Adaptive Multi Level Monte Carlo Simulation

The topic of this talk is a work that generalizes a multilevel Forward Euler Monte Carlo method
introduced in [1] for the approximation of expected values depending on the solution to an Ito stochastic differential equation. The work [1] 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. The present work introduces and analyzes an adaptive hierarchy of non uniform time discretizations, generated by adaptive algorithms introduced in [3, 2]. These adaptive algorithms apply either deterministic time steps or stochastic time
steps and are based on a posteriori error expansions first developed in [4]. Under sufficient regularity conditions, both our analysis and numerical results, which include one case with singular drift and one with stopped diffusion, exhibit savings in the computational cost to achieve an accuracy of O(TOL), from
O(TOL^-3) to O(TOL^-1*log(TOL))^2.

In the talk I will give a background on the Multi Level Monte Carlo method of uniform time discretizations in [1] and show how to extend this idea to the setting of adaptive time stepping.
This talk presents joint work with H. Hoel, A. Szepessy, and R. Tempone.

Key words: computational finance, Monte Carlo, multi-level, adaptivity, weak approximation, error
control, Euler-Maruyama method, a posteriori error estimates, backward dual functions, adjoints

References
[1] Michael B. Giles. Multilevel Monte Carlo path simulation. Oper. Res., 56(3):607-617, 2008.
[2] Kyoung-Sook Moon, Anders Szepessy, Raul Tempone, and Georgios E. Zouraris. Convergence rates for adaptive weak approximation of stochastic dierential equations. Stoch. Anal. Appl., 23(3):511-558, 2005.
[3] Kyoung-Sook Moon, Erik von Schwerin, Anders Szepessy, and Raul Tempone. An adaptive algorithm
for ordinary, stochastic and partial dierential equations. In Recent advances in adaptive computation, volume 383 of Contemp. Math., pages 325-343. Amer. Math. Soc., Providence, RI, 2005.
[4] Anders Szepessy, Raul Tempone, and Georgios E. Zouraris. Adaptive weak approximation of stochastic dierential equations. Comm. Pure Appl. Math., 54(10):1169-1214, 2001.