Non-nested Adaptive Timesteps in Multilevel Monte Carlo

Abstract This paper shows that it is relatively easy to incorporate adaptive timesteps into multilevel Monte Carlo simulations without violating the telescoping sum on which multilevel Monte Carlo is based. The numerical approach is presented for both SDEs and continuous-time Markov processes. Numerical experiments are given for each, with the full code available for those who are interested in seeing the implementation details. 1 Multilevel Monte Carlo and Adaptive Simulations Multilevel Monte Carlo methods [8, 4, 6] are a very simple and general approach to improving the computational efficiency of a wide range of Monte Carlo applications. Given a set of approximation levels ℓ = 0,1,...,L giving a sequence of approxima-tions Pℓ of a stochastic output P, with the cost and accuracy both increasing as ℓ increases, then a trivial telescoping sum gives E[PL] = E[P0]+ L ℓ=