15 research outputs found
Importance Sampling for Multiscale Diffusions
We construct importance sampling schemes for stochastic differential
equations with small noise and fast oscillating coefficients. Standard Monte
Carlo methods perform poorly for these problems in the small noise limit. With
multiscale processes there are additional complications, and indeed the
straightforward adaptation of methods for standard small noise diffusions will
not produce efficient schemes. Using the subsolution approach we construct
schemes and identify conditions under which the schemes will be asymptotically
optimal. Examples and simulation results are provided
Black-Scholes formulae for Asian options in local volatility models
We develop approximate formulae expressed in terms of elementary functions for the density, the price and the Greeks of path dependent options of Asian style, in a general local volatility model. An algorithm for computing higher order approximations is provided. The proof is based on a heat kernel expansion method in the framework of hypoelliptic, not uniformly parabolic, partial differential equations.Asian Options, Degenerate Diffusion Processes, Transition Density Functions, Analytic Approximations, Option Pricing
Importance sampling for McKean-Vlasov SDEs
This paper deals with the Monte-Carlo methods for evaluating expectations of
functionals of solutions to McKean-Vlasov Stochastic Differential Equations
(MV-SDE) with drifts of super-linear growth. We assume that the MV-SDE is
approximated in the standard manner by means of an interacting particle system
and propose two importance sampling (IS) techniques to reduce the variance of
the resulting Monte Carlo estimator. In the \emph{complete measure change}
approach, the IS measure change is applied simultaneously in the coefficients
and in the expectation to be evaluated. In the \emph{decoupling} approach we
first estimate the law of the solution in a first set of simulations without
measure change and then perform a second set of simulations under the
importance sampling measure using the approximate solution law computed in the
first step.
For both approaches, we use large deviations techniques to identify an
optimisation problem for the candidate measure change. The decoupling approach
yields a far simpler optimisation problem than the complete measure change,
however, we can reduce the complexity of the complete measure change through
some symmetry arguments. We implement both algorithms for two examples coming
from the Kuramoto model from statistical physics and show that the variance of
the importance sampling schemes is up to 3 orders of magnitude smaller than
that of the standard Monte Carlo. The computational cost is approximately the
same as for standard Monte Carlo for the complete measure change and only
increases by a factor of 2--3 for the decoupled approach. We also estimate the
propagation of chaos error and find that this is dominated by the statistical
error by one order of magnitude.Comment: 29 pages, 2 Table
Efficient large deviation estimation based on importance sampling
We present a complete framework for determining the asymptotic (or
logarithmic) efficiency of estimators of large deviation probabilities and rate
functions based on importance sampling. The framework relies on the idea that
importance sampling in that context is fully characterized by the joint large
deviations of two random variables: the observable defining the large deviation
probability of interest and the likelihood factor (or Radon-Nikodym derivative)
connecting the original process and the modified process used in importance
sampling. We recover with this framework known results about the asymptotic
efficiency of the exponential tilting and obtain new necessary and sufficient
conditions for a general change of process to be asymptotically efficient. This
allows us to construct new examples of efficient estimators for sample means of
random variables that do not have the exponential tilting form. Other examples
involving Markov chains and diffusions are presented to illustrate our results.Comment: v1: 34 pages, 8 figures; v2: Typos corrected; v3: More mathematical
version containing technical modifications in Assumption 2, Assumption 3, and
Eq. (53) needed in some of the proof