I would like to ask for help/advice with following problem. I am solving the Hugget type of model (stationary distribution/transition) using the transition matrix approach (one AR(1) and one IID shock). In the first version, I solved for stationary distribution just by brute-force (rising the matrix to power 3000). Unfortunately, this approach becomes very costly for finer discretizations of shock processes, when the size of matrix grows to (7000*7000). Also classical eigendecomposition using LinearAlgebra.eigen is pretty slow (it solves for all eigenvalues).
I would like to ask which method of solving for stationary distribution (finding eigenvector of the transpose of the transition matrix associated with unit eigenvalue) would be most efficient, given following sparsity pattern of the transition matrix.
Any advice/suggestion would be welcome!