Some relationships involving subordination

Abstract

If X(t) is a stochastic process taking values in some space SC and g is a real-valued function on SC then we may consider the process g(X(t)). Often from some knowledge about the process X(t) and the function g one can deduce properties of the process g(X(t)). For example: if X(t) is Brownian motion in Rn and g is a subharmonic function on Rn that doesn't grow too fast then g(X(t)) is a semimartingale (Doob [2, p. 92]). Sometimes the implication can be reversed in the sense that from some knowledge of the processes X(t) and g(X(t)) one can deduce properties of the function g. For example: if X(t) is Brownian motion in Rn and g(X(t)) is a martingale then g is equal almost everywhere to a harmonic function (Theorem 2, below). In this note we will consider implications of both sorts in several rather special cases, making use of the notion of subordination of one process to another.