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Double-Loop Importance Sampling for McKean--Vlasov Stochastic Differential Equation

Nadhir Ben Rached, Abdul-Lateef Haji-Ali, Shyam Mohan Subbiah Pillai, Raúl Tempone

Abstract

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with solutions to the $d$-dimensional McKean-Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting $P$-particle system, which is a set of $P$ coupled $d$-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a $P \times d$-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in [dos Reis et al., 2023], generating a $d$-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ with a significantly reduced constant to achieve a prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given $\mathrm{TOL}_{\mathrm{r}}$ compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.

Double-Loop Importance Sampling for McKean--Vlasov Stochastic Differential Equation

Abstract

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with solutions to the -dimensional McKean-Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting -particle system, which is a set of coupled -dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a -dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in [dos Reis et al., 2023], generating a -dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of with a significantly reduced constant to achieve a prescribed relative error tolerance . Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.
Paper Structure (20 sections, 5 theorems, 76 equations, 5 figures, 1 table, 5 algorithms)

This paper contains 20 sections, 5 theorems, 76 equations, 5 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The McKean--Vlasov stochastic differential equation
  4. Example: Fully Connected Kuramoto Model for Synchronized Oscillators
  5. IS using Stochastic Optimal Control for SDEs
  6. Problem Setting
  7. DLMC with IS
  8. Decoupling Approach for MV-SDEs
  9. Optimal IS for Decoupled MV-SDE using Stochastic Optimal Control
  10. Error Analysis
  11. Work Analysis
  12. Adaptive DLMC Algorithm
  13. Numerical Results
  14. Objective function $G(x) = \cos{x}$
  15. Objective function $G(x) = \mathbbm{1}_{\{x>K\}}$
  16. ...and 5 more sections

Key Result

Lemma 1

Let process $Y$ solve the SDE eqn:sde_defn, and controlled process $Y_\zeta$ solve the SDE eqn:sde_sde_is, where $\zeta:[0,T] \cross \mathbb{R}^d \rightarrow \mathbb{R}^d$ is the control. Assume the value function $u:[0,T] \cross \mathbb{R}^d \rightarrow \mathbb{R}^d$, defined in sde_value_fxn, has where

Figures (5)

  • Figure 1: Verifying Assumptions \ref{['ass:decoupling_err']} and \ref{['ass:time_err']} for Kuramoto model \ref{['eqn:kuramoto_model']} for $G(x)=\cos{x}$.
  • Figure 2: Verifying Assumptions \ref{['ass:var_1']} and \ref{['ass:var_2']} for Kuramoto model \ref{['eqn:kuramoto_model']}.
  • Figure 3: Adaptive DLMC Algorithm \ref{['alg:adlmc']} applied to Kuramoto example \ref{['eqn:kuramoto_model']} for $G(x) = \cos{x}$.
  • Figure 4: Variance reduction of the DLMC estimator using IS on Kuramoto example \ref{['eqn:kuramoto_model']} for $G(x) = \mathbbm{1}_{\{x>K\}}$.
  • Figure 5: Algorithm \ref{['alg:adlmc']} applied to Kuramoto example \ref{['eqn:kuramoto_model']} for $G(x) = \mathbbm{1}_{\{x>K\}}$.

Theorems & Definitions (14)

  • Lemma 1: Dynamic Programming for Standard SDEs
  • proof
  • Theorem 1: Optimal Control to Minimize Variance for Standard SDEs
  • proof
  • Corollary 1: HJB PDE
  • Remark 1: Existence and uniqueness of HJB solutions
  • Remark 2: Equivalent HJB Formulation
  • Remark 3: Using KBE for IS
  • Remark 4: Time Extension of the Empirical Law
  • Corollary 2: HJB PDE for decoupled MV-SDE
  • ...and 4 more