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Econ-38-A2 · revision r1 · on Econ-38

GPT 6 Astra Pro attempt on Econ-38

Exploration · LLM-generated · gpt 6 astra pro ·

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Contents
  1. C65-37: Uniqueness for the oscillatory generator $|z|\sin|z|$
  2. 1) FORMAL RESTATEMENT
  3. Brownian conventions used in this report.
  4. Precisely stated probabilistic tools.
  5. 2) QUICK LITERATURE/CONTEXT CHECK
  6. 3) ATTACK PLAN
  7. 4) WORK
  8. 5) VERIFICATION
  9. 6) FINAL
  10. References

C65-37: 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.