BENES CONDITION FOR A DISCONTINUOUS EXPONENTIAL MARTINGALE

摘要

It is known that the Girsanov exponent ξ_t, which is a solution of the Doleans-Dade equation ξ_i = 1 + ∫ ~t_0 ξ_s α(s)dB_s generated by a Brownian motion Bt and a random process α(t) with ∫~t_0 α~2(s)ds < ∞ a.s., is a martingale provided that the Benes condition, |α(t)|~2 ≤ const.[1 + sup_(s∈[0,t]) B~2_s] for any t > 0, holds. In this paper, we show that ∫~t_0 α(s)dB_s can be replaced by a purely discontinuous square integrable martingale Mt with paths from the Skorokhod space D_([0,∞) having jumps α(s) △ M_t > - 1. The method of proof differs from the original Benes proof.