\epsilonStrong 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2011
 arXiv:
 arXiv:1110.0110
 Bibcode:
 2011arXiv1110.0110B
 Keywords:

 Statistics  Computation;
 Mathematics  Statistics Theory
 EPrint:
 Published in at http://dx.doi.org/10.3150/11BEJ383 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)