Stochastic Processes and their Applications 15 (1983) 203-209 North-Holland 203 SHORT COMMUNICATION zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK STOCHASTIC INTEGRATION W.R.T. CONTINUOUS LOCAL MARTINGALES Rajeeva L. KARANDIKAR Statistics & Mathematics Division, Indian Statistical Institute, Calcutta 700-035, India Received 20 kay 1981 Revised 30 October 1981 In this note we develop the theory of stochastic integration w.r.t. continuous local martingales using a simple time change technique. We allow progressively measurable integrands. Let (0,9S, P) be a probability space. A right continuous increasing family 9 = (S*)r=o of sub m-fields of 3 is called a reference family if So contains all P-null sets in P-completion of 94. Let Z(9) be the class of continuous 9-local martingales starting from zero. For M belonging to X(.9), let (M) denote the unique continuous increasing process such that M’-(M) belongs to Z(9). A simple proof of the existence of (M) for continuous local martingales M is given in [5]. Now for M E L(9) and for a ‘simple function’ f, &f dM can be defined as in the Brownian motion case. It can be checked that In one of the traditional approaches (e.g., [2,3]) the next step is to show that any ‘predictable’ process g such that can be approximated by simple functions in the ‘norm’ given by (2). Thus, using (l), 1 g dM can be defined for such a g. Then the usual stopping time arguments 0304.4149/83/$03,00 @ Elsevier Science Publishers B.V. (North-Holland)