\epsilon-Strong simulation of the Brownian path

TL;DR

This paper introduces an psilon-strong iterative sampling method for simulating Brownian paths with unbiased bounds, enabling improved Monte Carlo estimations in financial applications and new simulation techniques for Brownian distributions.

Contribution

The paper develops an psilon-strong algorithm that provides unbiased, almost surely convergent bounds for Brownian paths, enhancing Monte Carlo simulations and distribution sampling.

Findings

Convergence rate in L1 norm is (\u00a0K^{-1/2})

Algorithm delivers unbiased estimators for path expectations

Application to option pricing demonstrates practical effectiveness

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.