不连续指数鞅的Bene $\check{\bf S}$条件
Bene$\check{\bf S}$ condition for discontinuous exponential martingale
摘要：
已知Girsanov指数$\mathfrak{z}_t$是由布朗运动$B_t$和具有$\int_0^tα^2(ω,s)ds<\infty$ a.s.的随机过程$α(ω,t)$生成的Doleans-Dade方程$ \mathfrak{z_t}=1+\int_0^tα(ω,s)dB_s $的解，是鞅，前提是Bene ${\rm \check{s}}$条件$$ |α(ω,t)|^2\le \text{\rm const.}\big[1+\sup_{s\in[0,t]}B^2_s\big], \forall t>0, $$成立。本文证明了$B_t$可以由一个来自Skorokhod空间$ \mathbb{D}_{[0,\infty)} $的具有正跳$\triangle M_t$和$\E\sum_{s\in[0,t]}(\triangle M_s)^3<\infty$的具有独立增量和路径的齐次纯间断平方可积鞅$M_t$代替.假设函数$α(ω,t)$是非负的且可预测。在此设置下$\mathfrak{z}_t$是鞅提供$$ α^2(ω,t)\le \text{\rm const.}\big[1+\sup_{s\in[0,t]}M^2_{s-}\big], \ \forall t>0. $$的证明方法不同于原来的贝尼${\rm \check{s}}$之一，是兼容的两个设置与$B_t$和$M_t$。
英文摘要：
It is known the Girsanov exponent $\mathfrak{z}_t$, being solution of Doleans-Dade equation $ \mathfrak{z_t}=1+\int_0^tα(ω,s)dB_s $ generated by Brownian motion $B_t$ and a random process $α(ω,t)$ with $\int_0^tα^2(ω,s)ds<\infty$ a.s., is the martingale provided that the Bene${\rm \check{s}}$ condition $$ |α(ω,t)|^2\le \text{\rm const.}\big[1+\sup_{s\in[0,t]}B^2_s\big], \forall t>0, $$ 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 $\E\sum_{s\in[0,t]}(\triangle M_s)^3<\infty$. A function $α(ω,t)$ is assumed to be nonnegative and predictable. Under this setting $\mathfrak{z}_t$ is the martingale provided that $$ α^2(ω,t)\le \text{\rm const.}\big[1+\sup_{s\in[0,t]}M^2_{s-}\big], \ \forall t>0. $$ 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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