Functional quantization-based stratified sampling methods
Abstract
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 factor
Keywords
path-dependent option
functional quantization
vector quantization
stratification
variance reduction
Monte Carlo
Karhunen-Loève
Gaussian process
Brownian motion
Brownian bridge
Ornstein-Uhlenbeck process
Ornstein-Uhlenbeck bridge
principal component analysis
numerical integration
option pricing
Voronoi diagram
product quantizer
path-dependent option.
Domains
Probability [math.PR]
Origin : Files produced by the author(s)