Time-changed Markov processes and space-time coupled non-local equations

In this paper we study coupled fully non-local equations, where a linear non-local operator jointly acts on the time and space variables. We establish existence and uniqueness of the solution. A maximum principle is proved and used to derive uniqueness. Existence is established by providing a stochastic representation based on anomalous processes constructed as a time change via the undershooting of an independent subordinator. This leads to general non-stepped processes with intervals of constancy representing a sticky or trapping effect. Our theory allows these intervals to be dependent on the immediately subsequent jump. These processes include scaling limit of suitable coupled continuous time random walks previously studied in applications, in particular in the context of anomalous diffusion and option pricing. Here we exploit our general theory to obtain a non-local analog of the Black and Scholes equation, addressing the problem of determining the seasoned price of a derivative security, in case the price fluctuations are described by a process whose jumps are dependent on the previous interval.