Construction of Exponential Martingales for Counting Processes

摘要

Let N(t) be a counting process with continuous compensator A(t) and f(t) a bounded predictabler process. If E(exp(2/f/N(t))) < infinity and E(exp (2(1 + exp/f/)A (t))) < infinity then it is shown that z(+) = exp (-integral from 0 to t (f(u)dN(u)) - integral from 0 to t (exp (-f(u)) - 1)dA(u) is a martingale. This is a partial extension of a theorem of Kabanov, Liptser, Shiryaev (1980) who assumed A(t) < or = but did not assume A(t) is continuous.