## Quasi-Monte Carlo Simulation

Quasi-Monte Carlo simulation is a Monte Carlo simulation that uses quasi-random sequences instead of pseudo random numbers. The quasi-random sequences, also called low-discrepancy sequences, are deterministic, uniformly distributed sequences that are specifically designed to place sample points as uniformly as possible. In many cases, this distributed sequences improves the performance of Monte Carlo simulations with faster computational times and sometimes higher accuracy.

The standard Monte Carlo simulation using pseudo random numbers has a convergence rate of only O(N-1/2), while the quasi-Monte Carlo rate of convergence can be much faster with an error of O(N-1) in the best cases. For example, for a standard Monte Carlo simulation, it is necessary to increase 100 times the number of simulations NTrials to reduce the error by a factor of 10, whereas a quasi-Monte Carlo simulation requires less, or much less, than 100 times to achieve the same goal.

Quasi-Monte Carlo simulation produces a purely deterministic result. Therefore, when computing the variance and constructing a confidence band for the estimates, randomized quasi-Monte Carlo simulation is useful because of faster computational times and sometimes higher accuracy. You can also use randomized quasi-Monte Carlo to introduce randomization into the low-discrepancy sequences.

The starting point for a Monte Carlo simulation is the construction of a Brownian motion sample path (or Wiener path). Such paths are built from a set of independent Gaussian variates, using either standard discretization, Brownian-bridge construction, or principal components construction.

To perform Quasi-Monte Carlo simulation, the name-value arguments for `MonteCarloMethod`, `QuasiSequence`, and `BrownianMotionMethod` by using the supported methods for the following SDE objects. For example, ```[paths,time,z] = simByEuler(cir_obj,10,'ntrials',4096,'method','basic','montecarlomethod','quasi','quasisequence','sobol','brownianmotionmethod','principal-components')``` performs a Quasi-Monte Carlo simulation using a principal components construction method with a `cir` model.