Stochastic Simulation

Monte Carlo simulation is useful in a wide variety of situations. For example, for option pricing, risk analysis, modelling queueing systems etc. Also stochastic simulation plays a profound role in Bayesian statistics, where the goal is to sample from a posterior distribution.

Emphasis in this course is on understanding (the workings of) simulation techniques and how simulation can be used to provide insight into stochastic problems; the most important aspect of the "doing simulations" part is that you can derive the simulation algorithm that results when one of the covered methods is applied to a problem. Since the key in efficient simulation almost always lies with the stochastic specifics of the problem, we focus on the stochastic methods that play a role in this process. Some simulations will be done in the mini-projects, to supplement and illustrate the theory. Here, you are encouraged to "bring your own problem to work on".

Topics:

• introduction,

• general aspects of stochastic simulation;

• generating random objects, univariate and multivariate random variables, and stochastic processes;

• analysis of simulation output: how to obtain estimates for quantities of interest, as well as confidence intervals; bias and small sample issues;

• theory of general state space Markov chains (stationary distribution, ergodicity, asymptotic variance)

• variance reduction, especially their stochastic background and optimization;

• rare event simulation;

• Markov Chain Monte Carlo.

• depending on time, a selection of advanced topics.