abstract:
This work 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. This work introduces and analyzes an adaptive hierarchy of non uniform time discretizations, generated
by adaptive algorithms introduced in [2, 3]. These adaptive algorithms apply either deterministic time steps or stochastic time steps and are based on adjoint
weighted 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 $O \left( \mbox{TOL} \right)$, from $O \left( \mbox{TOL}^{-3}\right)$ to $O \left( \left(
\mbox{TOL}^{-1}\mbox{ log(TOL)} \right) ^2 \right)$. \\
This is a joint work with H. Hoel, E. von Schwerin and A. Szepessy.
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\textbf{References}
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[1] & Giles, M. B., Multilevel Monte Carlo path simulation, \emph{Oper. Res}, $\mathbf{56}$, no. 3, 607-617, (2008). \\
[2] & Moon, K-S. ; von Schwerin, E. Szepessy, A.; and Tempone, R., An adaptive
algorithm for ordinary, stochastic and partial differential equations, \emph{Recent advances in adaptive computation, Contemp. Math}, $\mathbf{383}$, 325-343,
(2005). \\
[3] & Moon, K-S.; Szepessy, A.; Tempone, R. and Zouraris, G. E., Convergence rates
for adaptive weak approximation of stochastic differential equations, \emph{Stoch. Anal. Appl.}, $\mathbf{23}$, no. 3, 511-558, (2005). \\
[4] & Szepessy, A.; Tempone, R. and Zouraris, G. E.: Adaptive weak approximation
of stochastic differential equations, \emph{Comm. Pure Appl. Math}, $\mathbf{54}$, no. 10, 1169-1214, (2001). \bigskip
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