Quasi-martingales

Statistics meetings at the University of Washington in WS1). In this paper we investigate necessary and sufficient conditions for a stochastic process XT to have a decomposition into the sum of a martingale process and a process having almost every sample function of bounded variation on T. Such a process is called a quasi-martingale. Necessary and sufficient conditions for such a decomposition have already been obtained by P. Meyer 3 when the process is a sub-martingale. Johnson and Helms 4 have given conditions equivalent to Meyer's when the sub-martingale is sample continuous. Our main result, Theorem 3. 3, gives necessary and sufficient conditions for a sample continuous process XT to have the above decomposition, where both the processes in the decomposition are sample continuous and the process of sample bounded variation has finite expected variation. When the process is a sample continuous sub-martingale, the conditions reduce to those given in 4. It is further proved that the decomposition of Theorem 3. 3 is unique. The uniqueness follows from Lemma 3. 3. 1 where we have proved that a martingale which is sample continuous, and of sample bounded variation has constant sample functions. This property, known true for Brownian motion, is seen to be true for all sample continuous martingales. The dominating technique used throughout the paper is random stopping times defined in terms of the sample functions of the process. The major result involving stopping times is Theorem 2. 2 which allows us to approximate a sample continuous process by a sequence of sample equicontinuous and uniformly bounded processes. 1. Notation, definitions and examples. Let (Q, F, P) be a probability space on which is defined a family of random variables (r. v. 's) X (t) ; teT where Tis a subset of the real line. Let F (t) ; te T be a family of sub cr-fields of F with