  For a real Borel measurable function b, which satisfies certain integrability
conditions, it is possible to define a stochastic integral of the process b(Y)
with respect to a Brownian motion W, where Y is a diffusion driven by W. It is
well know that the stochastic exponential of this stochastic integral is a
local martingale. In this paper we consider the case of an arbitrary Borel
measurable function b where it may not be possible to define the stochastic
integral of b(Y) directly. However the notion of the stochastic exponential can
be generalized. We define a non-negative process Z, called generalized
stochastic exponential, which is not necessarily a local martingale. Our main
result gives deterministic necessary and sufficient conditions for Z to be a
local, true or uniformly integrable martingale.
