Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A multilevel Monte Carlo algorithm for SDEs driven by countably dimensional Wiener process and Poisson random measure

Michał Sobieraj

TL;DR

The work addresses efficient weak approximation of $\mathbb{E}(f(X(T)))$ for SDEs driven by a countably dimensional Wiener process and Poisson random measure. It introduces a truncated dimension randomized Euler scheme and develops a comprehensive MLMC framework with detailed complexity bounds that depend on both grid density and truncation parameters, supported by numerical GPU-accelerated experiments. The main contributions include an information-based complexity model, coupled MLMC analysis for infinite-dimensional noise, adaptive parameter selection, and practical Python/CUDA implementations. Overall, the results extend MLMC to infinite-dimensional noise settings, offering substantial reductions in computational cost for accurate weak approximations in stochastic models relevant to finance and applied sciences.

Abstract

In this paper, we investigate the properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by the infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which is determined by two parameters, i.e grid density $n \in \mathbb{N}_{+}$ and truncation dimension parameter $M \in \mathbb{N}_{+},$ is of the order $n^{-1/2}+δ(M)$ such that $δ(\cdot)$ is positive and decreasing to $0$. We derive complexity model and provide proof for the upper complexity bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both $n$ and $M.$ The complexity is measured in terms of upper bound for mean-squared error and compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation are also reported.

A multilevel Monte Carlo algorithm for SDEs driven by countably dimensional Wiener process and Poisson random measure

TL;DR

The work addresses efficient weak approximation of for SDEs driven by a countably dimensional Wiener process and Poisson random measure. It introduces a truncated dimension randomized Euler scheme and develops a comprehensive MLMC framework with detailed complexity bounds that depend on both grid density and truncation parameters, supported by numerical GPU-accelerated experiments. The main contributions include an information-based complexity model, coupled MLMC analysis for infinite-dimensional noise, adaptive parameter selection, and practical Python/CUDA implementations. Overall, the results extend MLMC to infinite-dimensional noise settings, offering substantial reductions in computational cost for accurate weak approximations in stochastic models relevant to finance and applied sciences.

Abstract

In this paper, we investigate the properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by the infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which is determined by two parameters, i.e grid density and truncation dimension parameter is of the order such that is positive and decreasing to . We derive complexity model and provide proof for the upper complexity bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both and The complexity is measured in terms of upper bound for mean-squared error and compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation are also reported.
Paper Structure (11 sections, 3 theorems, 87 equations, 2 figures)

This paper contains 11 sections, 3 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complexity model
  4. Monte Carlo method
  5. Multilevel Monte Carlo method
  6. Numerical experiments
  7. Multilevel Monte Carlo parameters
  8. Example equation
  9. Details on the implementation
  10. Conclusions
  11. Acknowledgments

Key Result

Theorem 1

There exists a constant $\kappa > 1$, depending only on the parameters of the class ${\mathcal{F}}(C,D,D_L,\Delta,\varrho_1, \varrho_2, \nu)$, such that for every $(a,b,c, \eta)\in{\mathcal{F}}( C,D,D_L,\Delta,\varrho_1, \varrho_2, \nu)$ and $M,n\in\mathbb{N}$ it holds where $\alpha:=\min\{\varrho_1, \varrho_2, 1/2\}.$

Figures (2)

  • Figure 1: Error vs informational cost comparison.
  • Figure 2: Estimated MLMC error and cost comparison with standard Monte Carlo

Theorems & Definitions (7)

  • Theorem 1: SIAM
  • Proposition 1
  • Remark 1
  • Definition 1
  • Theorem 2
  • proof
  • Example : Merton model with Call option payoff