### Abstract

We provide a mathematical study of the modified diffusion Monte Carlo (DMC) algorithm introduced in the companion article [3]. DMC is a simulation technique that uses branching particle systems to represent expectations associated with Feynman-Kac formulae. We provide a detailed heuristic explanation of why, in cases in which a stochastic integral appears in the Feynman-Kac formula (e.g., in rare event simulation, continuous time filtering, and other settings), the new algorithm is expected to converge in a suitable sense to a limiting process as the time interval between branching steps goes to 0. The situation studied here stands in stark contrast to the "naï ve" generalization of the DMC algorithm, which would lead to an exponential explosion of the number of particles, thus precluding the existence of any finite limiting object. Convergence is shown rigorously in the simplest possible situation of a random walk, biased by a linear potential. The resulting limiting object, which we call the "Brownian fan," is a very natural new mathematical object of independent interest.

Original language | English (US) |
---|---|

Pages (from-to) | 1-60 |

Number of pages | 60 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 68 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2015 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*68*(1), 1-60. https://doi.org/10.1002/cpa.21544

**The brownian fan.** / Hairer, Martin; Weare, Jonathan.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 68, no. 1, pp. 1-60. https://doi.org/10.1002/cpa.21544

}

TY - JOUR

T1 - The brownian fan

AU - Hairer, Martin

AU - Weare, Jonathan

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We provide a mathematical study of the modified diffusion Monte Carlo (DMC) algorithm introduced in the companion article [3]. DMC is a simulation technique that uses branching particle systems to represent expectations associated with Feynman-Kac formulae. We provide a detailed heuristic explanation of why, in cases in which a stochastic integral appears in the Feynman-Kac formula (e.g., in rare event simulation, continuous time filtering, and other settings), the new algorithm is expected to converge in a suitable sense to a limiting process as the time interval between branching steps goes to 0. The situation studied here stands in stark contrast to the "naï ve" generalization of the DMC algorithm, which would lead to an exponential explosion of the number of particles, thus precluding the existence of any finite limiting object. Convergence is shown rigorously in the simplest possible situation of a random walk, biased by a linear potential. The resulting limiting object, which we call the "Brownian fan," is a very natural new mathematical object of independent interest.

AB - We provide a mathematical study of the modified diffusion Monte Carlo (DMC) algorithm introduced in the companion article [3]. DMC is a simulation technique that uses branching particle systems to represent expectations associated with Feynman-Kac formulae. We provide a detailed heuristic explanation of why, in cases in which a stochastic integral appears in the Feynman-Kac formula (e.g., in rare event simulation, continuous time filtering, and other settings), the new algorithm is expected to converge in a suitable sense to a limiting process as the time interval between branching steps goes to 0. The situation studied here stands in stark contrast to the "naï ve" generalization of the DMC algorithm, which would lead to an exponential explosion of the number of particles, thus precluding the existence of any finite limiting object. Convergence is shown rigorously in the simplest possible situation of a random walk, biased by a linear potential. The resulting limiting object, which we call the "Brownian fan," is a very natural new mathematical object of independent interest.

UR - http://www.scopus.com/inward/record.url?scp=84910670664&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84910670664&partnerID=8YFLogxK

U2 - 10.1002/cpa.21544

DO - 10.1002/cpa.21544

M3 - Article

VL - 68

SP - 1

EP - 60

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 1

ER -