## Abstract

Consider a multidimensional diffusion process X = {X(t) : t ∈ [0, 1]}. Let ε > 0 be a deterministic, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of X, we construct a probability space, supporting both X and an explicit, piecewise constant, fully simulatable process X_{ε} such that sup ∥X_{ε}(t) - X(t)∥_{∞} < ε 0≤t≤1 with probability one. Moreover, the user can adaptively choose ε′ ∈ (0, ε) so that X_{ε′} (also piecewise constant and fully simulatable) can be constructed conditional on X_{ε} to ensure an error smaller than ε′ with probability one. Our construction requires a detailed study of continuity estimates of the Itô map using Lyons' theory of rough paths. We approximate the underlying Brownian motion, jointly with the Lévy areas with a deterministic ε error in the underlying rough path metric.

Original language | English (US) |
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Pages (from-to) | 275-336 |

Number of pages | 62 |

Journal | Annals of Applied Probability |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2017 |

## Keywords

- Brownian motion
- Lévy area
- Monte Carlo method
- Rough path
- Stochastic differential equation

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty