• Bernoulli
  • Volume 18, Number 4 (2012), 1223-1248.

$\varepsilon$-Strong simulation of the Brownian path

Alexandros Beskos, Stefano Peluchetti, and Gareth Roberts

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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 $\mathcal{O}(\mathcal{K}^{-1/2})$, $\mathcal{K}$ denoting the computing cost. The a.s. enfolding of the Brownian path can be exploited in Monte Carlo applications involving Brownian paths whence our algorithm (termed the $\varepsilon$-strong algorithm) can deliver unbiased Monte Carlo estimators over path expectations, overcoming discretisation errors characterising standard approaches. We will show analytical results from applications of the $\varepsilon$-strong algorithm for estimating expectations arising in option pricing. We will also illustrate that individual steps of the algorithm can be of separate interest, giving new simulation methods for interesting Brownian distributions.

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Bernoulli, Volume 18, Number 4 (2012), 1223-1248.

First available in Project Euclid: 12 November 2012

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Brownian bridge intersection layer iterative algorithm option pricing pathwise convergence unbiased sampling


Beskos, Alexandros; Peluchetti, Stefano; Roberts, Gareth. $\varepsilon$-Strong simulation of the Brownian path. Bernoulli 18 (2012), no. 4, 1223--1248. doi:10.3150/11-BEJ383.

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