论文标题
单跳过滤和当地的赛车
Single jump filtrations and local martingales
论文作者
论文摘要
单个跳跃过滤$（{\ mathscr {f}} _ t）_ {t \ in \ mathbb {r} _+} $由随机变量$γ$生成的，$ \ overline in $ \ overline {\ mathbb {r}} _+y+$ in的概率$ in compline compluatiaged $+} $ $（ω，{\ Mathscr {f}}，\ Mathsf {p}）$定义如下：in {\ Mathscr {f}} $ in {\ Mathscr {f}} $属于$ {\ Mathscr {f}}} $ \ {γ> t \} $。当且仅当其具有表示$ m_t = f（t）{\ Mathbb {1}} _ {\ {\ {t <γ\}}}+l {\ mathb {\ mathb {1}} _ $ geq {是确定性函数，$ l $是一个随机变量\ Mathbb {r} _+：{\ Mathsf {p}}}（γ\ geqslant t）> 0 \} $。即使在文献中已经研究过的特殊情况，即$ {\ mathscr {f}} $是最小的$σ$ - field，而$γ$是可测量的（然后，过滤是最小的$γ$是停止时间）。结果，给出了所有本地群众的完整描述，并根据其全球行为进行分类。
A single jump filtration $({\mathscr{F}}_t)_{t\in \mathbb{R}_+}$ generated by a random variable $γ$ with values in $\overline{\mathbb{R}}_+$ on a probability space $(Ω,{\mathscr{F}},\mathsf{P})$ is defined as follows: a set $A\in {\mathscr{F}}$ belongs to ${\mathscr{F}}_t$ if $A\cap \{γ>t\}$ is either $\varnothing$ or $\{γ>t\}$. A process $M$ is proved to be a local martingale with respect to this filtration if and only if it has a representation $M_t=F(t){\mathbb{1}}_{\{t<γ\}}+L{\mathbb{1}}_{\{t\geqslant γ\}}$, where $F$ is a deterministic function and $L$ is a random variable such that $\mathsf{E}|M_t|<\infty$ and $\mathsf{E}(M_t)=\mathsf{E}(M_0)$ for every $t\in \{t\in \mathbb{R}_+:{\mathsf{P}}(γ\geqslant t)>0\}$. This result seems to be new even in a special case that has been studied in the literature, namely, where ${\mathscr{F}}$ is the smallest $σ$-field with respect to which $γ$ is measurable (and then the filtration is the smallest one with respect to which $γ$ is a stopping time). As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour.
