ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A scalar Ornstein–Uhlenbeck SDE satisfies the strong well-posedness hypotheses

Example

Fix $T > 0$ , $θ \geq 0$ , and $σ_{0} \in R ∖ {0}$ . On a complete usual filtered probability space, let $W$ be a filtration-relative standard scalar Brownian motion and let $X_{0}$ be $F_{0}$ -measurable, square-integrable, and independent of the future Brownian increments. Consider $d X_{t} = - θ X_{t} d t + σ_{0} d W_{t}, 0 \leq t \leq T .$ In dimension one, a constant diffusion is nondegenerate when its variance $σ_{0}^{2}$ is positive.

Facts & Assumptions

Given: The data above, the coefficient maps $b (t, x) = - θ x$ and $σ (t, x) = σ_{0}$ , and the Itô SDE conventions of the A page.

[F1]

The global well-posedness contract requires jointly Borel coefficients, uniform global state Lipschitz and linear-growth bounds, a filtration-relative Brownian motion, and an independent $F_{0}$ initial value in $L^{2}$ . Itô SDE data and strong solution

[F2]

Under those hypotheses the SDE has an adapted continuous strong solution, unique up to indistinguishability, with finite $E \sup_{t \leq T} ∣ X_{t} ∣^{2}$ . Global strong well-posedness for Lipschitz Itô SDEs

Verification

1.1givenF1algebra

The coefficients are jointly Borel. For all $x, y$ , $∣ b (t, x) - b (t, y) ∣ + ∣ σ (t, x) - σ (t, y) ∣ = θ ∣ x - y ∣$ , so the uniform Lipschitz constant can be $L = θ$ . Also $∣ b (t, x) ∣ + ∣ σ (t, x) ∣ = θ ∣ x ∣ + ∣ σ_{0} ∣ \leq (θ + ∣ σ_{0} ∣) (1 + ∣ x ∣)$ , so the linear-growth constant can be $K = θ + ∣ σ_{0} ∣$ . Finally $σ_{0}^{2} > 0$ , so the constant scalar diffusion is nondegenerate.

2.1givenF1F2step 1.1∎

The initial value and Brownian motion satisfy the filtration and moment assumptions by the example's hypotheses. Step 1.1 verifies both coefficient bounds, so [F2] gives a unique adapted continuous strong solution on $[0, T]$ and its finite path-sup second moment.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ramon van Handel, Stochastic Calculus, Filtering, and Stochastic Control: Lecture Notes (ACM 217, Spring 2007), Chapter 5.1 (standard reference, not scraped)
A.W. van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Section 7.1 (standard reference, not scraped)