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Mar 26

High Dimensional Sobol Sequences

Posted by abiao at 13:54 | Others | Comments(0) | Reads(10715)
Quasi-Monte Carlo methods are very efficient in solving low dimensional integration problems, for example, pricing a path-dependent exotic option, among those low discrepancy sequences, Sobol is undoubtedly one of the best and most widely used due to its high convergence speed, how to generate Sobol Sequences was shared at posts Sobol sequence generator, Primitive polynomials for Sobol sequences, halton and sobol sequences and Sobol and Generalised Faure sequences.

However, empirical research have shown Quasi-Monte Carlo Sobol sequences perform poorly for high dimensional integration problems, it can never be more efficient than the ordinary Monte Carlo simulation, some researchers suggest to use Sobol sequences with Brownian bridge for a better result. A recent working paper shared at Articles uses a small trick - randomized Sobol sequences to avoid the poor performance high dimensional problem, indeed it is straightforward to add this trick into your codes.  Below is a comparison graph demonstrating the performance, very impressive.

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(Source from A note on multidimensional sobol sequences)

read the paper for detail if interested, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1558186

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