2011/12/02 by Klebaner, F., Liptser, R.
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1112.0430
Let \mathfrakz be a stochastic exponential, i.e., \mathfrakzt=1+∫0t\mathfrakzs-dMs, of a local martingale M with jumps \triangle Mt>-1. Then \mathfrakz is a nonnegative local martingale with \E\mathfrakzt≤ 1. If \E\mathfrakzT= 1, then \mathfrakz is a martingale on the time interval [0,T]. Martingale property plays an important role in many applications. It is therefore of interest to give natural and easy verifiable conditions for the martingale property. In this paper, the property \E\mathfrakzT=1 is verified with the so-called linear growth conditions involved in the definition of parameters of M, proposed by Girsanov \citeGirs. These conditions generalize the Beneŝ idea, \citeBenes, and avoid the technology of piece-wise approximation. These conditions are applicable even if Novikov, \citeNovikov, and Kazamaki, \citeKaz, conditions fail. They are effective for Markov processes that explode, Markov processes with jumps and also non Markov processes. Our approach is different to recently published papers \citeCFY and \citeMiUr.