Reflected backward stochastic differential equations under stopping with an arbitrary random time.
Bibliographic record
Abstract
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 a stopping time for the filtration $\mathbb F$ generated by the Brownian motion $W$. We consider the filtration $\mathbb G$ resulting from the progressive enlargement of $\mathbb F$ with $\tau$ where this becomes a stopping time, and study the RBSDE under $\mathbb G$. Precisely, we focus on answering the following 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 norm using 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 are related to each other? We prove that for any random time, having a positive Az\'ema supermartingale, there exists a positive discount factor ${\widetilde{\cal E}}$ that is vital in answering our questions without assuming any further assumption on $\tau$, and determining the space for the triplet-data $(f, \xi, S)$ and the space for the solution of the RBSDE as well.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".