8.2 Predictable Processes

In this section we describe the class of predictable processes. This class of processes has a central role in the theory. In particular, only predictable processes can be integrated with respect to a semimartingale. Recall that in discrete time a process H is predictable if H _n is measurable, that is, H is known with certainty at time n on the basis of information up to time n ? 1. Predictability in continuous time is harder to define. We recall some general definitions of processes starting with the class of adapted processes.

In construction of the stochastic integral ? ^t ₀ H ( u) dS( u), processes H and S are taken to be adapted to . For a general semimartingale S, the requirement that H is adapted is too weak, it fails to assure measurability of some basic constructions. H must be predictable. The exact definition of predictable processes involves ?-fields generated on ? and is given later in Section 8.13. Note that left-continuous processes are predictable, in the sense that H( t) = lim _{s ?. t} H( s) = H( s ?). So that if the values of the process before t are known, then the value...