14 - Basic Monte Carlo methods

Summary

Some problems arising from loss modeling may be analytically intractable. Many of these problems, however, can be formulated in a stochastic framework, with a solution that can be estimated empirically. This approach is called Monte Carlo simulation. It involves drawing samples of observations randomly according to the distribution required, in a manner determined by the analytic problem.

To solve the stochastic problem, samples of the specified distribution have to be generated, invariably using computational algorithms. The basic random number generators required in Monte Carlo methods are for generating observations from the uniform distribution. Building upon uniform random number generators, we can generate observations from other distributions by constructing appropriate random number generators, using methods such as inverse transformation and acceptance–rejection. We survey specific random number generators for some commonly used distributions, some of which are substantially more efficient than standard methods. An alternative method of generating numbers resembling a uniformly distributed sample of observations is the quasi-random number generator or the low-discrepancy sequence.

The accuracy of the Monte Carlo estimates depends on the variance of the estimator. To speed up the convergence of the Monte Carlo estimator to the deterministic solution, we consider designs of Monte Carlo sampling schemes and estimation methods that will produce smaller variances. Methods involving the use of antithetic variable, control variable, and importance sampling are discussed.