Abstract
A single jump filtration $({\mathscr{F}}_t)_{t\in \mathbb{R}_+}$ generated by a random variable $\gamma$ with values in $\overline{\mathbb{R}}_+$ on a probability space $(\Omega ,{\mathscr{F}},\mathsf{P})$ is defined as follows: a set $A\in {\mathscr{F}}$ belongs to ${\mathscr{F}}_t$ if $A\cap \{\gamma >t\}$ is either $\varnothing$ or $\{\gamma >t\}$. A process $M$ is proved to be a local martingale with respect to this filtration if and only if it has a representation $M_t=F(t){\mathbb{1}}_{\{t<\gamma \}}+L{\mathbb{1}}_{\{t\geqslant \gamma \}}$, 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 $t\in \{t\in \mathbb{R}_+:{\mathsf{P}}(\gamma \geqslant t)>0\}$. 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 $\sigma$-field with respect to which $\gamma$ is measurable (and then the filtration is the smallest one with respect to which $\gamma$ 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.
Highlights
Starting with Dellacherie [4], the following simple model has been studied and intensively used in applications
A single jump filtration (Ft )t∈R+ generated by a random variable γ with values in R+ on a probability space (, F, P) is defined as follows: a set A ∈ F belongs to Ft if A ∩ {γ > t} is either ∅ 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 Mt = F (t)1{t 0}
Summary
Starting with Dellacherie [4], the following simple model has been studied and intensively used in applications. Let us mention some related papers [1, 15, 16, 3, 8, 21, 12], where, in particular, local martingales with respect to the filtrations generated by jump processes or measures of certain kind are studied. A description of all local martingales via a full description of all possible solutions to a functional equation of type (2) is a simple consequence of this necessary and sufficient condition. If we are given just an adapted process X with almost all paths càdlàg, we define ρ and Y from values of X on a countable set exactly as is done in [6] in the case where X is a supermartingale.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.