Mean-field backward stochastic differential equations and applications
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Bibliographic record
Abstract
In this paper we study the linear mean-field backward stochastic differential equations (mean-field BSDE) of the form (0.1)dY(t)=−[α1(t)Y(t)+β1(t)Z(t)+∫R0η1(t,ζ)K(t,ζ)ν(dζ)+α2(t)E[Y(t)]+β2(t)E[Z(t)]+∫R0η2(t,ζ)E[K(t,ζ)]ν(dζ)+γ(t)]dt+Z(t)dB(t)+∫R0K(t,ζ)Ñ(dt,dζ),t∈0,T,Y(T)=ξ.where (Y,Z,K) is the unknown solution triplet, B is a Brownian motion, Ñ is a compensated Poisson random measure, independent of B. We prove the existence and uniqueness of the solution triplet (Y,Z,K) of such systems. Then we give an explicit formula for the first component Y(t) by using partial Malliavin derivatives. To illustrate our result we apply them to study a mean-field recursive utility optimization problem in finance.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it