On a hyperbolic PDE describing the forward evolution of a class of randomly alternating systems

For a class of randomly switched linear systems a transition functional is introduced. It is shown that the expectation of this function at time t satisfies a hyperbolic partial differential equation (PDE), which plays a similar role as the backward Kolmogorov equation for diffusions. Its formal adjoint leads to the forward equation, and their complexity is determined by the Lie algebra associated with the set of values assumable by the dynamic matrix A(t, w). The usual parabolic Kolmogorov equation is derived from this as a limiting case. The result leads to Monte-Carlo simulation methods for solving hyperbolic PDE.<>