  We present several Monte Carlo strategies for simulating discrete-time Markov
chains with continuous multi-dimensional state space; we focus on stratified
techniques. We first analyze the variance of the calculation of the measure of
a domain included in the unit hypercube, when stratified samples are used. We
then show that each step of the simulation of a Markov chain can be reduced to
the numerical integration of the indicator function of a subdomain of the unit
hypercube. Our approach for Markov chains simulates N copies of the chain in
parallel using stratified sampling and the copies are sorted after each step,
according to their successive coordinates. We analyze variance reduction on
examples of pricing of European and Asian options: enhanced efficiency of
stratified strategies is shown.
