Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations

We consider compositions of stochastic processes that are governed by higher-order partial differential equations. The processes studied include compositions of Brownian motions, stable-like processes with Brownian time, Brownian motion whose time is an integrated telegraph process, and an iterated integrated telegraph process. The governing higher-order equations that are obtained are shown to be either of the usual parabolic type or, as in the last example, of hyperbolic type.