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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

This paper addresses reflected backward stochastic differential equations(RBSDE hereafter) that take the form of \begin{eqnarray*} \begin{cases}dY_t=f(t,Y_t, Z_t)d(t\wedge\tau)+Z_tdW_t^{\tau}+dM_t-dK_t,\quad Y_{\tau}=\xi,  Y\geq S\quad\mbox{on}\quad \Lbrack0,\tau\Lbrack,\quad\displaystyle\int_0^{\tau}(Y_{s-}-S_{s-})dK_s=0\quad P\mbox{-a.s..}\end{cases}\end{eqnarray*} Here $\tau$ is an arbitrary random time that might not be astopping time for the filtration $\mathbb F$ generated by the Brownian motion$W$. We consider the filtration $\mathbb G$ resulting from the progressiveenlargement of $\mathbb F$ with $\tau$ where this becomes a stopping time, andstudy the RBSDE under $\mathbb G$. Precisely, we focus on answering thefollowing problems: a) What are the sufficient minimal conditions on the data$(f, \xi, S, \tau)$ that guarantee the existence of the solution of the$\mathbb G$-RBSDE in $L^p$ ($p>1$)? b) How can we estimate the solution in normusing the triplet-data $(f, \xi, S)$? c) Is there an RBSDE under $\mathbb F$that is intimately related to the current one and how their solutions arerelated to each other? We prove that for any random time, having a positiveAz\'ema supermartingale, there exists a positive discount factor${\widetilde{\cal E}}$ that is vital in answering our questions withoutassuming any further assumption on $\tau$, and determining the space for thetriplet-data $(f,\xi, S)$ and the space for the solution of the RBSDE as well.

Reflected backward stochastic differential equations under stopping with an arbitrary random time