 computational cost. In this work, we instead consider biased
multilevel approximations of U1(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.Comment: 28 pages, 2 figure