PROBABILITY AND MATHEMATICAL STALlSTLCS DRIVING NOISE OF A IFJINITX STATE MARROY PROCESS BY

Absrract. Given a finite state Markov process {X,), t 3 0, a global "driving noise " process is constructed on a larger probability space using non-standard analysis. Following ideas suggested by Kunita [3], forward and back-ward equations related to the Markov process are obtained. Given a finite state Markov process (X,), t 2 0, a global "driving noise" process is constructed on a larger probability space using non-standard analysis. The global driving process describes what happens at all times to all states of the process, and a particular realization of the Markov process is obtained by specifying a particular starting time and state; the evolution of the process from then takes place under the action of the driving noise. The driving process, therefore, plays a similar role to the diffusion term in a stochastic differential equation. Following ideas suggested by Kunita [3], forward and backward equations related to the Markov process are obtai-ned. The forward equation is derived in the next section, and does not need the global driving process. The formulation of the backward equation needs something like a global process; the backward equation is derived in section 3, assuming the basic properties derived from the global process that is constructed in section 4. In this section we describe the finite state Markov process and derive some related "forward " equations. Write ei = (0, 0,..., 1,..., 0) ' for the Ch unit column vector in P. Consider a Markov process {X,), t 2 0, defined on a probability space (a, 9, 2') whose state space is S = (el, ..., e, ). Write pi = P ( X, = e, ) ; we shall suppose that p, = ( p: , ..., pf) ' satisfies the forwar