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}
}
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… 

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References

SHOWING 1-10 OF 26 REFERENCES
Continuous martingales and Brownian motion
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
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
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
TLDR
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
TLDR
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
TLDR
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
We describe and implement a novel methodology for Monte Carlo simulation of one-dimensional killed diffusions. The proposed estimators represent an unbiased and efficient alternative to current Monte
Exact simulation of diffusions
We describe a new, surprisingly simple algorithm, that simulates exact sample paths of a class of stochastic differential equations. It involves rejection sampling and, when applicable, returns the
One-Shot CFTP; Application to a Class of Truncated Gaussian Densities
TLDR
A new method for perfect simulation of multivariate densities by simulating efficiently from high-dimensional truncated normal distributions using the Gibbs sampler is introduced.
...
1
2
3
...