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CS-636 · computer science · mathematics · Numerical analysis and scientific computing

Threshold-Avoiding Control for Bank Capital Ratios

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Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

For a ratio process $R_t$, the first threshold-crossing time is $\tau=\inf\{t\geq0:R_t<0.08\}$. A control law chooses allowed actions using the information available up to the current time.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The ratio dynamics, admissible controls, almost-sure versus probabilistic avoidance, and delay objective are not present in the supplied question; they must be taken from the cited model.

Source-record limitation: The stochastic ratio dynamics, admissible controls, and optimal-delay criterion of the named model are absent.

Statement recorded in the supplied source.
Does there exist a control law for the stochastic model that prevents a bank's capital adequacy ratio from dropping below $0.08$? If not, determine a control law that postpones the first threshold crossing for as long as possible. [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Find an admissible policy preventing capital-ratio failure, or one that delays the first failure optimally.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30000198 and code OWR-792-014 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-636-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-636

Threshold-Avoiding Control for Bank Capital Ratios

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

For a ratio process $R_t$, the first threshold-crossing time is $\tau=\inf\{t\geq0:R_t<0.08\}$. A control law chooses allowed actions using the information available up to the current time.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The ratio dynamics, admissible controls, almost-sure versus probabilistic avoidance, and delay objective are not present in the supplied question; they must be taken from the cited model.

Source-record limitation: The stochastic ratio dynamics, admissible controls, and optimal-delay criterion of the named model are absent.

Statement recorded in the supplied source.
Does there exist a control law for the stochastic model that prevents a bank's capital adequacy ratio from dropping below $0.08$? If not, determine a control law that postpones the first threshold crossing for as long as possible. [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Find an admissible policy preventing capital-ratio failure, or one that delays the first failure optimally.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30000198 and code OWR-792-014 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

A bounded-drift comparison route with an explicit model gap

The original report explicitly omits the stochastic system for lack of space. Consequently the bank-specific optimization problem cannot be reconstructed from that report alone. We test one restricted mathematical model, not asserted to be the omitted system. After subtracting a threshold, let \[ X_t^a=x+\int_0^t a_s\,ds+\sigma W_t, \qquad x>0,\ \sigma>0, \] where $W$ is standard Brownian motion and admissible adapted measurable drifts satisfy $|a_s|\le M$ almost everywhere, with a fixed $M\ge0$. The constant drift $a_s=M$ is allowed. Define $\tau_a=\inf\{t\ge0:X_t^a<0\}$, with infinity if the set is empty.
Proposition: On a common Brownian path, $\tau_a\le\tau_M$ for every admissible drift. Nevertheless every admissible drift has positive probability of threshold crossing by any fixed time $T>0$; specifically \[ \Pr(\tau_a\le T)\ge \Phi_{\rm N}\!\left(-\frac{x+MT}{\sigma\sqrt T}\right)>0, \] where $\Phi_{\rm N}$ is the standard normal distribution function.
Proof: The drift bound gives $\int_0^t a_sds\le Mt$ for every $t$ outside a single probability-zero exceptional set, so $X_t^a\le X_t^M$ simultaneously by continuity. Each time with $X_t^M<0$ also has $X_t^a<0$; taking infima proves the crossing-time order. Thus $M$ maximizes every survival probability and every bounded increasing function of the crossing time in this model. In particular it is a genuine pathwise optimum, even though an unbounded expected-time objective could be infinite and uninformative. The event $x+MT+\sigma W_T<0$ implies $X_T^a<0$ and hence $\tau_a\le T$. Since $W_T/\sqrt T$ is standard normal, this event has exactly the displayed probability. The Gaussian density is positive everywhere, so its probability is strictly positive for every finite argument. ∎
The comparison requires that control affect only an additively ordered drift with a common fixed diffusion. It can fail as a design argument if the actual control changes volatility, constraints, other state variables, or the set of feasible drifts. The attempted step from this ordered model to the source model stops at precisely those absent equations and admissibility conditions.

Verification and remaining gaps

For the illustrative values $x=\sigma=M=T=1$, the Gaussian lower bound is $\Phi_{\rm N}(-2)\approx0.0227501$. No real capital ratio, regulation or financial policy is assessed. Positive crossing probability is not probability-one eventual crossing; the duplicate source fragment's stronger implication is separately examined in record 30000199. The original bank-specific avoidance and optimal-delay targets remain unresolved.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2005/11. Relevant passages read: 2005/11 p. 639, capital-ratio question and explicit omission of the stochastic equations.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch016/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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