Optimality Conditions in Optimal Control of Jump Processes — Extended Abstract

Abstract

The notations in this paper are mainly taken from [1 and 2] , as that paper is basic to ours.Let there be given the finite time interval [0, 1] -unless otherwise specified- and a familiy of probability spaces (Ω, F_t, P)_{t∈ [0, 1]}with the usual properties. We are considering stochastic processes (x_t, F_t, P) which are fundamental jump processes (f.j.p.) in the sense of [2] that are described by

$$ {P^x}(B,t): = \mathop \pi \limits_{s \leqslant t} [{x_{s - }} \ne {x_s}][{x_s} \in {\text{B}}] $$

where B is a subset of a measurable Blackwell space (Z,Z).Henceforth we assume F_t to be the completed ς-algebra generated by the f.j.p. (x_t,P). Furthermore martingales (M¹) ,twice integrable martingales (M₂) ,local martingales (M ¹_loc , M ²_loc ) ,integrable processes (A⁺) ,processes with integrable variation (A) and analogously (A ⁺_loc , A_loc) are denoted as in [2]. In order to define an integral with respect to elements from A we must define the associated sets of integrable functions. This will here only be done for a special case: P^x(B,t) turns out to be a local semi martingale with some further properties.