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
20%