Continuous-time random walks for the numerical solution of stochastic differential equations

Nawaf Bou-Rabee, Eric Vanden Eijnden

Research output: Contribution to journalArticle

Abstract

This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the jump size of the approximation can be bounded uniformly in space, which often guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs. By directly analyzing the infinitesimal generator of the approximation, we prove that the approximation has a sharp stochastic Lyapunov function when applied to an SDE with a drift field that is locally Lipschitz continuous and weakly dissipative. We use this stochastic Lyapunov function to extend a local semimartingale representation of the approximation. This extension makes it possible to quantify the computational cost of the approximation. Using a stochastic representation of the global error, we show that the approximation is (weakly) accurate in representing finite and infinite-time expected values, with an order of accuracy identical to the order of accuracy of the infinitesimal generator of the approximation. The proofs are carried out in the context of both fixed and variable spatial step sizes. Theoretical and numerical studies confirm these statements, and provide evidence that these schemes have several advantages over standard methods based on time-discretization. In particular, they are accurate, eliminate nonphysical moves in simulating SDEs with boundaries (or confined domains), prevent exploding trajectories from occurring when simulating stiff SDEs, and solve first exit problems without time-interpolation errors.

Original languageEnglish (US)
Pages (from-to)1-136
Number of pages136
JournalMemoirs of the American Mathematical Society
Volume256
Issue number1228
DOIs
StatePublished - Nov 1 2018

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Continuous Time Random Walk
Stochastic Equations
Differential equations
Numerical Solution
Differential equation
Approximation
Lyapunov functions
Infinitesimal Generator
Lyapunov Function
Biophysics
Epidemiology
Population dynamics
Exit Problem
Stochastic Representation
Markov Jump Processes
Mathematical Finance
Interpolation Error
Finance
Kolmogorov Equation
Chemical Kinetics

Keywords

  • Discrete maximum principle
  • Fokker-Planck equation
  • Geometric ergodicity
  • Invariant measure
  • Kolmogorov equation
  • Markov jump process
  • Monotone operator
  • Non-symmetric diffusions
  • Parabolic partial differential equation
  • Stochastic differential equations
  • Stochastic Lyapunov function
  • Stochastic simulation algorithm

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Continuous-time random walks for the numerical solution of stochastic differential equations. / Bou-Rabee, Nawaf; Vanden Eijnden, Eric.

In: Memoirs of the American Mathematical Society, Vol. 256, No. 1228, 01.11.2018, p. 1-136.

Research output: Contribution to journalArticle

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