42 research outputs found
On the Acceleration of the Multi-Level Monte Carlo Method
The multi-level Monte Carlo method proposed by M. Giles (2008) approximates
the expectation of some functionals applied to a stochastic process with
optimal order of convergence for the mean-square error. In this paper, a
modified multi-level Monte Carlo estimator is proposed with significantly
reduced computational costs. As the main result, it is proved that the modified
estimator reduces the computational costs asymptotically by a factor
if weak approximation methods of orders and are
applied in case of computational costs growing with same order as variances
decay
Semi-Lagrangian schemes for linear and fully non-linear Hamilton-Jacobi-Bellman equations
We consider the numerical solution of Hamilton-Jacobi-Bellman equations
arising in stochastic control theory. We introduce a class of monotone
approximation schemes relying on monotone interpolation. These schemes converge
under very weak assumptions, including the case of arbitrary degenerate
diffusions. Besides providing a unifying framework that includes several known
first order accurate schemes, stability and convergence results are given,
along with two different robust error estimates. Finally, the method is applied
to a super-replication problem from finance.Comment: to appear in the proceedings of HYP201
General order conditions for stochastic partitioned Runge-Kutta methods
In this paper stochastic partitioned Runge-Kutta (SPRK) methods are
considered. A general order theory for SPRK methods based on stochastic
B-series and multicolored, multishaped rooted trees is developed. The theory is
applied to prove the order of some known methods, and it is shown how the
number of order conditions can be reduced in some special cases, especially
that the conditions for preserving quadratic invariants can be used as
simplifying assumptions
Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation
The multilevel Monte Carlo path simulation method introduced by Giles ({\it
Operations Research}, 56(3):607-617, 2008) exploits strong convergence
properties to improve the computational complexity by combining simulations
with different levels of resolution. In this paper we analyse its efficiency
when using the Milstein discretisation; this has an improved order of strong
convergence compared to the standard Euler-Maruyama method, and it is proved
that this leads to an improved order of convergence of the variance of the
multilevel estimator. Numerical results are also given for basket options to
illustrate the relevance of the analysis.Comment: 33 pages, 4 figures, to appear in Discrete and Continuous Dynamical
Systems - Series