  For the 2D matrix Langevin dynamics that corresponds to the continuous-time
limit of the product of some $2 \times 2$ random matrices, the finite-time
Lyapunov exponent can be written as an additive functional of the associated
Riccati process submitted to some Langevin dynamics on the infinite periodic
ring. Its large deviations properties can be thus analyzed from two points of
view that are equivalent in the end by consistency but give different
perspectives. In the first approach, one starts from the large deviations at
level 2.5 for the joint probability of the empirical density and of the
empirical current of the Riccati process and one performs the appropriate
Euler-Lagrange optimization in order to compute the cumulant generating
function of the Lyapunov exponent. In the second approach, this cumulant
generating function is obtained from the spectral analysis of the appropriate
tilted Fokker-Planck operator. The associated conditioned process obtained via
the generalization of Doob's h-transform allows to clarify the equivalence with
the first approach. Finally, applications to one-dimensional Anderson
Localization models are described in order to obtain explicitly the first
cumulants of the finite-size Lyapunov exponent.
