When a Stochastic Exponential Is a True Martingale. Extension of the Beneš Method

Abstract

Let ${\frak z}$ be a stochastic exponential, i.e., ${\frak z}_t=1+\int_0^t{\frak z}_{s-}\,dM_s,$ of a local martingale $M$ with jumps $\triangle M_t>-1$. Then ${\frak z}$ is a nonnegative local martingale with ${\bf E}\,{\frak z}_t\le 1$. If ${\bf E}\,{\frak z}_{_T}= 1$, then ${\frak z}$ 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 ${\bf E}\,{\frak z}_{_T}=1$ 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 Benes 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 [Tohoku Math. J., 29 (1977),...