### 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 language | English (US) |
---|---|

Pages (from-to) | 1-136 |

Number of pages | 136 |

Journal | Memoirs of the American Mathematical Society |

Volume | 256 |

Issue number | 1228 |

DOIs | |

State | Published - Nov 1 2018 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Memoirs of the American Mathematical Society*, vol. 256, no. 1228, pp. 1-136. https://doi.org/10.1090/memo/1228

}

TY - JOUR

T1 - Continuous-time random walks for the numerical solution of stochastic differential equations

AU - Bou-Rabee, Nawaf

AU - Vanden Eijnden, Eric

PY - 2018/11/1

Y1 - 2018/11/1

N2 - 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.

AB - 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.

KW - Discrete maximum principle

KW - Fokker-Planck equation

KW - Geometric ergodicity

KW - Invariant measure

KW - Kolmogorov equation

KW - Markov jump process

KW - Monotone operator

KW - Non-symmetric diffusions

KW - Parabolic partial differential equation

KW - Stochastic differential equations

KW - Stochastic Lyapunov function

KW - Stochastic simulation algorithm

UR - http://www.scopus.com/inward/record.url?scp=85054908130&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054908130&partnerID=8YFLogxK

U2 - 10.1090/memo/1228

DO - 10.1090/memo/1228

M3 - Article

AN - SCOPUS:85054908130

VL - 256

SP - 1

EP - 136

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

SN - 0065-9266

IS - 1228

ER -