Monte Carlo and Quasi-Monte Carlo Methods by Art B. Owen & Peter W. Glynn

Author:Art B. Owen & Peter W. Glynn
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(17)

where with being the numerical estimator of which uses adaptive function with for some positive integer Then the standard MLMC estimator is the following telescoping sum:

where is the terminal value of the nth numerical path in the time interval [0, T] using a suitable adaptive function with

Unlike the standard MLMC with fixed time interval [0, T], we now allow different levels to have a different length of time interval satisfying which means that as level increases, we obtain a better approximation not only by using smaller timesteps but also by simulating a longer time interval. However, the difficulty is how to construct a good coupling on each level since the fine path and coarse path have different lengths of time interval and

Following the idea of Glynn and Rhee [8] to estimate the invariant measure of some Markov chains, we perform the coupling by starting a level fine path simulation at time and a coarse path simulation at time and terminating both paths at Since the drift f and volatility g do not depend explicitly on time t, the distribution of the numerical solution simulated on the time interval is the same as one simulated on The key point here is that the fine path and coarse path share the same driving Brownian motion during the overlap time interval Owing to the result of Lemma 4, two solutions to the SDE satisfying Assumption 9, starting from different initial points and driven by the same Brownian motion will converge exponentially. Therefore, the fact that different levels terminate at the same time is crucial to the variance reduction of the multilevel scheme.

Our new multilevel scheme still has the identity (17) but with with being the terminal value of the numerical path approximation on the time interval using adaptive function with The corresponding new MLMC estimator is

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