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Register a new userBene$check{bf S}$ condition for discontinuous exponential martingale
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R. Liptser
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It is known the Girsanov exponent $mathfrak{z}_t$, being solution of Doleans-Dade equation $ mathfrak{z_t}=1+int_0^talpha(omega,s)dB_s $ generated by Brownian motion $B_t$ and a random process $alpha(omega,t)$ with $int_0^talpha^2(omega,s)ds<infty$ a.s., is the martingale provided that the Bene${rm check{s}}$ condition $$ |alpha(omega,t)|^2le text{rm const.}big[1+sup_{sin[0,t]}B^2_sbig], forall t>0, $$ holds true. In this paper, we show $B_t$ can be replaced by by a homogeneous purely discontinuous square integrable martingale $M_t$ with independent increments and paths from the Skorokhod space $ mathbb{D}_{[0,infty)} $ having positive jumps $triangle M_t$ with $Esum_{sin[0,t]}(triangle M_s)^3<infty$. A function $alpha(omega,t)$ is assumed to be nonnegative and predictable. Under this setting $mathfrak{z}_t$ is the martingale provided that $$ alpha^2(omega,t)le text{rm const.}big[1+sup_{sin[0,t]}M^2_{s-}big], forall t>0. $$ The method of proof differs from the original Bene${rm check{s}}$ one and is compatible for both setting with $B_t$ and $M_t$.
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