Abstract: In this talk, I will introduce the concept of stochastic dynamical systems and relate it with the theory of stationary measures, linear cocycles and Lyapunov exponents. The main goal is to present a result on the H ̈older joint continuity of the top Lyapunov exponent with respect to both the Markov cocycle and the transition kernel. The approach is via the regularity of the stationary measures and Furstenberg’s Formula. This method provides a computable H ̈older exponent. [Based on joint work with Ao Cai, Silvius Klein and Aline Melo.]