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

Abstract

We build a sequence of empirical measures on the space $\mathbb{D}(\mathbb{R}_{+},\mathbb{R}^{d})$ of $ℝ^d$-valued cadlag 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 on the convergence of this sequence. We then apply them to Brownian diffusions and solutions to Lévy-driven SDE’s under some Lyapunov-type stability assumptions. As a numerical application of this work, we show that this procedure provides an efficient means of option pricing in stochastic volatility models.