The first infinity

Abstract

We return to the M.C. {x _t , t∈T} which is assumed to be well-separable and Borel measurable. Furthermore we assume that for each i, q _i , > 0 and that (p’ _ij (0)) = (q _ij ) is a conservative Q-matrix. By Theorem 18.2 this is equivalent to the validity of the first system of differential equations. In probabilistic language, all states are stable and non-absorbing and the discontinuity of the sample functions at any exit time is an ordinary jump with probability one (see Theorems 15.2 and 15.6). It follows that we can enumerate the successive jumps until the first discontinuity that is not a jump, if such a discontinuity exists. By Theorem 7.4, since only case (c) there is possible, the sample function tends to infinity as this discontinuity is approached from the left.