## Bernoulli

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

### $\varepsilon$-Strong simulation of the Brownian path

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 12 November 2012

https://projecteuclid.org/euclid.bj/1352727808

Digital Object Identifier
doi:10.3150/11-BEJ383

Mathematical Reviews number (MathSciNet)
MR2995793

Zentralblatt MATH identifier
1263.65007

#### Citation

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. https://projecteuclid.org/euclid.bj/1352727808

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