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Contents
  1. C65-26: Critical exponential uniqueness for convex quadratic BSDEs
  2. 1) FORMAL RESTATEMENT
  3. Brownian conventions used in this report.
  4. Precisely stated probabilistic tools.
  5. 2) QUICK LITERATURE/CONTEXT CHECK
  6. 3) ATTACK PLAN
  7. 4) WORK
  8. 5) VERIFICATION
  9. 6) FINAL
  10. References

C65-26: Critical exponential uniqueness for convex quadratic BSDEs

Outcome: UNRESOLVED; subgradient densities and several uniqueness subclasses proved.

Tools: web browsing available; Python computation available. Literature checked on 9 October 2026.

1) FORMAL RESTATEMENT

Brownian conventions used in this report.
Fix $T>0$ and an integer $d\ge1$. The probability space carries a $d$-dimensional Brownian motion $B$ with its completed, right-continuous natural filtration; ${\mathcal F}={\mathcal F}_T$ and ${\mathcal F}_0$ is trivial modulo null sets. A scalar solution consists of a continuous adapted real process $Y$ and a predictable ${\mathbb{R}}^d$-valued process $Z$ such that \[ \int_0^T |Z_t|^2dt+\int_0^T|g(t,Y_t,Z_t)|dt<\infty\quad{\mathbb{P}}\text{-a.s.}, \] and the displayed BSDE holds simultaneously for all times outside one null set. Equality of solutions means indistinguishability of $Y$ and $dt\otimes{\mathbb{P}}$-a.e. equality of $Z$. A process $X$ is of class (D) when $\{X_\tau:\tau\le T\text{ a stopping time}\}$ is uniformly integrable. Logs are natural, norms Euclidean, and ${\operatorname{sgn}}(0)=0$. At $T=0$ the assertions reduce to $Y_0=\xi$, with no nontrivial $Z$ component.

Precisely stated probabilistic tools.
We use Itô's formula and its Tanaka version for continuous semimartingales, Brownian martingale representation, and the following forms of change of measure. If $\theta$ is predictable and ${\mathbb{E}}\exp(\frac12\int_0^T|\theta|^2)<\infty$, Novikov's theorem says that $D={\mathcal E}(\int\theta\,dB)$ is a uniformly integrable martingale; under $d{\mathbb{Q}}=D_Td{\mathbb{P}}$, $B-\int\theta dt$ is Brownian. Here ${\mathcal E}(M)=\exp(M-\langle M\rangle/2)$. Also, for a continuous Brownian martingale, the BMO hypothesis \[ {\operatorname*{ess\,sup}}_{\tau\le T}{\mathbb{E}}\left[\left.\int_\tau^T|\theta_t|^2dt\right|{\mathcal F}_\tau\right]<\infty \] implies that $D$ is uniformly integrable and $D_T\in L^q$ for some $q>1$; this is the continuous-martingale BMO exponential theorem [Kazamaki]. Each use below verifies the relevant integrability hypothesis. A local submartingale or supermartingale of class (D) is a true submartingale or supermartingale: apply the localized conditional inequalities and pass to the limit using uniform integrability. A local martingale of class (D) therefore equals the conditional expectation of its terminal value.

For every finite convex $g:{\mathbb{R}}^d\to{\mathbb{R}}$ with $0\le g(z)\le |z|^2/2$, and every ${\mathcal F}_T$-measurable $\xi$ with ${\mathbb{E}} e^{|\xi|}<\infty$, must any two solutions of \[ Y_t=\xi+\int_t^Tg(Z_s)ds-\int_t^TZ_s\cdot dB_s \] for which $e^{|Y|}$ is of class (D) coincide? This is an at-most-one assertion, not an assertion that every datum has a solution. The constants $1/2$ and $1$ are fixed. A stronger terminal exponential moment, or a positive lower curvature bound, may not be silently substituted.

The subdifferential is $\partial g(z)=\{q:g(w)\ge g(z)+q\cdot(w-z)\text{ for every }w\}$. The stress points are unbounded terminal data with exactly one exponential moment, flat regions of $g$, and change of measure at the boundary of entropy integrability.

2) QUICK LITERATURE/CONTEXT CHECK

This is Problem 5.4 in Fan--Hu--Tang [FHT]. Delbaen--Hu--Richou [DHR] prove a critical-case result with strong convexity; their Proposition 3.1 already establishes the uniformly integrable subgradient density. We reprove the latter fact below in the present sign convention. Wang--Zhang--Fan [WZF] impose additional structure and do not supply a general-convex endpoint theorem applicable here. These checks are not a claim of an exhaustive absence of later results.

3) ATTACK PLAN

Proof routes. Use convex subgradients to change measure and compare solutions; alternatively transform entropic generators explicitly; finally identify what extra integrability makes the comparison rigorous. The first route is pursued up to its precise optional-sampling gap.

Disproof routes. Test zero and bounded terminal values, generators with flat cores, and terminal tails for which a change-of-measure density has infinite entropy. The first two do not produce a counterexample; the last falsifies a tempting auxiliary claim, not uniqueness itself.

Landscape and tools. This is critical stochastic comparison. The seven main tools are supporting hyperplanes (to bound subgradients), convex conjugacy (to interpret the bound), exponential Itô transforms (to construct densities), Novikov after an intermediate change of measure (to avoid an unavailable moment), class-(D) localization (to justify sampling), conditional Jensen inequalities (for the entropic subclass), and entropy--exponential duality (to isolate sufficient extra integrability).

4) WORK

Lemma (A sharp subgradient inequality): For every $z\in{\mathbb{R}}^d$ and $q\in\partial g(z)$, \[ |q-z|^2\le |z|^2-2g(z),\qquad |q|\le2|z|. \]
Proof: The supporting inequality at the point $w=q$ and the quadratic upper bound give $g(z)+q\cdot(q-z)\le g(q)\le|q|^2/2$. Rearranging proves the first assertion. Its right side is at most $|z|^2$, so the triangle inequality proves the second. Finite convex functions on ${\mathbb{R}}^d$ have nonempty subdifferentials. Their subdifferential graphs are closed, and here the values are compact by the second bound. A Borel measurable selection exists for a nonempty compact-valued Borel correspondence of this kind (the finite-dimensional measurable selection theorem). Composing such a selection with a predictable $Z$ gives a predictable $q_t\in\partial g(Z_t)$. The bound ensures $\int|q|^2<\infty$ a.s. ∎
Proposition (The critical subgradient density is a true density): For any admissible solution $(Y,Z)$ and any predictable $q_t\in\partial g(Z_t)$, the stochastic exponential ${\mathcal E}(\int q\,dB)$ is a uniformly integrable martingale.
Proof: Define the finite, increasing process \[ A_t=\int_0^t\left(|Z_s|^2/2-g(Z_s)\right)ds\ge0. \] The BSDE gives \[ D^0_t={\mathcal E}\left(\int Z\,dB\right)_t =\exp(Y_t-Y_0-A_t)\le e^{-Y_0}e^{Y_t}. \] The stopping family on the right is uniformly integrable, since it is dominated by $e^{-Y_0}e^{|Y_\tau|}$. Thus the positive local martingale $D^0$ is of class (D), is a true uniformly integrable martingale, and ${\mathbb{E}} D^0_T=1$. Its terminal value is strictly positive. Set $d{\mathbb{Q}}^0=D^0_Td{\mathbb{P}}$ and $B^0=B-\int Zdt$.

Put $r=q-Z$. The previous lemma yields $\frac12\int_0^T|r|^2dt\le A_T$. Moreover, \[ {\mathbb{E}}_{{\mathbb{Q}}^0}e^{A_T} ={\mathbb{E}}[D^0_Te^{A_T}] =e^{-Y_0}{\mathbb{E}} e^\xi<\infty. \] Novikov's condition therefore holds for $\int r\,dB^0$ under ${\mathbb{Q}}^0$. Its exponential $D^1$ is a uniformly integrable ${\mathbb{Q}}^0$-martingale of mean one. Direct addition of the logarithms gives \[ D^0_tD^1_t =\exp\left(\int_0^tq_s\,dB_s-\frac12\int_0^t|q_s|^2ds\right). \] Indeed the cross term $-\int r\cdot Z$ in $\int r\,dB^0$ completes the square. Consequently the terminal expectation of this positive local ${\mathbb{P}}$-martingale is ${\mathbb{E}}[D^0_TD^1_T]={\mathbb{E}}_{{\mathbb{Q}}^0}D^1_T=1$. A nonnegative local martingale is a supermartingale; equal expectations at zero and $T$ force equality of all its conditional supermartingale inequalities. It is thus the martingale closed by its integrable terminal value, hence uniformly integrable. ∎

This removes a frequently proposed but incorrect first gap: it is not necessary to prove Novikov directly for $q$ under ${\mathbb{P}}$.

Theorem (Uniqueness for bounded terminal data): If $|\xi|\le K$ a.s., then the original solution class contains at most one solution, without strong convexity.
Proof: Since $g\ge0$, $Y$ is a local supermartingale. It is of class (D), because $|Y_\tau|\le e^{|Y_\tau|}$, so $Y_t\ge{\mathbb{E}}[\xi\mid{\mathcal F}_t]\ge-K$. Itô's formula shows that $e^Y$ is a local submartingale, its drift being $e^Y(|Z|^2/2-g(Z))dt\ge0$. Its class-(D) property gives $e^{Y_t}\le{\mathbb{E}}[e^\xi\mid{\mathcal F}_t]\le e^K$. Thus every admissible solution satisfies $|Y_t|\le K$.

Let $(\widetilde Y,\widetilde Z)$ be a second solution and choose $q\in\partial g(Z)$. The preceding proposition gives an equivalent probability ${\mathbb{Q}}$. With $\Delta Y=\widetilde Y-Y$ and $\Delta Z=\widetilde Z-Z$, Girsanov's theorem gives \[ d\Delta Y_t= \left[q_t\cdot\Delta Z_t-g(\widetilde Z_t)+g(Z_t)\right]dt +\Delta Z_t\,dB_t^{{\mathbb{Q}}}. \] The bracket is nonpositive by convexity. The process $\Delta Y$ is bounded, so localization makes it a true ${\mathbb{Q}}$-supermartingale. Since $\Delta Y_T=0$, it follows that $\Delta Y_t\ge0$. Reversing the two solutions gives the reverse inequality. Equality first at every rational time, then continuity, gives indistinguishability. The difference equation now has zero quadratic variation, so $\int_0^T|\Delta Z|^2dt=0$ a.s. ∎

Theorem (The entire entropic ray is solved): For $g(z)=\lambda|z|^2/2$, $0\le\lambda\le1$, the original question has a unique solution for every allowed terminal value. For $\lambda>0$ it is \[ Y_t=\lambda^{-1}\log{\mathbb{E}}[e^{\lambda\xi}\mid{\mathcal F}_t], \] and for $\lambda=0$ it is $Y_t={\mathbb{E}}[\xi\mid{\mathcal F}_t]$.
Proof: For an admissible solution and $\lambda>0$, Itô's formula makes $e^{\lambda Y}$ a local martingale. It is dominated in absolute value by $1+e^{|Y|}$, and hence is of class (D). Its terminal conditional expectation proves uniqueness of the displayed $Y$.

Conversely let $M_t={\mathbb{E}}[e^{\lambda\xi}\mid{\mathcal F}_t]$. Brownian representation supplies $dM=H\,dB$. The continuous martingale $M$ is strictly positive throughout $[0,T]$ a.s.: if it hit zero, optional sampling of the nonnegative martingale would force its strictly positive terminal value to vanish on that hitting event. A continuous strictly positive path on $[0,T]$ has a positive minimum. Local martingale representation and localization therefore make $Z=H/(\lambda M)$ square-integrable in time a.s. Itô's logarithm formula verifies the BSDE.

Conditional power-mean and Jensen inequalities imply \[ e^{Y_t}\le{\mathbb{E}}[e^\xi\mid{\mathcal F}_t],\qquad e^{-Y_t}\le e^{-{\mathbb{E}}[\xi\mid{\mathcal F}_t]}\le{\mathbb{E}}[e^{-\xi}\mid{\mathcal F}_t]. \] The first inequality is conditional concavity of $x\mapsto x^\lambda$ on $(0,\infty)$; the second uses $\lambda^{-1}\log{\mathbb{E}} e^{\lambda\xi}\ge{\mathbb{E}}\xi$ conditionally. Therefore $e^{|Y|}\le e^Y+e^{-Y}$ is dominated by the sum of two uniformly integrable closed martingales and is of class (D). The case $\lambda=0$ follows by martingale representation and conditional Jensen directly. In each case the $Z$ component is unique by quadratic variation. ∎

Proposition (A sufficient, but nonautomatic, entropy condition): For two admissible solutions, suppose a subgradient density can be chosen for each solution with finite entropy ${\mathbb{E}}[D_T\log D_T]<\infty$. Then the two solutions coincide.
Proof: Fix one such density $D_T$. Young's inequality $ab\le a\log a-a+e^b$, for $a>0,b\ge0$, gives \[ D_T|Y_\tau|\le D_T\log D_T-D_T+e^{|Y_\tau|}. \] The first two terms form a single integrable variable (the negative part of $a\log a$ is bounded), and the last family is uniformly integrable. Thus the family $D_T|Y_\tau|$ is uniformly integrable under ${\mathbb{P}}$. Also $\sup_\tau{\mathbb{E}}|Y_\tau|<\infty$, so ${\mathbb{P}}(|Y_\tau|>K)\to0$ uniformly in $\tau$. Uniform absolute continuity of the preceding integrals proves $\sup_\tau{\mathbb{E}}[D_T|Y_\tau|{\mathbf 1}_{\{|Y_\tau|>K\}}]\to0$. Hence $Y$ is of class (D) under ${\mathbb{Q}}$. This argument also applies to $\widetilde Y$ using the same density. The convexity comparison from the bounded theorem is therefore valid with class (D) in place of boundedness. It gives one order; the density chosen for the other solution gives the other order. ∎
Example (Infinite entropy really can occur): Let $U=\Phi(B_T^{(1)}/\sqrt T)$, where $\Phi$ is the standard normal distribution function, so $U$ is uniform on $(0,1)$. Set \[ \xi=\begin{cases} \displaystyle\log\frac{1}{U(\log(1/U))^2},&U<e^{-4},\\ 0,&U\ge e^{-4}. \end{cases} \] Then $\xi\ge0$ and, by $r=\log(1/U)$, \[ {\mathbb{E}} e^{|\xi|}=1-e^{-4}+\int_4^\infty r^{-2}dr =1-e^{-4}+\tfrac14<\infty, \] whereas \[ {\mathbb{E}}[e^\xi\xi]=\int_4^\infty\frac{r-2\log r}{r^2}dr=\infty. \] For $g(z)=|z|^2/2$, the solved entropic formula gives the subgradient density $D_T=e^\xi/{\mathbb{E}} e^\xi$, whose entropy is infinite. This example belongs to a uniqueness subclass; it refutes automatic finite entropy, not the original uniqueness statement.

5) VERIFICATION

Tiny cases and computation. Constant terminal values are covered by bounded-data uniqueness. Both $g=0$ and $g=|z|^2/2$ are covered at the exact endpoint. The script checks the subgradient inequality for the flat-core convex generator $g(z)=\frac12((|z|-1)^+)^2$ on 20,001 points; its minimum slack is zero. Truncated entropy integrals at upper limits $10,100,10^4$ are about $0.384,2.138,6.633$, consistent with the analytically proved divergence.

Adversarial review. Uniform integrability of a density alone does not transfer class (D) to the changed probability. The first proposition proves only that the probability exists; it does not prove the comparison step for unbounded differences. No positive curvature is used in the bounded-data theorem. All stochastic integrands remain pathwise square-integrable after equivalent changes of measure. Convex subgradient selection, the signs of the drift, and both comparison directions are explicitly accounted for.

6) FINAL

{LABEL: UNRESOLVED.}

Strongest checked partial result. Every admissible solution has a uniformly integrable subgradient exponential; uniqueness holds for bounded terminal data and, for arbitrary permitted terminal data, for every $g(z)=\lambda|z|^2/2$, $0\le\lambda\le1$. Finite entropy for comparison densities is another proved sufficient condition.

Exact first remaining gap in the selected route. For two arbitrary admissible solutions, prove that the convexity-induced local supermartingale $\widetilde Y-Y$, under the first solution's subgradient probability, satisfies the terminal comparison inequality $\widetilde Y_t-Y_t\ge{\mathbb{E}}_{{\mathbb{Q}}}[\widetilde Y_T-Y_T\mid{\mathcal F}_t]=0$, without imposing a stronger moment or curvature hypothesis. It would suffice to establish the required one-sided uniform integrability; it has not been established here.

Next three moves. (1) Prove a one-sided sampling estimate exploiting the Bregman divergence $g(\widetilde Z)-g(Z)-q\cdot(\widetilde Z-Z)$, rather than entropy alone. (2) Decompose directions with positive and zero curvature and test whether entropic comparison can be applied only in the former. (3) Investigate flat-core generators with explicit borderline terminal tails by stopped, finite-horizon approximations, checking the exact exponential class-(D) condition.

Likely structure of a counterexample, if one exists. It must escape bounded terminal data and the quadratic-ray generators. A plausible candidate would combine unbounded critical tails, nonuniform curvature, and loss of the particular changed-measure sampling property. The infinite-entropy example alone is not such a counterexample.

References

  • [Kazamaki] Norihiko Kazamaki. Continuous Exponential Martingales and BMO. Lecture Notes in Mathematics 1579, Springer, 1994. DOI: 10.1007/BFb0073585. Continuous BMO stochastic exponentials and reverse Hölder inequalities. https://doi.org/10.1007/BFb0073585.
  • [FHT] Shengjun Fan, Ying Hu and Shanjian Tang. 1D nonlinear backward stochastic differential equations: a unified theory and applications. arXiv:2309.06233v2, 11 February 2026. In particular Problems 5.1--5.5, PDF pp. 32--33; for C65-45 see (A1), (4.13)--(4.14), PDF p. 31. https://arxiv.org/pdf/2309.06233v2.
  • [DHR] Freddy Delbaen, Ying Hu and Adrien Richou. On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: The critical case. Discrete and Continuous Dynamical Systems 35(11) (2015), 5273--5283. DOI: 10.3934/dcds.2015.35.5273. Proposition 3.1 and Theorem 4.1. Preprint: https://arxiv.org/abs/1303.4859; publisher: https://www.aimsciences.org/article/doi/10.3934/dcds.2015.35.5273.
  • [WZF] Yan Wang, Yaqi Zhang and Shengjun Fan. On the uniqueness of solutions to quadratic BSDEs with non-convex generators and unbounded terminal conditions: the certain exponential moment case. arXiv:2409.13463v2, 3 August 2026. https://arxiv.org/abs/2409.13463v2.