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I am studying Monte-Carlo simulations using quasi random numbers and encounter the following problem: I am given a set of 1D quasi-random numbers $(X_i)$ over $[0,1)$, and would like to generate another set of 1D quasi-random numbers $(Y_i)$ over $[0,1)$, that jointly form a set $(X_i,Y_i)$ of quasi-random numbers over $[0,1)^2$. In other words, I would like Y to be "independent" of X. This is trivial with pseudo-random numbers, but for quasi numbers this is usually not the case. In fact, Y may be exactly the same as X if one generates Y using the same algorithm as X.

More generally, how do I sequentially generate sets of independent quasi-numbers? This is necessary for simulation a Brownian motion, but so far I have not seen how this can be done. Even if one could generate 2 set of independent quasi numbers, I don't see how one could generate indefinitely many sets. Any ideas?

Thank you for your inputs.

user138668
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quasi-randoms have to be created with a dimension in mind. Thus you need to develop a vector of pairs of quasi-randoms. Quasi random generators generally come with an ability to specify the dimensionality.

For developing independent sets, the usual solutions are randomized quasi Monte Carlo and scrambled Quasi Monte Carlo. Here you add pseudo-random uniforms to get independence. One uniform per dimension set.

Mark Joshi
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  • Dear Mark, thank you for your fast response. Are you saying if I randomize the SAME set of quasi numbers by adding independent sets of pseudo-numbers then modulo 1, I get effectively independent sets of quasi-numbers? In this case, do I still get the low discrepancy of the quasi numbers and independence? Sounds too good to be true or not? – user138668 Apr 04 '16 at 00:51
  • Dear Mark, I just experimented. Randomized quasi-numbers does not appear to be of low-discrepancy, they look just like pseudo-numbers! How do I make use of the major advantage of quasi numbers if I would like to simulate an option that does not have a solvable underlying SDE? Thank you for your expertise. – user138668 Apr 04 '16 at 01:07
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    you draw say a 1000 of them and add the same uniform to all of them. Then keep the 1000 but use a different uniform. This is a trick to get statistical estimates. It is not useful for increasing dimensionality. There is a whole chapter on this stuff in book More Mathematical Finance. – Mark Joshi Apr 04 '16 at 01:11