English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we associate, to any submartingale of class $(\Sigma)$,defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P},(\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical conditions, a$\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such that forall $t \geq 0$, and for all events $\Lambda_t  \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] =\mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t]$$ where $g$ is the lasthitting time of zero of the process $X$. This measure $\mathcal{Q}$ has alreadybeen defined in several particular cases, some of them are involved in thestudy of Brownian penalisation, and others are related with problems inmathematical finance. More precisely, the existence of $\mathcal{Q}$ in thegeneral case solves a problem stated by D. Madan, B. Roynette and M. Yor, in apaper studying the link between Black-Scholes formula and last passage times ofcertain submartingales. Moreover, the equality defining $\mathcal{Q}$ remainstrue if one replaces the fixed time $t$ by any bounded stopping time. Thisgeneralization can be viewed as an extension of Doob's optional stoppingtheorem.

On some universal sigma finite measures and some extensions of Doob's optional stopping theorem