The chaotic-representation property for a class of normal martingales

Abstract

Suppose \(Z=(Z_t)_{t\ge0}\) is a normal martingale which satisfies the structure equation $$d[Z]_t = (\alpha(t)+\beta(t)Z_{t-}) dZ_t + dt$$ . By adapting and extending techniques due to Parthasarathy and to Kurtz, it is shown that, if α is locally bounded and β has values in the interval [-2,0], the process Z is unique in law, possesses the chaotic-representation property and is strongly Markovian (in an appropriate sense). If also β is bounded away from the endpoints 0 and 2 on every compact subinterval of [0,∞] then Z is shown to have locally bounded trajectories, a variation on a result of Russo and Vallois.