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Econ-27 · economics · Economics

Quadratic BSDE uniqueness with only one exponential moment

LLM status Solved unverified

Statement

SC-04 |  Stochastic control and dynamic optimization

Published question

Definitions and assumptions

Use the scalar Brownian BSDE conventions above. Let $g:\R^d\to\R$ be a finite convex function satisfying \[ 0\leq g(z)\leq\tfrac12|z|^2\qquad(z\in\R^d). \] Convexity means $g(\theta z+(1-\theta)z')\leq\theta g(z)+(1-\theta)g(z')$ for $\theta\in[0,1]$; no positive lower bound on the curvature is assumed. Let $\xi$ satisfy $\E e^{|\xi|}<\infty$. The relevant solution class consists of scalar BSDE solutions for which $(e^{|Y_t|})_{0\leq t\leq T}$ is of class (D).

Formal question

For every generator and terminal value satisfying these hypotheses, does the equation \[ Y_t=\xi+\int_t^T g(Z_s)\,ds-\int_t^T Z_s\cdot dB_s \] have at most one solution in this exponential class-(D) class? Equivalently, must any two solutions in this class have indistinguishable $Y$ components and equal $Z$ components $dt\otimes\Pr$ almost everywhere?

Scope and status

The coefficient $1/2$ in the quadratic upper bound and the exponent $1$ in both integrability requirements fix the normalization of the question. A theorem requiring $\E e^{q|\xi|}<\infty$ and class (D) for $e^{q|Y|}$ with $q>1$ would address a stronger-data regime. A theorem requiring strong convexity would address a smaller generator class. Neither modification answers this endpoint formulation. The question is explicitly recorded in the cited source version; no unverified assertion of current universal openness is made here.

References for the source of the problem

Fan, Hu and Tang, 1D nonlinear backward stochastic differential equations: a unified theory and applications, arXiv:2309.06233v2, §5, Problem 5.4 and equation (5.3). That paragraph identifies the strong-convexity and stronger-exponential-moment results against which the endpoint question is posed. https://arxiv.org/html/2309.06233v2#S5.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Common conventions: BSDE conventions for SC-01--SC-06

Fix $T>0$ and an integer $d\geq1$. The probability space $(\Omega,\mathcal F,\Pr)$ is complete, $B$ is a standard $d$-dimensional Brownian motion, and $(\mathcal F_t)_{0\leq t\leq T}$ is its usual augmented natural filtration, with $\mathcal F=\mathcal F_T$. A scalar BSDE solution is a pair $(Y,Z)$, where $Y$ is continuous and adapted, $Z$ is predictable and $\R^d$-valued, and, almost surely, \[ \int_0^T\left(|g(s,Y_s,Z_s)|+|Z_s|^2\right)\,ds<\infty, \qquad Y_t=\xi+\int_t^T g(s,Y_s,Z_s)\,ds-\int_t^T Z_s\cdot dB_s. \] The identity holds simultaneously for every $t\in[0,T]$. Uniqueness means indistinguishability of $Y$ and equality of $Z$ for $dt\otimes\Pr$ almost every $(t,\omega)$. A process $U$ is of class (D) when $\{U_\tau:\tau\text{ is an }(\mathcal F_t)\text{-stopping time},\ 0\leq\tau\leq T\}$ is uniformly integrable. For $p>0$, $L^p$ denotes $\{\xi:\E|\xi|^p<\infty\}$; for $\mu>0$, $\exp(\mu L)$ denotes $\{\xi:\E e^{\mu|\xi|}<\infty\}$. These conventions remove ambiguity about filtration, stochastic integration, and integrability; they do not add an $L^2$ expectation requirement to the scalar solution definition. The conventions follow Fan, Hu and Tang, 1D nonlinear backward stochastic differential equations: a unified theory and applications, arXiv:2309.06233v2 (11 February 2026), §§1 and 1.3, https://arxiv.org/html/2309.06233v2.

Statement licence: CC-BY-4.0 · Source file

Contributions 3

Econ-27-A2 · Exploration · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-27

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.

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Econ-27-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-27

Endpoint quadratic BSDE uniqueness: finite-entropy comparison and its exact gap

Problem ID: C65-26. Research attempt; independent review pending.

Target, conventions, and scope

Fix $T>0$ and the usual augmented natural filtration of a $d$-dimensional Brownian motion $B$. The question concerns finite convex generators $0\leq g(z)\leq |z|^2/2$ and solutions \[ dY_t=-g(Z_t)\,dt+Z_t\cdot dB_t,\qquad Y_T=\xi, \qquad {\mathbb{E}} e^{|\xi|}<\infty, \] for which $e^{|Y|}$ is of class (D). A solution has continuous adapted $Y$, predictable $Z$, and $\int_0^T(|Z|^2+|g(Z)|)dt<\infty$ almost surely. Class (D) means uniform integrability over all stopping times bounded by $T$. No unstated expected-energy assumption is imposed.

We prove a comparison criterion using finite-entropy subgradient changes of measure, and identify the endpoint cancellation that prevents the usual entropy estimate from verifying that criterion. We also prove the complete constant-coefficient entropic subcase. These are partial results; they do not resolve the arbitrary convex endpoint question.

The source question and the distinction between one exponential moment, strong convexity, and stronger moments were checked in [FHT, Problem 5.4]. The relevant earlier results are [DHR11,DHR15]. Our elementary arguments are presented in full; no novelty of their ingredients or exhaustive literature survey is claimed.

The stochastic-calculus results used below (Itô integration, Brownian martingale representation, and Girsanov change of measure) use their standard forms in [RY]; their additional integrability requirements are checked explicitly in the proofs.

Changing measure without losing class (D)

For a density $D\geq0$ with ${\mathbb{E}} D=1$, write ${\operatorname{Ent}}(D)={\mathbb{E}}[D\log D]$, with $0\log0=0$.
Lemma: Let $D_T>0$ almost surely, ${\mathbb{E}} D_T=1$, and ${\operatorname{Ent}}(D_T)<\infty$. Set $D_t={\mathbb{E}}[D_T\mid{\mathcal F}_t]$ and $d{\mathbb{Q}}=D_Td{\mathbb{P}}$. If $e^{|Y|}$ is of class (D) under ${\mathbb{P}}$, then $Y$ is of class (D) under ${\mathbb{Q}}$.
Proof: Conditional Jensen gives $D_\tau\log D_\tau\leq{\mathbb{E}}[D_T\log D_T\mid{\mathcal F}_\tau]$ for every stopping time $\tau$. Since $x\log x\geq-1/e$, the positive parts $(D_\tau\log D_\tau)^+$ are bounded by the conditional expectations of $(D_T\log D_T)^+$ plus $1/e$. Conditional expectations of a fixed integrable variable form a uniformly integrable family: for example, truncate that variable at a constant and bound the remaining expectations by its $L^1$ truncation error. Thus these positive parts are uniformly integrable.

The elementary Young inequality $ab\leq e^b+a\log a-a$ for $a>0$, $b\geq0$ shows \[ D_\tau|Y_\tau|\leq e^{|Y_\tau|}+(D_\tau\log D_\tau)^+. \] Consequently the products on the left are uniformly integrable under ${\mathbb{P}}$. Also $\sup_\tau{\mathbb{P}}(|Y_\tau|>R)\leq C e^{-R}$, where $C=\sup_\tau{\mathbb{E}} e^{|Y_\tau|}<\infty$. Uniform integrability of the products therefore yields \[ \sup_\tau{\mathbb{E}}_{\mathbb{Q}}[|Y_\tau|{\mathbf 1}_{\{|Y_\tau|>R\}}] =\sup_\tau{\mathbb{E}}[D_\tau|Y_\tau|{\mathbf 1}_{\{|Y_\tau|>R\}}]\longrightarrow0. \] This is exactly class (D) under ${\mathbb{Q}}$. ∎

Lemma: For a finite convex $g$ with $0\leq g(z)\leq|z|^2/2$, every $q\in\partial g(z)$ satisfies \[ |q|\leq2|z|,\qquad q\cdot z-g(z)=g^*(q)\geq\tfrac12|q|^2, \] where $g^*(q)=\sup_w(q\cdot w-g(w))$.
Proof: The subgradient inequality makes the supremum defining $g^*(q)$ attain its value at $z$. The quadratic upper bound gives $g^*(q)\geq\sup_w(q\cdot w-|w|^2/2)=|q|^2/2$. If $z\ne0$ and $q\ne0$, apply the subgradient inequality at $w=z+|z|q/|q|$ to obtain $|z||q|\leq g(w)-g(z)\leq|w|^2/2\leq2|z|^2$. At $z=0$, $g(0)=0$ and $q\cdot w\leq|w|^2/2$ for all $w$; taking $w=\varepsilon q$ and sending $\varepsilon$ to zero gives $q=0$. The cases $q=0$ need no further argument. ∎
Theorem (A finite-entropy sufficient condition): Let $(Y^1,Z^1)$ and $(Y^2,Z^2)$ be two solutions in the target class with the same terminal value. Suppose, for $i=1,2$, there is a predictable selection $q^i_t\in\partial g(Z^i_t)$ such that \[ D^i_t=\exp\left(\int_0^t q^i_s\cdot dB_s -\tfrac12\int_0^t|q^i_s|^2ds\right) \] is a uniformly integrable martingale and ${\operatorname{Ent}}(D^i_T)<\infty$. Then $Y^1,Y^2$ are indistinguishable and $Z^1=Z^2$ $dt\otimes{\mathbb{P}}$ almost everywhere.
Proof: The quadratic variations of the displayed stochastic integrals are finite almost surely by Lemma Lemma; their terminal exponentials are strictly positive. Under $d{\mathbb{Q}}_1=D^1_Td{\mathbb{P}}$, Girsanov's theorem makes $B^1=B-\int q^1ds$ Brownian. For $\Delta Y=Y^2-Y^1$, $\Delta Z=Z^2-Z^1$, we have \[ d\Delta Y=-A_tdt+\Delta Z\cdot dB^1, \quad A_t=g(Z^2)-g(Z^1)-q^1\cdot(Z^2-Z^1)\geq0. \] The finite-variation integral exists pathwise: the subgradient bound and Cauchy--Schwarz control $\int |q^1||\Delta Z|$. Lemma Lemma makes $\Delta Y$ class (D) under ${\mathbb{Q}}_1$. Localize the stochastic integral and the finite-variation integral so that stopped conditional expectations are legitimate. The resulting inequality is the supermartingale inequality. Let the localizing times increase to $T$; class (D) gives $L^1({\mathbb{Q}}_1)$ convergence of the stopped values, so $\Delta Y$ is a true supermartingale. As $\Delta Y_T=0$, $\Delta Y_t\geq0$ for each $t$. Reversing the roles of the two solutions gives $Y^1_t-Y^2_t\geq0$. Equivalence of the measures and continuity give simultaneous equality of the $Y$ components. Subtracting the two semimartingale decompositions now shows that $\int\Delta Z\,dB$ is a continuous finite-variation local martingale, hence zero; its quadratic variation is $\int|\Delta Z|^2dt$, which therefore vanishes. ∎

The missing entropy estimate at the exact endpoint

Proposition: Let $(Y,Z)$ be any solution in the target class and let $q_t\in\partial g(Z_t)$ be predictable. Choose stopping times $\tau_n\uparrow T$ that bound $\sup_{t\leq\tau_n}|Y_t|$ and $\int_0^{\tau_n}|Z_t|^2dt$ by deterministic constants, and put $D^n=\mathcal E(\int_0^{\cdot\wedge\tau_n}q\,dB)_T$. Then $D^n$ is a density with finite entropy and \begin{equation}\label{wsref-0-65-6e-74-72-6f-70-79-62-6f-75-6e-64} {\operatorname{Ent}}(D^n)\leq{\mathbb{E}}[D^nY_{\tau_n}]-Y_0. \end{equation} For every $r>1$, \begin{equation}\label{wsref-0-67-61-70} (r-1){\operatorname{Ent}}(D^n) \leq \log{\mathbb{E}} e^{rY_{\tau_n}}-rY_0. \end{equation} In particular, if $e^{r|Y|}$ is class (D) for some $r>1$, then the unstopped exponential is a uniformly integrable martingale with finite terminal entropy. At $r=1$, the same calculation gives only $0\leq\log{\mathbb{E}} e^{Y_{\tau_n}}-Y_0$ and supplies no entropy bound.
Proof: The stopping times can be formed by hitting levels of the continuous processes $|Y|$ and $\int|Z|^2$; increase the bounds past $|Y_0|$. By Lemma Lemma, $\int_0^{\tau_n}|q|^2\leq4n$ if the energy cutoff is $n$. Bounded quadratic variation implies Novikov's condition for every scalar multiple of $q$ and finite moments of the stopped density. Under $d{\mathbb{Q}}_n=D^n d{\mathbb{P}}$, Itô's formula for $\log D^n$ and the square-integrable stopped Brownian integral give \[ {\operatorname{Ent}}(D^n)=\tfrac12{\mathbb{E}}_{{\mathbb{Q}}_n}\int_0^{\tau_n}|q|^2dt. \] The stopped BSDE, with its bounded energy, also permits expectation under ${\mathbb{Q}}_n$ and gives \[ {\mathbb{E}}_{{\mathbb{Q}}_n}Y_{\tau_n}-Y_0 ={\mathbb{E}}_{{\mathbb{Q}}_n}\int_0^{\tau_n}(q\cdot Z-g(Z))dt \geq\tfrac12{\mathbb{E}}_{{\mathbb{Q}}_n}\int_0^{\tau_n}|q|^2dt. \] This proves (entropybound). For any $r\geq1$, Jensen's inequality under ${\mathbb{Q}}_n$ applied to $\log(e^{rY_{\tau_n}}/D^n)$ gives \[ r{\mathbb{E}}_{{\mathbb{Q}}_n}Y_{\tau_n}-{\operatorname{Ent}}(D^n) \leq\log{\mathbb{E}} e^{rY_{\tau_n}}. \] Combining proves (gap). All expressions here are integrable because the stopped $Y$ is bounded and the stopped density has finite moments.

When $r>1$ and the stated class-(D) assumption holds, its exponential expectations are uniformly bounded, so (gap) bounds ${\operatorname{Ent}}(D^n)$ uniformly. Since $x\log x$ is superlinear, this makes $D^n$ uniformly integrable. Almost-sure convergence to the unstopped terminal exponential follows from finite quadratic variation. Thus the limit has expectation one, the unstopped positive local martingale is a uniformly integrable martingale, and Fatou applied to $x\log x+1/e$ gives finite terminal entropy. Substitution of $r=1$ verifies the final assertion exactly. ∎

This is an obstruction to this estimate, not evidence of actual nonuniqueness. At the endpoint the positive coefficient in front of the entropy disappears. No inference from pathwise finite quadratic variation to Novikov's condition has been made.

Endpoint checks where the argument closes

Proposition: Every solution in the target class satisfies \[ {\mathbb{E}}[\xi\mid{\mathcal F}_t]\leq Y_t\leq\log{\mathbb{E}}[e^\xi\mid{\mathcal F}_t] \quad(0\leq t\leq T). \] Consequently, for deterministic terminal value $\xi=c$, the only solution in that class is $Y_t=c$, $Z_t=0$.
Proof: The drift of $Y$ is nonpositive and $Y$ is class (D); localization and $L^1$ convergence of stopped values show it is a supermartingale. Itô's formula gives \[ d(e^Y)=e^Y\left(\tfrac12|Z|^2-g(Z)\right)dt+e^YZ\cdot dB. \] Its drift is nonnegative. Since $e^Y\leq e^{|Y|}$, it is a class-(D) local submartingale and hence a submartingale, by the same stopped conditional-expectation argument. Evaluating both inequalities at $T$ gives the bounds. For $\xi=c$ they coincide. Vanishing quadratic variation in the BSDE then gives $Z=0$ almost everywhere. ∎
Theorem (The entropic family, including its zero-curvature endpoint): For $0\leq a\leq1$ let $g_a(z)=a|z|^2/2$ and assume ${\mathbb{E}} e^{|\xi|}<\infty$. There is exactly one solution in the target class. For $a>0$ it is characterized by \[ Y_t=\frac1a\log{\mathbb{E}}[e^{a\xi}\mid{\mathcal F}_t], \] and for $a=0$ by $Y_t={\mathbb{E}}[\xi\mid{\mathcal F}_t]$.
Proof: For $a>0$, Itô's formula makes $e^{aY}$ a local martingale for any solution. It is dominated by $e^{|Y|}$, hence is a class-(D) martingale. Its terminal value determines it, proving uniqueness of $Y$ and then of $Z$ by quadratic variation.

For existence, the Brownian martingale representation theorem represents $M_t={\mathbb{E}}[e^{a\xi}\mid{\mathcal F}_t]$ as $M_0+\int_0^tH_s\cdot dB_s$, with a locally square-integrable integrand. The continuous nonnegative martingale $M$ cannot hit zero: after a zero hitting time its conditional terminal expectation is zero, contradicting $e^{a\xi}>0$. Thus its continuous path has a strictly positive minimum on $[0,T]$ almost surely. Set $Y=a^{-1}\log M$ and $Z=H/(aM)$. The martingale has finite quadratic variation on $[0,T]$, so $\int|Z|^2dt<\infty$ almost surely. Itô's formula verifies the BSDE. Conditional Jensen, first for $x^{1/a}$ and then for $x^{-1/a}$, yields \[ e^{Y_t}\leq{\mathbb{E}}[e^\xi\mid{\mathcal F}_t],\qquad e^{-Y_t}\leq{\mathbb{E}}[e^{-\xi}\mid{\mathcal F}_t]. \] Their sum is bounded by $2{\mathbb{E}}[e^{|\xi|}\mid{\mathcal F}_t]$, a class-(D) martingale. This proves the required exponential class (D). For $a=0$, use Brownian representation of ${\mathbb{E}}[\xi\mid{\mathcal F}_t]$; conditional Jensen gives $e^{|Y_t|}\leq{\mathbb{E}}[e^{|\xi|}\mid{\mathcal F}_t]$. Every other class-(D) local martingale with the same terminal value is this martingale, completing that case. ∎

Exact remaining question

The general endpoint target remains unresolved by this attempt. It would suffice to obtain the finite-entropy subgradient densities in Theorem Theorem (A finite-entropy sufficient condition) from the original assumptions alone. Proposition Proposition explains why importing the stronger-moment proof without change fails. The entropic family and deterministic terminal values are fully checked subcases, not a classification of all convex generators.

References

  • [FHT] S. Fan, Y. Hu and S. Tang, 1D nonlinear backward stochastic differential equations: a unified theory and applications, arXiv:2309.06233v2, 11 February 2026. https://arxiv.org/html/2309.06233v2.
  • [DHR11] F. Delbaen, Y. Hu and A. Richou, On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 47 (2011), 559--574. https://doi.org/10.1214/10-AIHP372.
  • [DHR15] F. Delbaen, Y. Hu and A. 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 (2015), 5273--5283. https://doi.org/10.3934/dcds.2015.35.5273.
  • [RY] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd edition, Springer, 1999. Chapters IV, V and VIII cover stochastic integration, Brownian martingale representation and Girsanov's theorem. https://doi.org/10.1007/978-3-662-06400-9.

Completion Estimate

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Econ-27-A3 · Full disproof claimed · LLM-generated · gpt-6 astra, claude opus 5.5, gpt-6 astra pro

Nonuniqueness for a convex BSDE with one exponential moment

Nonuniqueness for a convex BSDE
\protect
\ with one exponential moment

Ji Ho Bae
LawAll AI

10 October 2026

\begin{abstract} We construct a finite convex generator $0\leq g(z)\leq z^2/2$ and a terminal value $\xi$ with $\mathbb E e^{|\xi|}<\infty$ for which the scalar BSDE has a continuum of distinct solutions. Each constructed solution satisfies the required exponential class-(D) condition and has a square-integrable integrand. This gives a negative answer to Problem 5.4 of Fan, Hu and Tang. \end{abstract}

A process is of class (D) if its values at stopping times bounded by the terminal time form a uniformly integrable family. Fan, Hu and Tang [FHT, Problem 5.4, equation $5.3$] ask whether convexity, the bound $0\leq g(z)\leq |z|^2/2$, and one exponential moment ensure uniqueness among solutions for which $e^{|Y|}$ is of class (D). The following construction satisfies these conditions in one Brownian dimension.

Theorem: For every $T>0$, there exist a probability measure $\mathbb P$, a standard Brownian motion $B$ with its usual augmented natural filtration, a deterministic finite convex function $g:\mathbb R\to\mathbb R$ with $0\leq g(z)\leq z^2/2$, and a terminal value $\xi$ satisfying $\mathbb E_{\mathbb P}e^{|\xi|}<\infty$, such that \begin{equation}\label{wsref-0-65-71-3a-62-73-64-65} Y_t=\xi+\int_t^T g(Z_s)\,ds-\int_t^T Z_s\,dB_s \qquad(0\leq t\leq T) \end{equation} has a solution $(Y^c,Z^c)$ for every $c\in[0,1]$, with $Y^c_0=-c$. Each $e^{|Y^c|}$ is of class (D), and $\mathbb E_{\mathbb P}\int_0^T|Z^c_s|^2ds<\infty$.
The generator.
Set \begin{equation}\label{wsref-0-65-71-3a-67-65-6e-65-72-61-74-6f-72} K=\{0\}\cup\{2^n:n=0,1,\ldots\},\qquad g(z)=\sup_{p\in K}\left(pz-\frac{p^2}{2}\right). \end{equation} The term $p=0$ gives $g\geq0$, and completing the square gives $pz-p^2/2=z^2/2-(p-z)^2/2\leq z^2/2$. Thus $g$ is finite and convex, hence continuous. Its affine pieces have the following useful property: \begin{equation}\label{wsref-0-65-71-3a-66-61-63-65} g(\beta q)=(\beta-\tfrac12)q^2 \quad\text{for }q=2^n, n\geq0,\quad \tfrac34\leq\beta\leq1. \end{equation} Indeed, every $p\in K$ satisfies either $p\leq q/2$ or $p\geq q$. For $z\in[3q/4,q]$, \[ \left(qz-\frac{q^2}{2}\right)- \left(pz-\frac{p^2}{2}\right) =(q-p)\left(z-\frac{q+p}{2}\right)\geq0. \] So the term indexed by $q$ realizes the supremum. This argument also covers $q=1$; it does not require $q/2\in K$.
A stopped Brownian clock.
Work first under an auxiliary measure $\mathbb Q$ with a standard Brownian motion $W$ and its usual augmented natural filtration. Define \[ t_n=T(1-4^{-n}),\qquad h(t)=2^n\quad(t_n\leq t<t_{n+1}), \qquad h(T)=0, \] and, for $t<T$, let \[ a(t)=\int_0^t h(s)^2ds,\qquad M_t=\int_0^t h(s)\,dW_s. \] Each interval contributes $3T/4$ to the clock, so $a(t_n)=3Tn/4\to\infty$. The process $M$ is a Brownian motion under this deterministic time change. Consequently its first hitting time of $-4$ occurs before time $T$ almost surely. Denote this stopping time by $\tau$, taking $\tau=T$ on the null event of no hit, and put \begin{equation}\label{wsref-0-65-71-3a-73-74-6f-70-70-65-64} q_t=h(t)\mathbf1_{\{t\leq\tau\}},\qquad A_t=\int_0^t q_s^2ds,\qquad R_t=1+\frac14\int_0^t q_s\,dW_s. \end{equation} The deterministic function $h$ and the adapted left-continuous indicator $\mathbf1_{\{t\leq\tau\}}$ make $q$ predictable. Almost surely, $A_T=a(\tau)<\infty$, $R\geq0$, and $R_T=0$; $R$ and $A$ are constant after $\tau$, and $q$ vanishes there.
The probability measure.
The stochastic exponential \begin{equation}\label{wsref-0-65-71-3a-64-65-6e-73-69-74-79} N_t=\exp\left(-\int_0^t q_s\,dW_s-\frac12A_t\right) =\exp\left(4-4R_t-\frac12A_t\right) \end{equation} is a positive local martingale bounded above by $e^4$. It is therefore a uniformly integrable martingale on $[0,T]$, and \[ N_T=e^{4-A_T/2}>0,\qquad \mathbb E_{\mathbb Q}N_T=1. \] Define $d\mathbb P=N_T\,d\mathbb Q$. These measures are equivalent, and Girsanov's theorem makes \[ B_t=W_t+\int_0^t q_s\,ds \] a Brownian motion under $\mathbb P$. Moreover, Bayes' formula gives \begin{equation}\label{wsref-0-65-71-3a-72-65-63-69-70-72-6f-63-61-6c} D_t:=N_t^{-1}=\mathbb E_{\mathbb P}[N_T^{-1}\mid\mathcal F_t]. \end{equation} Thus $D$ is a uniformly integrable $\mathbb P$-martingale. The energy is integrable under this measure: \begin{equation}\label{wsref-0-65-71-3a-65-6e-65-72-67-79} \mathbb E_{\mathbb P}A_T=e^4\mathbb E_{\mathbb Q}[A_Te^{-A_T/2}] \leq e^4\sup_{x\geq0}xe^{-x/2}=2e^3. \end{equation}

We also check the filtration. For $t<T$, define the unstopped process \[ X_t=1+\frac14\int_0^t h(s)\,dB_s-\frac14a(t). \] It agrees with $R$ up to $\tau$, is positive before $\tau$, and vanishes at $\tau$. Hence $\tau=T\wedge\inf\{t<T:X_t=0\}$, with $\inf\varnothing=\infty$, is a stopping time of the usual natural filtration of $B$. Therefore $q$ is predictable in that filtration, and $W=B-\int q\,ds$ is adapted to it. The reverse inclusion follows from the construction of $B$ from $W$. Equivalence of the measures and continuity extend this equality of augmented natural filtrations to time $T$.

The solutions.
Under $\mathbb P$ we have \[ dR_t=\frac14q_t\,dB_t-\frac14q_t^2dt, \qquad d\log D_t=q_t\,dB_t-\frac12q_t^2dt. \] For $c\in[0,1]$, set \begin{equation}\label{wsref-0-65-71-3a-73-6f-6c-75-74-69-6f-6e-73} Y^c_t=\log D_t-cR_t=(4-c)R_t-4+\frac12A_t, \qquad Z^c_t=(1-c/4)q_t. \end{equation} Because $q$ takes values in $K$, equation (eq:face) and $g(0)=0$ imply \[ dY^c_t=(1-c/4)q_t\,dB_t-(1/2-c/4)q_t^2dt =Z^c_t\,dB_t-g(Z^c_t)dt. \] All these continuous processes have the same terminal value \[ \xi=\frac12A_T-4, \] so integration proves (eq:bsde). Equation (eq:energy) gives square integrability of every $Z^c$ and integrability of $g(Z^c)$. The processes stop before time $T$, so no limiting stochastic-integral identity is needed at the terminal time.

Finally, $R\geq0$ and $0\leq c\leq1$ give \[ Y^c_t\geq-4,\qquad e^{Y^c_t}=D_te^{-cR_t}\leq D_t, \qquad e^{|Y^c_t|}\leq D_t+e^4. \] The last inequality holds at every stopping time. By (eq:reciprocal), its right-hand side forms a uniformly integrable family. Hence $e^{|Y^c|}$ is of class (D), and $\mathbb E_{\mathbb P}e^{|\xi|}\leq1+e^4$. Since $Y^c_0=-c$, the solutions are distinct. This proves Theorem Theorem.$\square$

Scope.
The generator is deterministic, finite and convex; differentiability and strong convexity are not among the question's hypotheses. The construction therefore refutes the universal uniqueness assertion. Higher Brownian dimensions can be treated by using only the first coordinate and replacing $g(z)$ by $g(z_1)$.

Use of AI tools

The author used research AI agents that the author built and configured (GPT-6 Astra max, Claude Opus 5.5 max and GPT-6 Astra Pro) during the proof work, for proof search and construction, proof checking, translation, Python computation and similar tasks. The final manuscript was written, and the proofs were checked, by the human author. All responsibility for this work rests with the author personally.

References

  • [FHT] S. Fan, Y. Hu and S. Tang. 1D nonlinear backward stochastic differential equations: a unified theory and applications. arXiv:2309.06233v2, 11 February 2026, §1 and Problem 5.4. https://arxiv.org/html/2309.06233v2.

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