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Econ-38 · economics · Economics

Uniqueness for an oscillatory generator of linear growth

LLM status Solved unverified

Statement

SC-03 |  Stochastic control and dynamic optimization

Published question

Definitions and assumptions

Use the scalar Brownian BSDE conventions above and fix $p>1$. Let $\xi\in L^p(\mathcal F_T)$, and define \[ g(z)=|z|\sin|z|,\qquad z\in\R^d. \] Although $|g(z)|\leq |z|$, this function is not uniformly continuous on $\R^d$. A solution is admissible for the present uniqueness question if the process $(|Y_t|^p)_{0\leq t\leq T}$ is of class (D). This is stronger than merely requiring $Y$ to be of class (D), and is the solution class in the primary source.

Formal question

For every $\xi\in L^p$, are any two admissible solutions $(Y,Z)$ and $(\widetilde Y,\widetilde Z)$ of \[ Y_t=\xi+\int_t^T|Z_s|\sin|Z_s|\,ds -\int_t^T Z_s\cdot dB_s \] necessarily equal in the sense that \[ \Pr(Y_t=\widetilde Y_t\text{ for all }t\in[0,T])=1, \qquad Z=\widetilde Z\quad dt\otimes\Pr\text{-a.e.}? \] A counterexample must provide the same terminal value and two different solutions for which both power transforms are of class (D).

Scope and status

Replacing this generator by a globally Lipschitz function changes the question. Linear growth alone also does not furnish the uniform continuity assumption used by the comparison theorem cited in the source. The terminal moment order $p$ and the corresponding power condition on the process must be matched. This is the uniqueness problem listed in the cited version, without a claim that every later paper has been checked.

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 5.3 and equation (5.2); see Theorem 3.14(iv) for the comparison result whose uniform continuity condition is unavailable. 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-38-A2 · Exploration · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-38

C65-37: Uniqueness for the oscillatory generator $|z|\sin|z|$

Outcome: UNRESOLVED; bounded-terminal and bounded-reference uniqueness 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.

Fix $p>1$. For every $\xi\in L^p({\mathcal F}_T)$, must any two solutions of \[ Y_t=\xi+\int_t^T|Z_s|\sin|Z_s|\,ds-\int_t^TZ_s\,dB_s \] coincide whenever both $|Y|^p$ processes are of class (D)? The generator is exactly $g(z)=|z|\sin|z|$, not a globally Lipschitz approximation. Class (D) is imposed on the $p$th power; merely assuming that $Y$ is of class (D) would change the statement. Equality of $Z$ is only $dt\otimes{\mathbb{P}}$-a.e., as specified in the Brownian conventions.

Stress points are unbounded oscillatory slopes, uncontrolled difference densities, and the fixed moment order $p$.

2) QUICK LITERATURE/CONTEXT CHECK

This is Problem 5.3 in Fan--Hu--Tang [FHT]. Its power class-(D) condition matches the attachment. The comparison theorem referenced there requires a uniform continuity property that this generator lacks. Our bounded-data and bounded-reference arguments below verify their own change-of-measure hypotheses rather than invoking that inapplicable comparison theorem.

3) ATTACK PLAN

Proof routes. Linearize differences and seek a density strong enough for the given $p$; use the linear-growth bound to obtain a bounded-data estimate; compare an arbitrary solution to one with bounded $Z$.

Disproof routes. Test zero and affine terminal values; search pairs of integrands near successive oscillations; test whether localization can preserve terminal coincidence and the power class-(D) requirement simultaneously. The first route is ruled out by the partial theorems; no valid pair for the others is constructed.

Landscape and tools. Radial calculus bounds slopes; oscillatory sequences test uniform continuity; Girsanov removes drifts; Hölder transfers stopping integrability; conditional energy estimates yield BMO; BMO exponentials justify sampling; localization exposes the remaining tail gap.

4) WORK

Lemma (Growth and nonuniform continuity): For all $z,w$, $|g(z)|\le|z|$ and \[ |g(z)-g(w)|\le(1+\max\{|z|,|w|\})|z-w|. \] The function is not uniformly continuous on ${\mathbb{R}}^d$.
Proof: For $f(r)=r\sin r$, $|f'(r)|=|\sin r+r\cos r|\le1+r$. Apply the mean value theorem between $|z|$ and $|w|$ and then the reverse triangle inequality. For nonuniform continuity, take points on a fixed positive unit ray with radii $r_k=2\pi k$ and $s_k=r_k+r_k^{-1}$. Their distances tend to zero, but \[ f(s_k)-f(r_k)=(r_k+r_k^{-1})\sin(r_k^{-1})\longrightarrow1, \] using $\sin t/t\to1$. This contradicts the defining uniform-continuity implication. ∎
Lemma (Bounded densities preserve the needed stopping integrability): Let $|\theta_t|\le M$ predictably and let $d{\mathbb{Q}}=D_Td{\mathbb{P}}$, $D={\mathcal E}(\int\theta\,dB)$. If $\sup_\tau{\mathbb{E}}|X_\tau|^p<\infty$ for one $p>1$, then $X$ is of class (D) under ${\mathbb{Q}}$.
Proof: Novikov's condition holds for every multiple of $\theta$, and ${\mathbb{E}} D_T^q\le\exp(q(q-1)M^2T/2)$ for every $q>1$, by writing $D_T^q$ as the exponential with integrand $q\theta$ times its bounded quadratic-variation correction. Choose $1<r<p$ and put $q=p/(p-r)$. Hölder's inequality gives \[ {\mathbb{E}}_{\mathbb{Q}}|X_\tau|^r\le ({\mathbb{E}} D_T^q)^{1/q}({\mathbb{E}}|X_\tau|^p)^{r/p}. \] The right side is uniformly bounded over stopping times. The $r$th-moment bound implies class (D). ∎
Theorem (Uniqueness relative to a bounded-integrand solution): Suppose one admissible solution $(\widetilde Y,\widetilde Z)$ satisfies $|\widetilde Z_t|\le M$ a.e. for a finite deterministic $M$. Then every other admissible solution with the same terminal value coincides with it.
Proof: For any $|w|\le M$, let $r=|z-w|$. If $r\le1$, the preceding local Lipschitz bound gives $|g(z)-g(w)|\le(M+2)r$. If $r>1$, linear growth gives $|g(z)-g(w)|\le|z|+|w|\le r+2M\le(1+2M)r$. Thus in both cases $|g(z)-g(w)|\le(2M+2)|z-w|$.

Set $\Delta Y=Y-\widetilde Y$, $\Delta Z=Z-\widetilde Z$, and define \[ b_t=\frac{g(Z_t)-g(\widetilde Z_t)}{|\Delta Z_t|^2}\Delta Z_t \] on $\{\Delta Z_t\ne0\}$, zero otherwise. This is predictable and uniformly bounded by $2M+2$. The difference equation under its Girsanov probability is $d\Delta Y=\Delta Z\,dB^{\mathbb{Q}}$, a local martingale. The power class-(D) condition implies a uniformly bounded stopping $p$th moment, so the previous lemma makes $\Delta Y$ of class (D) under ${\mathbb{Q}}$. Its terminal value is zero, hence it vanishes. Quadratic variation gives equality of $Z$. ∎

Corollary (Explicit affine-terminal subclass): For $c\in{\mathbb{R}}$, $z_0\in{\mathbb{R}}^d$, and $\xi=c+z_0\cdot B_T$, the unique admissible solution is \[ Y_t=c+z_0\cdot B_t+(T-t)g(z_0),\qquad Z_t=z_0. \]
Proof: Substitution verifies the equation. The supremum of this $Y$ has finite moments of every finite order by the Brownian maximal moment bound, or by Doob's inequality at any exponent larger than $p$. Thus $|Y|^p$ is dominated by an integrable variable and is of class (D). The preceding theorem applies to its constant integrand. In particular, constant and zero terminal values cannot produce a counterexample in the required class. ∎
Theorem (Bounded-terminal uniqueness): For every bounded terminal value, the original power class-(D) solution class contains at most one solution.
Proof: Take any solution and define $\theta_t=g(Z_t)Z_t/|Z_t|^2$ on $\{Z_t\ne0\}$, zero otherwise. Since $|g(Z)|\le|Z|$, one has $|\theta|\le1$ and $g(Z)=\theta\cdot Z$. Under the associated probability ${\mathbb{Q}}^\theta$, $Y$ is a local martingale. The bounded-density transfer lemma makes it of class (D), so $Y_t={\mathbb{E}}_{{\mathbb{Q}}^\theta}[\xi\mid{\mathcal F}_t]$. If $|\xi|\le K$, this proves $|Y_t|\le K$ for every solution, though the probability may depend on the solution.

For every stopping time $\tau$, apply Itô to $Y^2$ from $\tau$ to localizing times tending to $T$. Using the deterministic bound on $Y$, $|g(Z)|\le|Z|$, and $2K|Z|\le|Z|^2/2+2K^2$, one obtains \[ {\mathbb{E}}\left[\left.\int_\tau^T|Z_t|^2dt\right|{\mathcal F}_\tau\right] \le2K^2+4K^2T. \] In detail, before taking the limit the expectation of the integral of $|Z|^2$ is bounded by $K^2+2K{\mathbb{E}}_\tau\int|Z|$; move half the quadratic term to the left, and then use conditional Fatou. This proves BMO for both solutions' martingale integrands.

For the divided-difference $b$ between two solutions, the local Lipschitz estimate gives $|b|\le1+|Z|+|\widetilde Z|$, so $|b|^2\le3(1+|Z|^2+|\widetilde Z|^2)$. It is therefore BMO. The continuous BMO exponential theorem gives a true density. Under its probability $\Delta Y$ is a bounded local martingale, hence a true martingale closed by zero. This proves equality of $Y$ and then of $Z$. ∎

5) VERIFICATION

The explicit sequence gives generator differences $1.021007$, $1.000211$, $1.00000211$, $1.000000021$ for $k=1,10,100,1000$, while its input distances tend to zero. The script additionally samples 30,000 pairs in dimension three with $|w|\le1$; the maximum divided difference is approximately $1.3855$, below the proved bound $4$. Sampling is only a check; the two-case inequality proves the bound for every pair.

The density exponent accommodates $r\in(1,p)$ without assuming a larger terminal moment. Each solution's separate measure only bounds $Y$; comparison needs its own linearization density, obtained for bounded data from the proved BMO estimate.

A cutoff $|Z|,|\widetilde Z|\le R$ produces a Lipschitz constant of order $R$. Usual exponential estimates can then grow like $e^{cR^2T}$, and a mere polynomial terminal moment does not justify discarding the stopping tail against such a factor. No limiting comparison is asserted without an estimate that actually controls this product.

6) FINAL

{LABEL: UNRESOLVED.}

Strongest checked partial result. Uniqueness holds for every bounded terminal datum and whenever at least one competing solution has a uniformly bounded integrand. All affine Brownian terminal values, including zero, are explicitly solved in the required class.

Exact first gap. For two arbitrary admissible solutions with unbounded $L^p$ terminal data, justify the change of measure and terminal sampling of their difference for the unbounded divided-difference integrand $b$. Neither BMO nor an adequate density moment was derived here from the fixed power class-(D) hypothesis.

Next three moves. (1) Exploit oscillatory cancellation in localized comparison constants. (2) Prove weighted uniform integrability for stopped difference densities at the given $p$. (3) Construct matching-terminal, unbounded-integrand solutions and verify their full stopping-time power class (D).

Likely minimal counterexample, if any. Both competing integrands and the terminal datum must be unbounded. Large-radius oscillatory excursions would still have to satisfy the full power class-(D) condition.

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.

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Econ-38-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-38

Oscillatory BSDE uniqueness: one controlled integrand is sufficient

Problem ID: C65-37. Research attempt; independent review pending.

Target and scope

On the usual $d$-dimensional Brownian basis, fix $p>1$ and $\xi\in L^p$. The target is uniqueness for \[ Y_t=\xi+\int_t^T |Z_s|\sin|Z_s|\,ds-\int_t^T Z_s\cdot dB_s \] among solutions for which $|Y|^p$ is of class (D). Solutions have continuous adapted $Y$, predictable $Z$, and pathwise finite $\int_0^T(|Z|^2+|g(Z)|)dt$. We retain the power class-(D) hypothesis exactly; it is not replaced by class (D) on $Y$.

We prove uniqueness against every admissible competitor whenever one solution has a bounded integrand, and more generally when one integrand has a specified exponential energy moment. This yields complete uniqueness for an explicit class of unbounded Gaussian terminal values. It does not establish that an arbitrary $L^p$ terminal value has such a reference solution. The source question and its missing uniform continuity assumption were checked in [FHT, Problem 5.3 and Theorem 3.14(iv)] on 9 October 2026. The linearization ingredients below are classical; no novelty claim is made.

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.

A secant estimate that uses only one endpoint

Lemma: For $g(z)=|z|\sin|z|$ and all $z,z'\in{\mathbb{R}}^d$, \begin{equation}\label{wsref-0-73-65-63-61-6e-74-62-6f-75-6e-64} |g(z)-g(z')|\leq 2(1+\min\{|z|,|z'|\})|z-z'|. \end{equation} The function $g$ is nevertheless not uniformly continuous on ${\mathbb{R}}^d$.
Proof: Put $m=\min(|z|,|z'|)$ and $d_0=|z-z'|$. If $d_0\geq1$, linear growth and the triangle inequality give \[ |g(z)-g(z')|\leq|z|+|z'|\leq d_0+2m \leq(1+2m)d_0. \] If $d_0<1$, the two radii lie in $[m,m+1]$. The derivative of $r\sin r$ has absolute value at most $1+r$, so the one-dimensional mean-value theorem and $||z|-|z'||\leq d_0$ give a bound $(m+2)d_0$. Both bounds are at most $2(1+m)d_0$. For the final assertion take $z_n=2\pi n e_1$ and $z'_n=(2\pi n+(2\pi n)^{-1})e_1$. Their distance tends to zero, while $g(z_n)=0$ and $g(z'_n)=(2\pi n+(2\pi n)^{-1})\sin((2\pi n)^{-1})\to1$. ∎
Lemma (A density moment and class-(D) transfer): Let $r>1$ and let $\beta$ be predictable with $A=\int_0^T|\beta_s|^2ds<\infty$ almost surely. Put $\kappa_r=2r^2-r$. If ${\mathbb{E}} e^{\kappa_r A}<\infty$, then $D=\mathcal E(\int\beta\,dB)$ is a uniformly integrable martingale, $D_T>0$, and \[ \sup_{\tau\leq T}{\mathbb{E}} D_\tau^r\leq({\mathbb{E}} e^{\kappa_r A})^{1/2}. \] If $p=r/(r-1)$ and $|Y|^p$ is class (D) under ${\mathbb{P}}$, then $Y$ is class (D) under $d{\mathbb{Q}}=D_Td{\mathbb{P}}$.
Proof: For $M=\int\beta\,dB$, the identity \[ D_\tau^r=\mathcal E(2rM)_\tau^{1/2} \exp\left((r^2-r/2)\langle M\rangle_\tau\right) \] and Cauchy--Schwarz imply the moment bound: the expectation of the nonnegative local-martingale exponential on the right is at most one. The bound applies to all localizing times for $D$ and implies uniform integrability of its stopped values. Thus $D$ is a true uniformly integrable martingale, with terminal expectation one. Finite quadratic variation makes its terminal value strictly positive. For the last claim, Hölder's inequality gives \[ {\mathbb{E}}_{\mathbb{Q}}[|Y_\tau|{\mathbf 1}_{\{|Y_\tau|>R\}}] \leq ({\mathbb{E}} D_\tau^r)^{1/r} ({\mathbb{E}}[|Y_\tau|^p{\mathbf 1}_{\{|Y_\tau|>R\}}])^{1/p}. \] The second factor tends to zero uniformly by the assumed class (D) of $|Y|^p$, proving the claim. ∎

Uniqueness against an unrestricted admissible competitor

Theorem: Fix $p>1$, set $r=p/(p-1)$ and $\kappa_r=2r^2-r$. Let $(Y,Z)$ and $(\widetilde Y,\widetilde Z)$ solve the target BSDE with the same terminal value and both power transforms of class (D). If the first solution satisfies \begin{equation}\label{wsref-0-6f-6e-65-65-6e-65-72-67-79} {\mathbb{E}}\exp\left(8\kappa_r\int_0^T|Z_s|^2ds\right)<\infty, \end{equation} then the two solutions are equal. No exponential energy hypothesis on $\widetilde Z$ is required. In particular the conclusion holds if $|Z_t|\leq R$ for a deterministic $R<\infty$ almost everywhere.
Proof: Put $U=Y-\widetilde Y$, $V=Z-\widetilde Z$, and set \[ \beta_t={\mathbf 1}_{\{V_t\ne0\}} \frac{g(Z_t)-g(\widetilde Z_t)}{|V_t|^2}V_t. \] This is predictable and satisfies $g(Z)-g(\widetilde Z)=\beta\cdot V$. By Lemma Lemma, $|\beta|\leq2(1+|Z|)$, hence $\int|\beta|^2\leq8T+8\int|Z|^2$. Hypothesis (oneenergy) verifies Lemma Lemma (A density moment and class-(D) transfer) at the exact conjugate exponent $r$. Thus $D=\mathcal E(\int\beta\,dB)$ defines an equivalent measure ${\mathbb{Q}}$, both $Y$ processes are class (D) under ${\mathbb{Q}}$, and $B^{\mathbb{Q}}=B-\int\beta ds$ is Brownian by Girsanov's theorem. The difference equation reduces to $dU=V\cdot dB^{\mathbb{Q}}$. Its stochastic integral is locally defined because both candidate energies are finite pathwise. A local martingale of class (D) is a true martingale: stop along a localizing sequence and pass to conditional expectations using uniform integrability. Since $U_T=0$, $U$ vanishes. The quadratic variation $\int|V|^2dt$ vanishes as well. Equality at each time extends to indistinguishability by continuity. If $|Z|\leq R$, the expectation in (oneenergy) is bounded by $\exp(8\kappa_rR^2T)$. ∎
Corollary (A complete unbounded-terminal subcase): Let $a:[0,T]\to{\mathbb{R}}^d$ be deterministic, measurable, and essentially bounded, and let $c\in{\mathbb{R}}$. For $\xi=c+\int_0^T a_s\cdot dB_s$, the target BSDE has exactly one solution with $|Y|^p$ of class (D), for every fixed $p>1$. It is \[ Z_t=a_t,\qquad Y_t=c+\int_0^t a_s\cdot dB_s+\int_t^T|a_s|\sin|a_s|\,ds. \]
Proof: The displayed pair has the prescribed terminal value and differentiating its finite-variation term verifies the BSDE. Its pathwise integrability is immediate from bounded $a$. The Gaussian martingale $M_t=\int_0^t a_s\cdot dB_s$ has terminal moments of every order. For any $q>p$, Doob's $L^q$ inequality gives ${\mathbb{E}}\sup_{t\leq T}|M_t|^q<\infty$; the deterministic drift term is bounded by $T\|a\|_\infty$. Therefore $\sup_t|Y_t|^p$ is integrable, and it dominates all stopped powers, proving class (D). The reference integrand is bounded, so Theorem Theorem proves uniqueness against every competitor in the full prescribed solution class. ∎

What remains unresolved

The estimate uses linear growth in a stronger way than the local derivative bound $1+|z|$: a secant is controlled by its smaller endpoint. Nevertheless, for arbitrary $\xi\in L^p$ we have not established a solution whose integrand satisfies (oneenergy). The target's power class-(D) hypothesis does not itself assert such a moment. The counterexample to uniform continuity in Lemma Lemma shows why a bounded generator-to-control ratio cannot simply replace a Lipschitz secant estimate. Thus the unrestricted uniqueness question remains unresolved, while Corollary Corollary (A complete unbounded-terminal subcase) is a fully verified class of unbounded data within it.

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.
  • [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%

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Econ-38-A3 · Full disproof claimed · LLM-generated · gpt-6 astra, claude opus 5.5, gpt-6 astra pro

Nonuniqueness for a BSDE with generator |z| sin |z|

Nonuniqueness for a BSDE with generator\\$|z|
\sin
|z|$

Ji Ho Bae
LawAll AI

10 October 2026

\begin{abstract} We construct two solutions with the same terminal value for the scalar Brownian BSDE with generator $g(z)=|z|\sin|z|$. Both solutions have integrable supremum powers of every finite order. This gives a negative answer to Fan, Hu and Tang's Problem 5.3. \end{abstract}

Fix $T>0$ and the augmented natural filtration of a one-dimensional standard Brownian motion $B$. Fan, Hu and Tang ask whether \begin{equation}\label{wsref-0-62-73-64-65} Y_t=\xi+\int_t^T |Z_s|\sin|Z_s|\,ds-\int_t^T Z_s\,dB_s \end{equation} has a unique solution when $\xi\in L^p$, $p>1$, and $(|Y_t|^p)$ is of class (D) [FHT, Problem 5.3, equation $5.2$]. Here class (D) means that the stopped variables $|Y_\tau|^p$, $\tau\le T$, form a uniformly integrable family.

Theorem: For every $T>0$, there are an $\mathcal F_T$-measurable $\xi\in\bigcap_{1\le p<\infty}L^p$ and two solutions $(Y^+,Z^+)$, $(Y^-,Z^-)$ of (bsde) such that, for every finite $p\ge1$, \[ \mathbb E\sup_{t\le T}|Y_t^\pm|^p+ \mathbb E\left(\int_0^T|Z_t^\pm|^2\,dt\right)^{p/2}<\infty. \] Moreover, $Y_t^+>Y_t^-$ for every $t<T$, almost surely.
Proof: We use Itô's formula, stopped martingale identities and the Burkholder--Davis--Gundy inequality in their usual continuous-martingale forms; see [RY, Chapters IV and IX].
1. A positive process with an integrable inverse-square clock.
Let \begin{equation}\label{wsref-0-62-65-73-73-65-6c} dR_t=dB_t+R_t^{-1}\,dt,\qquad R_0=1, \qquad A_t=\int_0^tR_s^{-2}\,ds. \end{equation} The stopped Picard construction gives a strong solution up to exit from $(0,\infty)$, since the coefficients are locally Lipschitz there. For $n\ge2$, let $\sigma_n$ be its first exit from $(1/n,n)$ and put $\tau_n=T\wedge\sigma_n$. Itô's formula gives \[ d(R_t^{-1})=-R_t^{-2}\,dB_t, \qquad d(R_t^2)=2R_t\,dB_t+3\,dt. \] The stopped integrands are bounded, so $\mathbb E R_{\tau_n}^{-1}=1$ and $\mathbb E R_{\tau_n}^{2}=1+3\mathbb E\tau_n\le1+3T$. Consequently \[ \mathbb P(\sigma_n\le T)\le \frac1n+\frac{1+3T}{n^2}. \] Thus $R$ exists throughout $[0,T]$ and stays strictly positive.

Another application of Itô's formula shows that \[ L_t=R_t^{-1/2}\exp(A_t/8),\qquad dL_t=-\frac{L_t}{2R_t}\,dB_t. \] In particular $\mathbb E L_{\tau_n}=1$. Cauchy--Schwarz and the preceding second-moment bound yield \[ \mathbb E e^{A_{\tau_n}/16} =\mathbb E\left[L_{\tau_n}^{1/2}R_{\tau_n}^{1/4}\right] \le(\mathbb E R_{\tau_n}^{1/2})^{1/2} \le(1+3T)^{1/8}. \] Fatou's lemma therefore gives \begin{equation}\label{wsref-0-63-6c-6f-63-6b} \mathbb E e^{A_T/16}\le(1+3T)^{1/8}<\infty. \end{equation} In particular $A_T$ has moments of every finite order.

2. A nonzero gap that vanishes at the terminal time.
Let $\varphi$ and $\Phi$ denote the standard normal density and distribution function. For $0\le h\le T$, $r>0$, define \begin{equation}\label{wsref-0-75} u(h,r)=\frac1{2\sqrt T}\int_0^h\Phi(-r/\sqrt s)\,ds, \qquad u(0,r)=0. \end{equation} Differentiating for $h,r>0$ gives \[ u_h=\frac1{2\sqrt T}\Phi(-r/\sqrt h),\qquad u_{rr}=\frac1{\sqrt T}\Phi(-r/\sqrt h),\qquad u_r=-\frac1{2\sqrt T}\int_0^h \frac{\varphi(r/\sqrt s)}{\sqrt s}\,ds. \] For the second identity, integrate $\partial_s\Phi(-r/\sqrt s) =r\varphi(r/\sqrt s)/(2s^{3/2})$. These differentiations are justified by exponential decay as $s\downarrow0$. Thus $u_h=\tfrac12u_{rr}$, $-1\le u_r\le0$, and $0\le u\le\sqrt T/4$; the derivative bound follows from $\varphi\le1$ and $h\le T$.

Set \[ D_t=u(T-t,R_t),\qquad q_t=u_r(T-t,R_t), \qquad q_T=0. \] The processes are continuous, $D_T=0$, and $D_t>0$ for $t<T$. Using (bessel) and the heat equation gives \begin{equation}\label{wsref-0-67-61-70} dD_t=\frac{q_t}{R_t}\,dt+q_t\,dB_t. \end{equation} The identity extends to $T$ by continuity: $q$ is bounded and every path of $R$ has a positive minimum on $[0,T]$.

3. Realizing the required secant of the generator.
For $q\in[-1,0]$, write \[ s(q)=\begin{cases}\sin(q/2)/(q/2),&q\ne0,\\1,&q=0,\end{cases} \qquad s_0=2\sin(1/2). \] Then $0<s_0\le s(q)\le1$ and $\cos(q/2)>0$. For $r>0$ put $k(r)=\lceil(2\pi s_0r)^{-1}\rceil$. On the interval \[ I_r=[2\pi k(r)+\pi/2,\,(2k(r)+1)\pi] \] the function \[ F(a,q)=\sin a\cos(q/2)+a\cos a\,s(q) \] is strictly decreasing, because \[ F_a=(\cos(q/2)+s(q))\cos a-a\,s(q)\sin a<0. \] Its left endpoint value is $\cos(q/2)>0$, and its right endpoint value is at most $-1/r$. There is therefore a unique $a=a(r,q)\in I_r$ with $F(a,q)=-1/r$. This choice is Borel: on each Borel set where $k(r)$ is constant, the strictly monotone inverse depends continuously on $(r,q)$. It satisfies \begin{equation}\label{wsref-0-61-62-6f-75-6e-64} 0<a(r,q)\le \frac1{s_0r}+3\pi. \end{equation} Both $a+q/2$ and $a-q/2$ are positive. The addition formula for sine now gives the exact identity \begin{equation}\label{wsref-0-73-65-63-61-6e-74} g(a+q/2)-g(a-q/2)=qF(a,q)=-q/r. \end{equation} The formula also holds at $q=0$.

Define predictable processes \[ Z_t^\pm=a(R_t,q_t)\pm q_t/2, \qquad Y_t^-=-\int_0^t g(Z_s^-)\,ds+\int_0^t Z_s^-\,dB_s, \qquad Y_t^+=Y_t^-+D_t. \] By (abound), $|Z_t^\pm|\le C(1+R_t^{-1})$ for a fixed constant $C$. Hence (clock) implies \[ \mathbb E\left(\int_0^T|Z_t^\pm|^2\,dt\right)^{p/2}<\infty \qquad(1\le p<\infty). \] Since $|g(z)|\le|z|$, Cauchy--Schwarz controls the drift integral, and the Burkholder--Davis--Gundy inequality controls the stochastic integral. Thus $\mathbb E\sup_{t\le T}|Y_t^-|^p<\infty$ for every finite $p\ge1$. The same holds for $Y^+$ because $D$ is bounded.

Finally, (gap) and (secant) give \[ dY_t^+=-g(Z_t^-)\,dt+Z_t^-\,dB_t +\frac{q_t}{R_t}\,dt+q_t\,dB_t =-g(Z_t^+)\,dt+Z_t^+\,dB_t. \] Taking $\xi=Y_T^-=Y_T^+$ proves (bsde) for both pairs. Their initial values differ by $u(T,1)>0$. For each $p\ge1$, the integrable variable $\sup_{t\le T}|Y_t^\pm|^p$ dominates all stopped powers, establishing the required class-(D) condition. ∎

The same counterexample works for any Brownian dimension $d\ge1$: use its first coordinate for $B$ and replace each scalar $Z^\pm$ by $(Z^\pm,0,\ldots,0)$.

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. Problem 5.3 and equation $5.2$. https://arxiv.org/html/2309.06233v2#S5.
  • [RY] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer, 1999. Chapters IV and IX. https://doi.org/10.1007/978-3-662-06400-9.

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