13.7 Random Number Generation

To model a discrete-time Stochastic Process using the Monte Carlo Method we need to use some random variables. Numbers generated by computers are not truly random. Standard random number generators produce uniformly distributed pseudo-random numbers that are made to have the statistical properties of truly random numbers.

Non-uniform variables are sampled through the transformation of uniform variables. There are several methods developed for this purpose, both general (to produce random variables of a given density) and special (to directly generate normal random variables): transformation, acceptance-rejection, Box-Muller, and so on. We use the so-called “polar algorithm” technique.

You have a choice to use Low Discrepancy Sequences (LDS), also called quasi-random numbers, as a substitute to pseudo-random numbers. One more important feature of LDS is to reach the consistency across all computers in generating rate paths (pseudo-random numbers generated by different machines are different).