When a stochastic exponential is a true martingale. Extension of the Benes method

Let z be a stochastic exponential, i.e., zt=1+∫t0zs−dMs, of a local martingale M with jumps △Mt>−1. Then z is a nonnegative local martingale with Ezt≤1. If EzT=1, then z is a martingale on the time interval [0,T]. The martingale property plays an important role in many applications. It is therefore of interest to give natural and easily verifiable conditions for the martingale property. In this paper, the property EzT=1 is verified with the so-called linear growth conditions involved in the definition of parameters of M, proposed by Girsanov [Theory Probab. Appl., 5 (1960), pp. 285--301]. These conditions generalize the Beneš idea [SIAM J. Control, 9 (1971), pp. 446--475] and avoid the technology of piecewise approximation. These conditions are applicable even if the Novikov [Theory Probab. Appl., 24 (1979), pp. 820--824] and Kazamaki [Tôhoku Math. J., 29 (1977), pp. 597--600] conditions fail. They are effective for Markov processes that explode, Markov processes with jumps, and also non-Markov processes. Our approach is different from the recently published paper [P. Cheredito D. Filipovićbͅ and M. Yor, Ann. Appl. Probab., 15 (2005), pp. 1713--1732] and preprint [A. Mijitović and M. Urusov, arXiv:0905.3701v1[math.PR], 2009].