Uniform integrability of exponential martingales and SDEs

Abstract

We consider local continuous martingales of the form , where a(s, W) depends only on the past of the Wiener process W = (W(t)) up to time s. The main result (Theorem 2) contains growth conditions on a(s, x) in x at infinity under which (et ) is uniformly integrable, that is the Girsanov equality e l= 1 holds. At the same time, the existence and unqueness of a weak (distribution sense) solution of the SDE , are established. These conditions are weaker than the classical ones of linear growth in the theory of SDEs and admit an additional logarithmic factor