On filtrations related to purely discontinuous martingales

General martingale theory shows that every martingale can be decomposed into continuous and purely discontinuous parts. In this paper specify a filtration for which the continuous part of the decomposition is 0 a.s. for any Ft martingale. It is a well-known fact that every martingale can be decomposed into continuous and purely discontinuous parts. It is of interest to study the filtrations that do not support continuous martingales (i.e. those for which every continuous martingale with respect to that filtration is constant a.s.). In a previous work J. Jacod and A.V. Skorokhod (1994) [5] introduced the notion of jumping filtration. A filtration .~t is jumping if there is a sequence of increasing stopping times {Tn} (we will call them loosely "jumps" ), such that the u-algebras 7t and coincide up to the null sets on {Tn t In other words 7t = n {Tn t} : A E They proved that a (7-algebra is jumping iff it supports only martingales of bounded variation. Under more restrictive conditions we generalize their result to filtrations supporting only purely discontinuous martingales. As opposed to the jumping filtrations that support only martingales of locally bounded variation with finitely many jumps on finite intervals, our filtrations can support a martingale that has infinitely many jumps on a finite interval. An example of such a filtration is the natural filtration of an Azema martingale (e.g. the filtration generated by the sign of Brownian motion) or a natural filtration of purely discontinuous Levy process with infinitely many jumps on finite intervals. To accommodate this change we replace the increasing sequence of stopping times with a countable set of totally inaccessible stopping times with disjoint graphs. Unless stated otherwise we always assume that the filtration 7t is complete, rightcontinuous, quasi-left-continuous, = {0, 03A9} a.s., and the 03C3-algebra F~ is countably generated. All martingales are considered to be in their càdlàg version. Let us introduce the following definitions. Definition 1. A filtration is called purely discontinuous if any continuous adapted martingale is constant a.s. *In this paper we present some of the results in my Ph.D. dissertation [3] completed under the supervision of Professor A. V. Skorokhod at Michigan State University