This work generalizes a multilevel forward Euler Monte Carlo method introduced in Michael B. Giles. (Michael Giles. Oper. Res. 56(3):607-617, 2008.) for the approximation of expected values depending on the solution to an ltd stochastic differential equation. The work (Michael Giles. Oper. Res. 56(3):607-617, 2008.) proposed and analyzed a forward Euler multilevel Monte Carlo method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level. Forward Euler Monte Carlo method. This work introduces an adaptive hierarchy of non uniform time discretizations, generated by an adaptive algorithm introduced in (Anna Dzougoutov et al. Raul Tempone. Adaptive Monte Carlo algorithms for slopped diffusion. In Multiscale methods in science and engineering, volume 44 of Led. Notes Comput. Sci. Eng., pages 59-88. Springer, Berlin, 2005; Kyoung-Sook Moon et al. Stoch. Anal. Appl. 23(3):511-558,2005; Kyoung-Sook Moon ct al. An adaptive algorithm for ordinary, stochastic and partial differential equations. In Recent advances in adaptive computation, volume 383 of Contemp. Math., pages 325-343. Amcr. Math. Soc, Providence, RI, 2005.). This form of the adaptive algorithm generates stochastic, path dependent, time steps and is based on a posteriori error expansions first developed in (Anders Szepessy et al. Comm. Pure Appl. Math. 54(10):l 169-1214, 2001). Our numerical results for a stopped diffusion problem, exhibit savings in the computational cost to achieve an accuracy of O(TOL), from O(TOL~(-3)) using a single level version of the adaptive algorithm to O((TOL~(-1) log (TOL))~2).
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