Beneš condition for a discontinuous exponential martingale

It is known that the Girsanov exponent ${{\mathfrak{z}}_t}$, which is a solution of the Doléans-Dade equation ${{\mathfrak{z}}_{{\, \mathfrak{t}}}}=1+\int\limits_0^t {{{\mathfrak{z}}_s}} \alpha (s)d{B_s}$ generated by a Brownian motion B _t and a random process α(t) with $\int\limits_0^t {{\alpha^2}(s)ds<\infty\,\,a.s}$., is a martingale provided that the Beneš condition,

$$ {{\left| {\alpha (t)} \right|}^2}\leq \mathrm{const}.\left[ {1 + \mathop{\sup}\limits_{{s\in \left[ {0,t} \right]}}\,\,B_s^2} \right]for\,\,any\,\,\,t>0, $$

holds. In this paper, we show that $\int\limits_0^t {\alpha (s)d{B_s}}$, can lie replaced by a purely discontinuous square integrable martingale M _t with paths from the Skorokhod space ${{\mathbb{D}}_{{\left[ {0,\infty } \right)}}}$ hailing jumps $\alpha (s)\varDelta {M_t}>-1$. The method of proof differs from the original Beneš proof. Bibliography: 13 titles.