Finding an NARE whose minimal nonnegative solution represents first passage quantities in the two-dimensional Brownian motion

The goal of this paper is to find a nonsymmetric algebraic Riccati equation(NARE) of which the minimal nonnegative solution can represent the Laplace transform of the total increment of one component during the first passage time of the other in the two-dimensional Brownian motion. For that purpose, we construct a sequence of two-dimensional Markov modulated fluid flow which converges to the two-dimensional Brownian motion and then derive various approximation results relevant to the NARE of our interest. This is the preliminary research for investigating first-passage-related quantities in the two-dimensional Markov modulated Brownian motion in which the parameters vary according to the states of an underlying Markov process.