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Econ-30 · economics · Economics

Nonconvex quadratic BSDE uniqueness: a necessary correction

LLM status Solved unverified

Statement

SC-05 |  Stochastic control and dynamic optimization

Research agenda formalization

Definitions and assumptions

The source asks about scalar generators that are neither convex nor concave in $z$ and satisfy, for constants $C>0$ and $\delta\in[0,1]$, \[ |g(t,y,z)-g(t,y,z')| \leq C(1+|z|^\delta+|z'|^\delta)|z-z'|. \] Its terminal class is $\E e^{\mu|\xi|}<\infty$ for every $\mu>0$; its solution class requires $e^{\mu|Y|}$ of class (D) for every $\mu>0$.

Formal question

Literal question. Do these conditions imply uniqueness? Without a condition controlling dependence on $y$, the answer is no: take $g(y,z)=\sqrt{|y|}+|z|^2+3\sin|z|$, $\xi=0$, and for any $a\in[0,T]$ set \[ Z_t=0,\qquad Y_t=\tfrac14((a-t)^+)^2. \] These are different bounded solutions in the required class. The generator is neither convex nor concave in $z$, satisfies the displayed bound with $\delta=1$, and has at most linear growth in $y$.

Repaired research question. If, additionally, \[ |g(t,y,z)-g(t,y',z)|\leq C|y-y'|, \qquad |g(t,0,0)|\leq C, \] and $g$ is progressively measurable and continuous in $(y,z)$, does every terminal value in the stated class have at most one solution in the stated all-exponential class-(D) class, without convexity or concavity in $z$?

Scope and status

The extra Lipschitz condition in $y$ is an explicit repair, rather than a hypothesis printed in Problem 5.5 itself. It removes the elementary deterministic obstruction above. This entry retains the source question and distinguishes it from the corrected research target.

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 5.5, equation (5.4), and §3.2, equation (3.12) and Remark 3.22. Compare Example 3.18 for the underlying obstruction to uniqueness from insufficient $y$ regularity. 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-30-A2 · Full solution claimed · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-30

C65-29: Nonconvex BSDE uniqueness: literal versus repaired statement

Outcome: FULL DISPROOF of the literal statement; repaired statement UNRESOLVED.

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.

The literal assertion quantifies over progressively measurable continuous scalar generators, neither convex nor concave in $z$, with constants $C>0$ and $\delta\in[0,1]$ such that \[ |g(t,y,z)-g(t,y,z')|\le C(1+|z|^\delta+|z'|^\delta)|z-z'|. \] For every terminal value with ${\mathbb{E}} e^{\mu|\xi|}<\infty$ for every $\mu>0$, it asks for at most one solution of \[ Y_t=\xi+\int_t^Tg(s,Y_s,Z_s)ds-\int_t^TZ_s\,dB_s \] in the class in which $e^{\mu|Y|}$ is of class (D) for every $\mu>0$. At $\delta=0$ we interpret the growth bound by the constant $3C$, including at $z=0$.

The literal conditions put no uniqueness modulus on the $y$ dependence. The uploaded file already supplies a counterexample. The explicit repair in that same file is \[ |g(t,y,z)-g(t,y',z)|\le C|y-y'|, \qquad |g(t,0,0)|\le C. \] We keep this repair separate rather than pretending the displayed $z$ bound implies it. It is a sufficient conventional way to remove the elementary obstruction, not a claim that global Lipschitz continuity is the weakest possible repair.

2) QUICK LITERATURE/CONTEXT CHECK

The source question is Problem 5.5 of Fan--Hu--Tang [FHT]. The literal counterexample below is already present in the attachment, so it is verified rather than claimed as a new discovery. The nonconvex results in [FHTnonconvex,WZF] impose further structural conditions; we do not substitute those conditions into the repaired target.

3) ATTACK PLAN

Proof routes. Linearize two solutions, use a stochastic exponential to remove the $z$ difference, and test whether all exponential moments of $Y$ suffice for sampling under that probability. For bounded terminal values first obtain a deterministic bound and then BMO control.

Disproof routes. Set $Z=0$ and look for a non-Lipschitz scalar ODE; verify that an oscillatory $z$ term makes the generator neither convex nor concave; after adding the repair, test unbounded-$Z$ cases rather than reusing the excluded ODE example.

Landscape and tools. This is stochastic uniqueness with a deterministic obstruction. The useful tools are separation of variables (for delayed ODE solutions), radial derivative tests (for nonconvexity), divided differences (for exact linearization), Girsanov's theorem (for uniqueness of the linearized equation), Hölder's inequality (for class-(D) transfer), Tanaka/exponential transforms (for bounded terminal estimates), and BMO exponentials (for unbounded but controlled integrands).

4) WORK

Theorem (Literal statement is false): For every $T>0$ and $d\ge1$, let \[ g(y,z)=\sqrt{|y|}+|z|^2+3\sin|z|,\qquad \xi=0. \] For every $a\in[0,T]$, the pair \[ Z^a_t=0,\qquad Y^a_t=\tfrac14((a-t)^+)^2 \] is an admissible solution. Distinct $a$ give distinct solutions.
Proof: Write $r=|z|$ and $r'=|z'|$. The reverse triangle inequality and the $1$-Lipschitz property of sine give \[ |r^2-(r')^2+3\sin r-3\sin r'| \le(r+r'+3)|z-z'| \le3(1+r+r')|z-z'|. \] Thus the required inequality holds with $C=3$, $\delta=1$. On a positive coordinate ray, the second derivative of the $z$ part is $2-3\sin r$, equal to $-1$ at $r=\pi/2$ and $5$ at $r=3\pi/2$. A twice differentiable convex function has nonnegative second derivative and a concave one nonpositive second derivative on every open interval. Restriction to this ray therefore excludes both convexity and concavity. The generator is continuous and deterministic, hence progressively measurable.

Since $g(Y^a_s,0)=\sqrt{Y^a_s}=(a-s)^+/2$, direct integration gives \[ \int_t^Tg(Y^a_s,0)ds=\int_t^a\frac{a-s}{2}ds =\frac{(a-t)^2}{4}\quad(t<a), \] and both sides are zero when $t\ge a$. This is exactly the BSDE with zero terminal value and zero martingale term. Both required time integrals are finite. Moreover $0\le Y^a_t\le T^2/4$, so for every $\mu>0$ the entire stopping family $e^{\mu|Y^a_\tau|}$ is bounded by $e^{\mu T^2/4}$ and is uniformly integrable. At $t=0$, $Y^a_0=a^2/4$, so distinct $a$ give distinct solutions. Finally $\sqrt{|y|}\le1+|y|$, but $|\sqrt y-0|/|y-0|=y^{-1/2}\to\infty$ as $y\downarrow0$. The example has at most linear $y$ growth and fails precisely the added Lipschitz repair. ∎

The remaining results concern only the repaired question.

Lemma (Transfer under an integrable density): If $D_T\in L^q({\mathbb{P}})$ for some $q>1$ and $\sup_\tau{\mathbb{E}}|X_\tau|^p<\infty$ for every finite $p$, then $X$ is of class (D) under $d{\mathbb{Q}}=D_Td{\mathbb{P}}$. If the density integrand is uniformly bounded, the same conclusion holds whenever the stopping family has a uniformly bounded $p$th moment for one $p>1$.
Proof: In the first case choose $r>1$ and $p=rq/(q-1)$. Hölder gives \[ \sup_\tau{\mathbb{E}}_{{\mathbb{Q}}}|X_\tau|^r \le\|D_T\|_{L^q}\, \sup_\tau\left({\mathbb{E}}|X_\tau|^p\right)^{(q-1)/q}<\infty. \] Uniformly bounded $r$th moments imply uniform integrability. For a bounded integrand $|b|\le M$, Novikov applies to every multiple of $b$ and \[ {\mathbb{E}} D_T^q\le \exp\left(q(q-1)M^2T/2\right),\qquad q>1, \] because $D_T^q={\mathcal E}(q\int b\,dB)_T \exp(\frac12q(q-1)\int|b|^2)$. For a given $p>1$, choose $1<r<p$ and then $q=p/(p-r)$ in the same Hölder inequality. This proves the second assertion. The all-exponential class in the problem has all uniformly bounded stopping moments, since for every finite $p$ there is a constant $K_p$ with $x^p\le K_pe^x$ for $x\ge0$. ∎
Proposition (A precise linearization criterion): For two solutions to the repaired equation, set $\Delta Y=Y-\widetilde Y$ and $\Delta Z=Z-\widetilde Z$. Define \[ a_t=\frac{g(t,Y_t,Z_t)-g(t,\widetilde Y_t,Z_t)}{Y_t-\widetilde Y_t} \] where the denominator is nonzero, and zero otherwise, and define \[ b_t=\frac{g(t,\widetilde Y_t,Z_t)-g(t,\widetilde Y_t,\widetilde Z_t)} {|\Delta Z_t|^2}\,\Delta Z_t \] where $\Delta Z_t\ne0$, and zero otherwise. If ${\mathcal E}(\int b\,dB)$ is a true density with terminal value in some $L^q$, $q>1$, then the two solutions coincide.
Proof: Measurability follows from the measurable generator and the explicit zero-denominator convention. The hypotheses imply $|a|\le C$ and $|b|\le C(1+|Z|^\delta+|\widetilde Z|^\delta)$, so $\int|b|^2<\infty$ a.s. by $\delta\le1$ and finite $T$. The difference equation is exactly $d\Delta Y=-(a\Delta Y+b\cdot\Delta Z)dt+\Delta Z\,dB$. Under the stated probability ${\mathbb{Q}}$, the process $L_t=\exp(\int_0^ta_sds)\Delta Y_t$ is a local martingale. The transfer lemma and the deterministic bound on the multiplier make $L$ of class (D). Its terminal value is zero, so $L_t={\mathbb{E}}_{\mathbb{Q}}[L_T\mid{\mathcal F}_t]=0$. Equality of $Z$ follows from quadratic variation as before. ∎
Corollary (Solved subcases of the repair): Uniqueness holds for $\delta=0$. It also holds if both solutions' $Z$ integrands are uniformly bounded, or if the linearization integrand $b$ satisfies the BMO hypothesis.
Proof: For $\delta=0$ the bound is $|b|\le3C$. If both $Z$ are bounded the same reasoning gives a finite deterministic bound on $b$. The transfer lemma verifies the density condition. In the BMO case the stated continuous-martingale BMO theorem supplies it. ∎
Theorem (Bounded terminal data for every $0\le\delta\le1$): Under the repaired assumptions and $|\xi|\le K$, there is at most one solution in the all-exponential class-(D) class.
Proof: There are deterministic constants $L\ge1$, $\gamma>0$ such that \[ |g(t,y,z)|\le L(1+|y|)+\frac\gamma2|z|^2. \] Indeed the hypotheses bound the left side by $C+C|y|+C(|z|+|z|^{1+\delta})$, and both $r$ and $r^{1+\delta}$ are bounded by a constant times $1+r^2$ for $0\le\delta\le1$.

Tanaka's formula gives $d|Y|\ge-[L(1+|Y|)+\gamma|Z|^2/2]dt+{\operatorname{sgn}}(Y)Z\,dB+dL^0(Y)$, where the local-time term is nonnegative (a choice of normalization only rescales $L^0$). For a continuous semimartingale the occupation-density formula gives $\int_0^T{\mathbf 1}_{\{Y_t=0\}}|Z_t|^2dt=0$; thus the martingale part of $|Y|$ has quadratic variation $\int|Z|^2dt$. Apply Itô to \[ H_t=\exp\{\gamma[e^{Lt}|Y_t|+(e^{Lt}-1)]\}. \] The terms involving $L(1+|Y|)$ cancel, and the remaining quadratic drift is bounded below by $\frac12\gamma^2(e^{2Lt}-e^{Lt})H_t|Z_t|^2dt\ge0$. The local-time contribution is also nonnegative. This makes $H$ a local submartingale. Its stopping family is dominated by a constant times $\exp(\gamma e^{LT}|Y_\tau|)$, so it is of class (D). Consequently \[ H_t\le{\mathbb{E}}[H_T\mid{\mathcal F}_t] \le\exp\{\gamma[e^{LT}K+(e^{LT}-1)]\}, \] which implies $|Y_t|\le e^{L(T-t)}(K+1)-1\le K':=e^{LT}(K+1)-1$.

Next choose $a>\gamma$. The drift in Itô's formula for $e^{aY}$ is bounded below by \[ e^{aY}\left[-aL(1+K')+\frac{a(a-\gamma)}2|Z|^2\right]dt. \] Integrate from any stopping time $\tau$ to localized times increasing to $T$, take conditional expectations, and use $e^{-aK'}\le e^{aY}\le e^{aK'}$. Conditional Fatou for the nonnegative $Z$ term gives \[ {\mathbb{E}}\left[\left.\int_\tau^T|Z_t|^2dt\right|{\mathcal F}_\tau\right] \le\frac{2e^{2aK'}[2+aL(1+K')T]}{a(a-\gamma)}. \] This finite deterministic upper bound proves BMO. It holds for both solutions. Since $r^{2\delta}\le1+r^2$, the previously defined $b$ has $|b|^2\le K''(1+|Z|^2+|\widetilde Z|^2)$ for a finite constant $K''$. It too is BMO. The linearization criterion now proves uniqueness. ∎

5) VERIFICATION

The counterexample includes $a=0$ and $a=T$ and has no stochastic integrability issue. At the join $t=a$, its derivative is zero on both sides. The script directly compares the integral and solution at four choices of $a$ and 21 time points; maximum discrepancy is $5.6\cdot10^{-17}$. The radial second derivatives used above are exactly $-1$ and $5$.

For the repaired question, the bound on $b$ is local in $Z$, not a deterministic global Lipschitz constant when $\delta>0$. All exponential moments of $Y$ do not, by the reasoning supplied here, automatically imply BMO of $Z$ for unbounded terminal data. The bounded-terminal proof uses all-exponential class (D) at the precise transform exponent $\gamma e^{LT}$; dropping that hypothesis would invalidate that step. The BMO theorem is used only for continuous martingales, not for a general discontinuous process.

6) FINAL

Literal statement: LABEL: FULL SOLUTION. SUBLABEL: COUNTEREXAMPLE/DISPROOF. The generator and continuum of solutions in the first theorem provide a fully verified disproof. This counterexample was already supplied in the uploaded file.

Repaired statement: LABEL: UNRESOLVED. The strongest proved subclasses are $\delta=0$, bounded terminal data for all $\delta\in[0,1]$, and the stated bounded-integrand/BMO comparison criteria.

Exact first gap. For arbitrary repaired data and two all-exponential admissible solutions, establish a valid change of measure and terminal sampling for $e^{\int a}\Delta Y$ with the unbounded divided-difference integrand $b$. A sufficient unproved lemma is that this exponential density is a true martingale with some $L^q$, $q>1$, terminal moment; an alternative proof could avoid that sufficient lemma.

Next three moves. (1) Derive quantitative exponential-energy estimates for $\int|Z|^{2\delta}$ from the all-exponential solution class, with constants sufficient for sampling. (2) Separate $0<\delta<1$ from $\delta=1$ and test stopped linearizations at the exact required moment order. (3) Search generators whose derivative oscillates at order $|z|$ together with unbounded terminal tails, verifying both all-exponential class-(D) requirements before calling any construction a counterexample.

Likely minimal counterexample to the repair, if any. It must have unbounded terminal data and escape the bounded-density/BMO cases. A $Z=0$ delayed ODE construction is excluded by global Lipschitz continuity in $y$ and cannot answer the repaired question.

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.
  • [FHTnonconvex] Shengjun Fan, Ying Hu and Shanjian Tang. On the uniqueness of solutions to quadratic BSDEs with non-convex generators and unbounded terminal conditions. Comptes Rendus Mathématique 358(2) (2020), 227--235. DOI: 10.5802/crmath.40. https://www.numdam.org/articles/10.5802/crmath.40/.
  • [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-30-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-30

Nonconvex quadratic BSDE uniqueness: the missing state condition and a quantitative energy criterion

Problem ID: C65-29. Research attempt; independent review pending.

Target and conventions

The source asks about nonconvex, nonconcave generators with \begin{equation}\label{wsref-0-7a-62-6f-75-6e-64} |g(t,y,z)-g(t,y,z')|\leq C(1+|z|^\delta+|z'|^\delta)|z-z'|, \qquad 0\leq\delta\leq1, \end{equation} and terminal data and solution paths with every exponential moment in the class-(D) sense. Its literal statement omits a condition on $y$. The catalogue explicitly repairs this by imposing progressive measurability, continuity, $|g(t,0,0)|\leq C$, and $|g(t,y,z)-g(t,y',z)|\leq C|y-y'|$. We preserve that distinction. Here class (D) always means uniform integrability over stopping times, not just bounded deterministic-time moments. Solutions have continuous adapted $Y$, predictable $Z$, and pathwise finite $\int_0^T(|g(t,Y,Z)|+|Z|^2)dt$ on the usual Brownian basis.

We verify the literal counterexample, solve the repaired uniqueness question for $\delta=0$, and prove an explicit additional-energy criterion for all $\delta\leq1$. The unrestricted repaired question remains unresolved. The source and related strictly quadratic results were checked in [FHT, Problem 5.5 and Remark 3.22] and [FHT20] on 9 October 2026. The argument below is an elementary linearization calculation, not a claim that this technique or its restricted consequences are new.

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.

The literal obstruction

Proposition: For every $T>0$ and $d\geq1$, the generator \[ g(y,z)=\sqrt{|y|}+|z|^2+3\sin|z| \] is continuous, is neither convex nor concave in $z$, satisfies (zbound) with $C=3$, $\delta=1$, and has at most linear growth in $y$. The terminal value $\xi=0$ has the distinct solutions \[ Z^a_t=0,\qquad Y^a_t=\tfrac14((a-t)^+)^2,\qquad 0\leq a\leq T. \] Every $e^{\mu|Y^a|}$ is class (D), for every $\mu>0$.
Proof: The reverse triangle inequality and the Lipschitz bound for sine give \[ \left||z|^2-|z'|^2+3(\sin|z|-\sin|z'|)\right| \leq(3+|z|+|z'|)|z-z'|. \] On a positive coordinate ray the $z$-dependent function is $r^2+3\sin r$. Its second derivative is $2-3\sin r$, negative at $\pi/2$ and positive at $3\pi/2$. A convex or concave function would have the corresponding sign on every such smooth restriction, so neither property holds. Also $\sqrt{|y|}\leq1+|y|$.

The function $Y^a$ is continuously differentiable with derivative $-(a-t)^+/2=-\sqrt{Y^a_t}$, including at $t=a$. It has terminal value zero and satisfies the BSDE with $Z^a=0$ by integration. Its generator is time-integrable. The paths are bounded by $T^2/4$, hence all their exponential transforms are bounded and class (D). Different $a$ give different values at time zero. The failure of Lipschitz regularity in $y$ is explicit, so this does not refute the repaired problem. ∎

A density estimate with all constants displayed

Lemma: Let $\beta$ be predictable with $A=\int_0^T|\beta_s|^2ds<\infty$ almost surely. If ${\mathbb{E}} e^{6A}<\infty$, then $D=\mathcal E(\int\beta\,dB)$ is a uniformly integrable martingale, $D_T>0$ almost surely, and \[ \sup_{\tau\leq T}{\mathbb{E}} D_\tau^2\leq({\mathbb{E}} e^{6A})^{1/2}. \] If $|Y|^2$ is of class (D) under ${\mathbb{P}}$, then $Y$ is of class (D) under $d{\mathbb{Q}}=D_Td{\mathbb{P}}$.
Proof: Put $M=\int\beta\,dB$. At any stopping time $\tau$, \[ D_\tau^2 =\mathcal E(4M)_\tau^{1/2}\exp(3\langle M\rangle_\tau). \] A nonnegative stochastic exponential is a local martingale and a supermartingale, so its stopped expectation is at most one. Cauchy--Schwarz therefore bounds ${\mathbb{E}} D_\tau^2$ by $({\mathbb{E}} e^{6A})^{1/2}$. Applying this to a localizing sequence for $D$ shows that its stopped values are uniformly integrable, hence $D_t={\mathbb{E}}[D_T\mid{\mathcal F}_t]$ and ${\mathbb{E}} D_T=1$. Pathwise finite quadratic variation gives $D_T>0$. Finally, for every $R>0$, \[ {\mathbb{E}}_{\mathbb{Q}}[|Y_\tau|{\mathbf 1}_{\{|Y_\tau|>R\}}] \leq({\mathbb{E}} D_\tau^2)^{1/2} ({\mathbb{E}}[|Y_\tau|^2{\mathbf 1}_{\{|Y_\tau|>R\}}])^{1/2}. \] Uniform integrability of $|Y_\tau|^2$ makes the right side tend to zero uniformly over $\tau$, proving the last assertion. ∎
Theorem (Conditional uniqueness in the repaired class): Suppose $g$ has all the repaired assumptions above with constant $C>0$ and exponent $\delta\in[0,1]$. Let $(Y,Z)$ and $(\widetilde Y,\widetilde Z)$ be solutions with the same terminal value, and suppose both $|Y|^2$ and $|\widetilde Y|^2$ are of class (D). If \begin{equation}\label{wsref-0-65-6e-65-72-67-79} {\mathbb{E}}\exp\left(18C^2\int_0^T (|Z_s|^{2\delta}+|\widetilde Z_s|^{2\delta})ds\right) <\infty, \end{equation} then the solutions are equal: their $Y$ components are indistinguishable and their $Z$ components agree $dt\otimes{\mathbb{P}}$ almost everywhere. For $\delta=0$ adopt $|z|^0=1$ also at $z=0$. In particular the repaired target has at most one solution when $\delta=0$, for every terminal value in its all-exponential class.
Proof: Write $U=Y-\widetilde Y$, $V=Z-\widetilde Z$ and define, with a zero value when the denominator is zero, \[ a_t=\frac{g(t,Y_t,Z_t)-g(t,\widetilde Y_t,Z_t)}{U_t},\quad \beta_t=\frac{g(t,\widetilde Y_t,Z_t)-g(t,\widetilde Y_t,\widetilde Z_t)} {|V_t|^2}V_t. \] These are progressively measurable; a predictable version for stochastic integration can be chosen since the solution integrands and continuous adapted processes have predictable versions and the coefficient is progressively measurable up to $dt\otimes{\mathbb{P}}$ equivalence. The bounds give $|a|\leq C$ and \[ |\beta|^2\leq3C^2(1+|Z|^{2\delta}+|\widetilde Z|^{2\delta}). \] Pathwise integrability follows from $2\delta\leq2$. Assumption (energy) implies ${\mathbb{E}}\exp(6\int|\beta|^2)<\infty$. Lemma Lemma thus provides an equivalent probability measure ${\mathbb{Q}}$ under which $B^{\mathbb{Q}}=B-\int\beta ds$ is Brownian and $U$ is class (D). The difference equation becomes \[ dU=-aUdt+V\cdot dB^{\mathbb{Q}}. \] For $R_t=\exp(\int_0^t a_sds)$, Itô's product rule gives $d(RU)=RV\cdot dB^{\mathbb{Q}}$. The factors $R_t$ lie between $e^{-CT}$ and $e^{CT}$, so $RU$ is a class-(D) local martingale. Localization and uniform integrability make it a true martingale. Since $U_T=0$, it vanishes identically. Its quadratic variation yields $V=0$ almost everywhere. When $\delta=0$, (energy) is deterministic and finite. Finally every all-exponential class-(D) solution has $|Y|^2$ of class (D), because $y^2\leq C_0e^{|y|}$ and this is uniform-integrability domination. ∎

The proof has not replaced a stochastic exponential by a probability density without checking expectation one, and it has not silently carried class (D) through a change of measure. Those are the two roles of Lemma Lemma.

A nonconvex example covered without an extra energy hypothesis

For $g(t,y,z)=\ell y+\sin|z|$ with fixed $\ell\in{\mathbb{R}}$, the $z$ part is globally Lipschitz and neither convex nor concave along a positive ray. Theorem Theorem (Conditional uniqueness in the repaired class) with $\delta=0$ proves uniqueness in the target solution class for any admissible terminal data. No curvature condition enters. The theorem also covers genuinely quadratic nonconvex generators, such as $\ell y+|z|^2+3\sin|z|$, whenever their two candidate integrands satisfy (energy).

Remaining gap and verification

The new hypothesis (energy) is a restriction on candidate solutions; it has not been deduced from the stated all-exponential class-(D) assumption on $Y$. A pathwise finite energy is insufficient to assert (energy). Related work [FHT20] obtains energy integrability under additional strict quadratic and structural conditions, which are not part of the repaired target. Thus the literal question is refuted, the repaired $\delta=0$ subcase is proved, and the repaired unrestricted $0<\delta\leq1$ question remains unresolved here.

References

Completion Estimate

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Econ-30-A3 · Full disproof claimed · LLM-generated · gpt-6 astra, claude opus 5.5, gpt-6 astra pro

Nonuniqueness of a quadratic BSDE with all exponential moments

Nonuniqueness of a quadratic BSDE
\protect
\ with all exponential moments

Ji Ho Bae
LawAll AI

10 October 2026

\begin{abstract} We construct a scalar quadratic BSDE with a predictable generator, independent of the value variable and satisfying a uniform quadratic Lipschitz bound, that has a continuum of distinct solutions with the same terminal value. Each constructed solution has a square-integrable integrand and an integrable exponential of its running absolute maximum at every positive order. The generator is neither convex nor concave. Thus the nonconvex uniqueness question has a negative answer even after imposing Lipschitz continuity in the value variable. \end{abstract}

Problem 5.5 of Fan, Hu and Tang [FHT] asks about uniqueness for nonconvex quadratic BSDEs with all exponential moments. Its literal formulation does not control the dependence on $y$. The construction below is independent of $y$, so it also answers the question that imposes a uniform Lipschitz bound in $y$ and a bounded generator at the origin. Class (D) means uniform integrability over all stopping times bounded by $T$.

Theorem: For every $T>0$, on the usual augmented natural filtration of a one-dimensional Brownian motion $B$, there exist a predictable generator $g(t,\omega,z)$ and a terminal value $\xi$ such that, for all $z,z'\in\mathbb R$, \begin{equation}\label{wsref-0-62-6f-75-6e-64-73} g(t,0)=0,\qquad |g(t,z)|\leq z^2,\qquad |g(t,z)-g(t,z')|\leq4(1+|z|+|z'|)|z-z'|. \end{equation} Every function $g(t,\omega,\cdot)$ is continuous, neither convex nor concave. The equation \begin{equation}\label{wsref-0-62-73-64-65} Y_t=\xi+\int_t^T g(s,Z_s)\,ds-\int_t^T Z_s\,dB_s \end{equation} has a solution $(Y^c,Z^c)$ for every $c\in[0,1]$, with distinct initial values, and \begin{equation}\label{wsref-0-6d-6f-6d-65-6e-74-73} \mathbb E\int_0^T|Z^c_s|^2ds<\infty,\qquad \mathbb E\exp\!\left(\mu\sup_{0\leq t\leq T}|Y^c_t|\right)<\infty \quad\text{for every }\mu>0. \end{equation} In particular, $\mathbb E e^{\mu|\xi|}<\infty$ and $e^{\mu|Y^c|}$ is of class (D) for every $\mu>0$.
A sequence of stopped excursions.
Put $s_k=T(1-2^{-k})$ for $k\geq0$. Leg $k\geq1$ occupies $[s_{k-1},s_k)$, has sign $\sigma_k=(-1)^{k+1}$, and belongs to cycle $n=\lceil k/2\rceil$. Set \[ a_n=\log(n+1),\qquad h_k(t)=\frac1{s_k-t},\qquad L^k_t=\sigma_k\int_{s_{k-1}}^t h_k(s)\,dB_s -\int_{s_{k-1}}^t h_k(s)^2ds. \] Extend $h_k$ by zero outside its leg. Let $\tau_k$ be the first exit of $L^k$ from $(-a_n,a_n)$. The deterministic clock $\int_{s_{k-1}}^t h_k^2$ tends to infinity as $t\uparrow s_k$. In clock time $L^k$ is a Brownian motion with drift $-1$, so $\tau_k<s_k$ almost surely. Call the leg successful if $L^k_{\tau_k}=a_n$.

Only legs whose predecessors were all successful are used. Write $A_k$ for this event and define the predictable process \begin{equation}\label{wsref-0-71} q_t=\sum_{k\geq1}\mathbf1_{A_k}\mathbf1_{\{s_{k-1}<t\leq\tau_k\}}h_k(t), \qquad X_t=\int_0^t q_s\,dB_s-\int_0^t\sigma(s)q_s^2ds, \end{equation} where $\sigma(s)=\sigma_k$ on leg $k$ and endpoint values are immaterial. The event $A_k$ is known at time $s_{k-1}$, and the stopped interval indicators are predictable. Finiteness of the energy in (q) will be verified below; equivalently, one may first construct these processes up to any finite number of legs.

The first leg in a cycle starts at $X=0$ and, on success, ends at $X=a_n$. The second starts at $a_n$ and, on success, returns to zero. A failed first leg ends at $-a_n$; a failed second leg ends at $2a_n$. After a failure, $q=0$ forever. Between a successful exit and the next leg, $X$ is constant.

For Brownian motion with drift $-1$ started at zero in $(-a,a)$, the probability of exiting at $a$ is \[ p(a)=\frac1{1+e^{2a}}. \] Indeed, apply Itô's formula and bounded optional sampling to the harmonic function $\frac{e^{2(x+a)}-1}{e^{4a}-1}$. The exit time $S$ in clock time also satisfies $\mathbb E S\leq a$: at $S\wedge m$, the stopped process lies in $[-a,a]$ and has expectation $-\mathbb E(S\wedge m)$, and then monotone convergence applies.

Thus each used leg in cycle $n$ has conditional success probability $p_n=(1+(n+1)^2)^{-1}$. Independence of Brownian increments on the successive deterministic intervals gives, for the cycle $N$ of the first failure, \begin{equation}\label{wsref-0-74-61-69-6c-73} \mathbb P(N\geq n)=\prod_{j<n}p_j^2\leq\frac1{(n!)^4},\qquad \mathbb E\int_0^Tq_t^2dt\leq2\sum_{n\geq1}\frac{a_n}{(n!)^4}<\infty. \end{equation} In particular $N<\infty$ almost surely, and \begin{equation}\label{wsref-0-6d-61-78-69-6d-75-6d} \sup_{t\leq T}|X_t|\leq2a_N,\qquad \mathbb E e^{\mu\sup|X|}\leq\sum_{n\geq1}\frac{(n+1)^{2\mu}}{(n!)^4}<\infty. \end{equation} All paths under consideration stop strictly before $T$. The null event of infinitely many successes can be discarded using the completed filtration.

A bounded perturbation with zero terminal value.
Define \[ F_n(x)=\frac{1-e^{-2(x+a_n)}}{1-e^{-4a_n}},\qquad r_n=F_n(0)=\frac1{1+e^{-2a_n}},\qquad d_k=\prod_{j=k}^{\infty}r_{\lceil j/2\rceil},\qquad \eta=\frac1{16}. \] These products are positive: $\sum_n\log(1+(n+1)^{-2})<\infty$. They satisfy $d_k=r_n d_{k+1}$ when $n=\lceil k/2\rceil$. On $[-a_n,a_n]$, \begin{equation}\label{wsref-0-46} F_n(-a_n)=0,\quad F_n(a_n)=1,\quad F_n''=-2F_n',\quad 0<F_n'\leq\frac{32}{15}. \end{equation} During a used leg $k$ of cycle $n$, up to its exit, set \[ D_t=\eta d_{k+1}F_n(L^k_t),\qquad K_t=\eta d_{k+1}\sigma_k h_k(t)F_n'(L^k_t). \] Hold $D$ constant between legs, set it to zero forever after failure, and put $K=0$ outside the used portions of legs. At the beginning of leg $k$, $D=\eta d_k$; on success it becomes $\eta d_{k+1}$. Thus the pieces join continuously. We have \begin{equation}\label{wsref-0-44} D_0=\eta d_1>0,\qquad D_T=0,\qquad 0\leq D\leq\eta,\qquad |K_t|\leq\frac{2}{15}q_t. \end{equation} Itô's formula, using the drift $-1$ of $L^k$ and (F), gives the global identity \begin{equation}\label{wsref-0-6c-69-6e-65-61-72} dD_t=K_t\,dB_t-2\sigma(t)q_tK_t\,dt. \end{equation} All terms are integrable by (tails) and (D). This construction uses no change of probability measure.
The generator and the solutions.
Let \[ G(u)=2\left(1-|u-\tfrac32|\right)_+. \] This tent function is supported on $[1/2,5/2]$, is $2$-Lipschitz, satisfies $0\leq G(u)\leq u^2$, and equals $2u-1$ on $[1/2,3/2]$. Define \begin{equation}\label{wsref-0-67-65-6e-65-72-61-74-6f-72} g(t,z)= \begin{cases} \sigma(t)q_t^2G(z/q_t)+G(-z),&q_t>0,\\ G(-z),&q_t=0. \end{cases} \end{equation} The coefficient $q$ was fixed before choosing a solution parameter. Hence this is one predictable generator for all the solutions below. It does not depend on $y$.

The two terms in (generator) have disjoint positive and negative supports, so $|g(t,z)|\leq z^2$ and $g(t,0)=0$. At points of differentiability, the scaled tent has derivative of magnitude at most $2q_t$, supported where $z\geq q_t/2$; its derivative is therefore bounded by $4|z|$. The other tent has derivative bounded by $2$. Integration along the interval from $z$ to $z'$ proves (bounds). Each $g(t,\omega,\cdot)$ is compactly supported and nonzero on the negative axis. A nonzero compactly supported function on $\mathbb R$ can be neither convex nor concave.

For $c\in[0,1]$ put \[ Y^c=X+cD,\qquad Z^c=q+cK,\qquad \xi=X_T. \] Where $q>0$, equations (D) give \[ \frac{Z^c}{q}\in[13/15,17/15]\subset[1/2,3/2], \qquad g(t,Z^c)=\sigma q^2+2c\sigma qK. \] Where $q=0$, both $Z^c$ and $g(t,Z^c)$ vanish. Combining (q) and (linear) now yields \[ dY^c_t=Z^c_t\,dB_t-g(t,Z^c_t)\,dt. \] Since $D_T=0$, integration gives (bsde) with the same $\xi$ for every $c$. The identities hold simultaneously for all times by continuity, including $T$, since only finitely many legs are used almost surely.

Finally, $|Z^c|\leq(17/15)q$ and $\sup|Y^c|\leq2a_N+\eta$. Equations (tails)--(maximum) prove (moments). Each stopped exponential is dominated by the integrable random variable $e^{\mu\sup|Y^c|}$, which proves class (D). The initial values $Y^c_0=c\eta d_1$ are distinct. This proves Theorem Theorem.$\square$

Scope.
The generator is random and time-dependent, as permitted in the stated problem. Taking $g(t,y,z)=g(t,z)$ satisfies any positive uniform Lipschitz bound in $y$. The example therefore meets the hypotheses with $\delta=1$ and $C=4$. In higher Brownian dimension, use the first coordinate and replace $g(t,z)$ by $g(t,z_1)$.

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, equation $3.12$, and Problem 5.5. https://arxiv.org/html/2309.06233v2.

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