# ε-Strong simulation of the Brownian path

@article{Beskos2011StrongSO,
title={$\epsilon$-Strong simulation of the Brownian path},
author={Alexandros Beskos and Stefano Peluchetti and Gareth O. Roberts},
journal={Bernoulli},
year={2011},
volume={18},
pages={1223-1248}
}
• Published 1 October 2011
• Computer Science, Mathematics
• Bernoulli
We present an iterative sampling method which delivers upper and lower bounding processes for the Brownian path. We develop such processes with particular emphasis on being able to unbiasedly simulate them on a personal computer. The dominating processes converge almost surely in the supremum and L 1 norms. In particular, the rate of converge in L 1 is of the order O(K −1/2 ) , K denoting the computing cost. The a.s. enfolding of the Brownian path can be exploited in Monte Carlo applications…

## Figures and Tables from this paper

Steady-state simulation of reflected Brownian motion and related stochastic networks
• Mathematics
• 2015
This paper develops the first class of algorithms that enable unbiased estimation of steady-state expectations for multidimensional reflected Brownian motion. In order to explain our ideas, we first
Exact simulation of multidimensional reflected Brownian motion
• Computer Science
Journal of Applied Probability
• 2018
This work applies recently developed so-called ε-strong simulation techniques to provide a piecewise linear approximation to RBM with ε (deterministic) error in uniform norm, and condition on a suitably designed information structure so that a feasible proposal distribution can be applied.
ε-STRONG SIMULATION FOR MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS VIA ROUGH PATH ANALYSIS By
• Mathematics
• 2014
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
On the Euler discretization error of Brownian motion about random times
• Mathematics
• 2017
In this paper we derive weak limits for the discretization errors of sampling barrier-hitting and extreme events of Brownian motion by using the Euler discretization simulation method. Specifically,
On the exact and ε-strong simulation of (jump) diffusions
• Mathematics, Computer Science
• 2015
This paper introduces a framework for simulating finite dimensional representations of (jump) diffusion sample paths over finite intervals, without discretisation error (exactly), in such a way that
ɛ-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations
• Mathematics
Math. Oper. Res.
• 2021
These algorithms enjoy the tolerance-enforcement feature, under which the error bounds can be updated sequentially in an efficient way and can be readily combined with other advanced simulation techniques to estimate the expectations of functionals of fBMs efficiently.
Simulating bridges using confluent diffusions.
• Mathematics, Computer Science
• 2019
A Markov chain Monte Carlo algorithm for simulating diffusion bridges which is both exact (in the sense that there is no discretisation error) and has computational cost that is linear in the duration of the bridge.
Unbiased Simulation of Rare Events in Continuous Time
• Mathematics, Computer Science
Methodology and Computing in Applied Probability
• 2021
This work specifies how such algorithms can be combined with the classical multilevel splitting method for rare event simulation, and provides unbiased estimations of the probability in question.
Perfect Simulation, Sample-path Large Deviations, and Multiscale Modeling for Some Fundamental Queueing Systems
This thesis introduces some techniques of asymptotic analysis that are relatively new to queueing applications in order to give more accurate probabilistic characterization of queueing models with large scale and complicated structure, and gives the first functional large deviation result for infinite-server system with general inter-arrival and service times.
Studies in Stochastic Networks: Efficient Monte-Carlo Methods, Modeling and Asymptotic Analysis
This dissertation performs detailed running time analysis under heavy traffic of the perfect sampling algorithms for infinite server queues and multi-server loss queues and proves that the algorithms achieve nearly optimal order of complexity.

## References

SHOWING 1-10 OF 26 REFERENCES
Continuous martingales and Brownian motion
• Mathematics
• 1990
0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.-
Boundary crossing probability for Brownian motion
• Mathematics
Journal of Applied Probability
• 2001
Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary
A Factorisation of Diffusion Measure and Finite Sample Path Constructions
• Mathematics
• 2008
In this paper we introduce decompositions of diffusion measure which are used to construct an algorithm for the exact simulation of diffusion sample paths and of diffusion hitting times of smooth
Brownian Motion and Stochastic Calculus
This chapter discusses Brownian motion, which is concerned with continuous, Square-Integrable Martingales, and the Stochastic Integration, which deals with the integration of continuous, local martingales into Markov processes.
Retrospective exact simulation of diffusion sample paths with applications
• Computer Science, Mathematics
• 2006
An algorithm for exact simulation of a class of Ito's diffusions and a method that exploits the properties of the algorithm to carry out inference on discretely observed diffusions without resorting to any kind of approximation apart from the Monte Carlo error is described.
Exact retrospective Monte Carlo computation of arithmetic average Asian options
• Computer Science
Monte Carlo Methods Appl.
• 2007
An exact simulation based technique for pricing continuous arithmetic average Asian options in the Black & Scholes framework is applied, no longer prone to the discretization bias resulting from the approximation of continuous time processes through discrete sampling.
ON THE CONVERGENCE OF DIFFUSION PROCESSES CONDITIONED TO REMAIN IN A BOUNDED REGION FOR LARGE TIME TO LIMITING POSITIVE RECURRENT DIFFUSION PROCESSES
obtain a new process yT(S), 0 S) (which are always satisfied if a-'b is a gradient function), we show that yT(S) is an inhomogeneous diffusion process and that as T -. oo, yT(S), 0 < s < t converges
Exact Monte Carlo simulation of killed diffusions
• Physics