1,025 research outputs found
Adaptive Stratified Sampling for Monte-Carlo integration of Differentiable functions
We consider the problem of adaptive stratified sampling for Monte Carlo
integration of a differentiable function given a finite number of evaluations
to the function. We construct a sampling scheme that samples more often in
regions where the function oscillates more, while allocating the samples such
that they are well spread on the domain (this notion shares similitude with low
discrepancy). We prove that the estimate returned by the algorithm is almost
similarly accurate as the estimate that an optimal oracle strategy (that would
know the variations of the function everywhere) would return, and provide a
finite-sample analysis.Comment: 23 pages, 3 figures, to appear in NIPS 2012 conference proceeding
Convenient Multiple Directions of Stratification
This paper investigates the use of multiple directions of stratification as a
variance reduction technique for Monte Carlo simulations of path-dependent
options driven by Gaussian vectors. The precision of the method depends on the
choice of the directions of stratification and the allocation rule within each
strata. Several choices have been proposed but, even if they provide variance
reduction, their implementation is computationally intensive and not applicable
to realistic payoffs, in particular not to Asian options with barrier.
Moreover, all these previously published methods employ orthogonal directions
for multiple stratification. In this work we investigate the use of algorithms
producing convenient directions, generally non-orthogonal, combining a lower
computational cost with a comparable variance reduction. In addition, we study
the accuracy of optimal allocation in terms of variance reduction compared to
the Latin Hypercube Sampling. We consider the directions obtained by the Linear
Transformation and the Principal Component Analysis. We introduce a new
procedure based on the Linear Approximation of the explained variance of the
payoff using the law of total variance. In addition, we exhibit a novel
algorithm that permits to correctly generate normal vectors stratified along
non-orthogonal directions. Finally, we illustrate the efficiency of these
algorithms in the computation of the price of different path-dependent options
with and without barriers in the Black-Scholes and in the Cox-Ingersoll-Ross
markets.Comment: 21 pages, 11 table
Toward Optimal Stratification for Stratified Monte-Carlo Integration
We consider the problem of adaptive stratified sampling for Monte Carlo
integration of a noisy function, given a finite budget n of noisy evaluations
to the function. We tackle in this paper the problem of adapting to the
function at the same time the number of samples into each stratum and the
partition itself. More precisely, it is interesting to refine the partition of
the domain in area where the noise to the function, or where the variations of
the function, are very heterogeneous. On the other hand, having a (too) refined
stratification is not optimal. Indeed, the more refined the stratification, the
more difficult it is to adjust the allocation of the samples to the
stratification, i.e. sample more points where the noise or variations of the
function are larger. We provide in this paper an algorithm that selects online,
among a large class of partitions, the partition that provides the optimal
trade-off, and allocates the samples almost optimally on this partition
Functional quantization-based stratified sampling methods
In this article, we propose several quantization-based stratified sampling methods to reduce the variance of a Monte Carlo simulation. Theoretical aspects of stratification lead to a strong link between optimal quadratic quantization and the variance reduction that can be achieved with stratified sampling. We first put the emphasis on the consistency of quantization for partitioning the state space in stratified sampling methods in both finite and infinite dimensional cases. We show that the proposed quantization-based strata design has uniform efficiency among the class of Lipschitz continuous functionals. Then a stratified sampling algorithm based on product functional quantization is proposed for path-dependent functionals of multi-factor diffusions. The method is also available for other Gaussian processes such as Brownian bridge or Ornstein-Uhlenbeck processes. We derive in detail the case of Ornstein-Uhlenbeck processes. We also study the balance between the algorithmic complexity of the simulation and the variance reduction facto
- …