Monte Carlo simulation
Monte Carlo simulation estimates properties of a stochastic model by generating many independent realisations.
Repeated trajectories
Suppose an event of interest occurs in \(M\) of \(n\) simulated trajectories. Its probability can be estimated by
\[\widehat p=\frac{M}{n}.\]Estimating expectations
If the simulation produces outcomes \(Y_1,\ldots,Y_n\), then
\[\widehat{E[Y]}=\frac1n\sum_{r=1}^nY_r.\]Sampling error
Monte Carlo estimates vary because only finitely many simulations are used. Increasing the number of independent runs generally reduces this sampling uncertainty.
Key idea. One stochastic trajectory is one possible outcome; Monte Carlo simulation uses many trajectories to estimate the distribution of possible outcomes.