Approximation of the distribution of a stationary Markov process with application to option pricing

We build a sequence of empirical measures on the space D(R_+,R^d) of R^d-valued c\`adl\`ag functions on R_+ in order to approximate the law of a stationary R^d-valued Markov and Feller process (X_t). We obtain some general results of convergence of this sequence. Then, we apply them to Brownian diffusions and solutions to L\'evy driven SDE's under some Lyapunov-type stability assumptions. As a numerical application of this work, we show that this procedure gives an efficient way of option pricing in stochastic volatility models.