Lévy systems and absolutely continuous changes of measure for a jump process

Abstract

As in [7] we consider below an elementary single jump process xt , t > 0, with values in a Blackwell space (X, 9). xt has initial value z,, E X and at the random time T jumps to the random position z E X. The Levy system of the process is a regular family of conditional probabilities (X, A) where, roughly speaking, rl describes the probability that the jump occurs at a certain time, given that it has not yet happened, and h describes where in X the jump goes, given that it happens at a certain time. The object of this paper is to determine the new Levy system if the basic probability measure describing the process “globally” is replaced by a second absolutely continuous measure. Results of this kind are important for discussing the control and filtering of jump processes, and martingale techniques were first applied to such problems by Boel et al. [I, 21. The representation formulas for the Levy system of a single totally inaccessible jump were obtained by Davis [4], and Davis’ techniques are extended to the general case below. The converse problem, of how to effect an absolutely continuous change of measure, is then discussed and related to the paper [9] by van Schuppen and Wong on local martingales under a change of probability measure. Finally we discuss, in a formal way, how our results can be thought of as describing solutions of stochastic differential equations driven by a jump process and indicate possible extensions to multijump processes. Applications will be treated in another paper.