  A Monte Carlo method for computing the action of a matrix exponential for a
certain class of matrices on a vector is proposed. The method is based on
generating random paths, which evolve through the indices of the matrix,
governed by a given continuous-time Markov chain. The vector solution is
computed probabilistically by averaging over a suitable multiplicative
functional. This representation extends the existing linear algebra Monte
Carlo-based methods, and was used in practice to develop an efficient algorithm
capable of computing both, a single entry or the full vector solution. Finally,
several relevant benchmarks were executed to assess the performance of the
algorithm. A comparison with the results obtained with a Krylov-based method
shows the remarkable performance of the algorithm for solving large-scale
problems.
