When studying random dynamical systems described by stochastic dif- ferential equations, Lyapunov exponents are powerful tools to measure qual- itative behavior. Moments Lyapunov exponents capture finer information, such as fluctuations of finite time Lyapunov exponents, and yield large de- viations estimates. These moments Lyapunov exponents can be obtained as principal eigenvalues of elliptic PDEs associated to the original SDEs. I will describe a computer-assisted approach that can be used to obtain rig- orous enclosures of such eigenvalues. This is joint work with Alex Blumen- thal (Georgia Tech), Maximilan Engel (U. of Amsterdam) and Alexandra Blessing-Neamtu (U. of Konstanz).