Chapter 4

Numerical methods

4.1 Monte Carlo Method

Suppose we are given a random variable X and are interested in the evaluation of where g( · ) is some known function. If we are able to draw n pseudo random numbers x1, … , xn from the distribution of X, then we can think about approximating with the sample mean of the g(xi),

The expression (4.1) is not just symbolic but holds true in the sense of the law of large numbers whenever . Moreover, the central limit theorem guarantees that

where N(m, s2) denotes the distribution of the Gaussian random variable with expected value m and variance s2. In the end, the number we estimate with simulations will have a deviation from the true expected value of order . Given that P(|Z| < 1.96)0.95, Z N(0, 1), one can construct an interval for the ...