TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge 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.

Strong order one for scalar and commutative-noise Milstein schemes

Statement

Assume AC. Let $d, m \geq 1$ be finite, let $W$ be an $m$ -dimensional Brownian motion relative to a filtration satisfying Continuous Brownian Ito processes, and let $X$ solve $d X_{t} = b (X_{t}) d t + \sum_{j = 1}^{m} σ_{j} (X_{t}) d W_{t}^{j}, X_{0} \in L^{2} (R^{d}) .$ Assume $b, σ_{1}, \dots, σ_{m} \in C_{b}^{3} (R^{d})$ as defined in Milstein updates, iterated Brownian integrals, and commutative noise. Put $L_{i} f = D f σ_{i}$ and assume commutative noise $L_{i} σ_{j} = L_{j} σ_{i}$ for all $i, j$ . On the uniform grid $t_{n} = n h$ , $h = T / N \leq 1$ , initialize $Y_{0} = X_{0}$ and use the product-increment Milstein update in the Definition. Then there is a constant $C_{T}$ , independent of $N$ , such that $\max_{0 \leq n \leq N} E ∣ Y_{n} - X_{t_{n}} ∣^{2} \leq C_{T} h^{2} .$ Thus the grid root-mean-square error is $O (h)$ on $[0, T]$ . For $m = 1$ the commutativity condition is automatic and the scheme is the scalar Milstein update. No claim is made for a noncommuting-noise scheme without an area sampler satisfying a separately proved approximation bound.

Facts & Assumptions

Given: AC, the SDE and coefficient hypotheses, a finite horizon $T > 0$ , an integer $N \geq 1$ with $h = T / N \leq 1$ , the exact solution $X$ , and the Milstein grid process $Y$ .

[L1]

Milstein updates, iterated Brownian integrals, and commutative noise: the commutative update is $Y_{n + 1} = Y_{n} + b (Y_{n}) h + \sum_{j} σ_{j} (Y_{n}) Δ W_{n}^{j} + \frac{1}{2} \sum_{i, j} L_{i} σ_{j} (Y_{n}) (Δ W_{n}^{i} Δ W_{n}^{j} - δ_{i j} h)$ .

[L2]

Product identity and conditional moments for iterated Brownian integrals: $I_{i j, n} + I_{j i, n} = Δ W_{n}^{i} Δ W_{n}^{j} - δ_{i j} h$ , so the commutative double-integral term equals the product-increment term.

[L3]

Conditional local defect bounds for the Milstein Taylor increment: on each step, $X_{t_{n + 1}} = X_{t_{n}} + b (X_{t_{n}}) h + \sum_{j} σ_{j} (X_{t_{n}}) Δ W_{n}^{j} + \sum_{i, j} L_{i} σ_{j} (X_{t_{n}}) I_{i j, n} + R_{n + 1}$ , with $∣ E [R_{n + 1} ∣ F_{t_{n}}] ∣ \leq C h^{2}$ and $E [∣ R_{n + 1} ∣^{2} ∣ F_{t_{n}}] \leq C h^{3}$ .

[L4]

Discrete stochastic local-to-global RMS stability bound: if a grid error has an adapted Lipschitz drift increment, martingale-difference terms with variance $O (h ∣ e_{n} ∣^{2})$ , a bounded local mean $O (h^{2})$ , and a centered local remainder with second moment $O (h^{3})$ , then its terminal and grid root-mean-square error is $O (h)$ .

[F1]

Continuous Brownian Ito processes: the Brownian increments after $t_{n}$ are independent of $F_{t_{n}}$ , and the SDE solution has the stated adapted continuous drift-plus-It^o decomposition.

[A1]

The Axiom of Choice: AC is inherited from the exact Brownian and It^o suppliers [F1], [L2], and [L3]; no choice is made in the error recurrence.

Proof

Given: The assumptions above.

1.1F1L1L2L3given

Define $e_{n} = Y_{n} - X_{t_{n}}$ . Apply [L3] to the exact solution on $[t_{n}, t_{n + 1}]$ . Using [L2] and commutativity, the exact Taylor increment has the same product form as the update [L1], except that its coefficients are evaluated at $X_{t_{n}}$ and it includes the remainder $R_{n + 1}$ .

1.2F1L1L3step 1.1

Subtract the exact update from [L1]. With $u_{n} = b (Y_{n}) - b (X_{t_{n}})$ , set $M_{n + 1} = \sum_{j} (σ_{j} (Y_{n}) - σ_{j} (X_{t_{n}})) Δ W_{n}^{j} + \frac{1}{2} \sum_{i, j} (L_{i} σ_{j} (Y_{n}) - L_{i} σ_{j} (X_{t_{n}})) (Δ W_{n}^{i} Δ W_{n}^{j} - δ_{i j} h) .$ Then $e_{n + 1} = e_{n} + h u_{n} + M_{n + 1} - R_{n + 1}$ . The increments after $t_{n}$ are independent of $F_{t_{n}}$ by [F1]; their first moments vanish, and each centered quadratic factor has mean zero. Hence $M_{n + 1}$ is orthogonal to every square-integrable $F_{t_{n}}$ -measurable vector.

2.1F1L1step 1.2algebra

The derivatives of $b$ are bounded, so $∣ u_{n} ∣ \leq C ∣ e_{n} ∣$ . Also each $L_{i} σ_{j}$ is globally Lipschitz: its derivative is a sum of products of first and second derivatives of the diffusion fields, all bounded by [L1]. The linear Gaussian term in $M_{n + 1}$ has conditional second moment at most $C h ∣ e_{n} ∣^{2}$ . For the quadratic term, $E [(Δ W_{i} Δ W_{j} - δ_{i j} h)^{2} ∣ F_{t_{n}}] \leq 2 h^{2}$ by independent centered normal coordinates; the finite sum and the Lipschitz bound therefore give second moment at most $C h^{2} ∣ e_{n} ∣^{2} \leq C h ∣ e_{n} ∣^{2}$ . Thus $E ∣ M_{n + 1} ∣^{2} \leq C h E ∣ e_{n} ∣^{2}$ .

2.2L3L4step 1.2algebra

Let $r_{n} = E [R_{n + 1} ∣ F_{t_{n}}]$ and $Q_{n + 1} = - (R_{n + 1} - r_{n})$ . By [L3], $∣ r_{n} ∣ \leq C h^{2}$ , $E ∣ R_{n + 1} ∣^{2} \leq C h^{3}$ , and hence $E ∣ Q_{n + 1} ∣^{2} \leq 2 E ∣ R_{n + 1} ∣^{2} + 2 E ∣ r_{n} ∣^{2} \leq C h^{3}$ because $h \leq 1$ . The defining event-integral property of conditional expectation makes $Q_{n + 1}$ orthogonal to every square-integrable $F_{t_{n}}$ -measurable vector. Therefore the recurrence in step 1.2 has the form required by [L4], with local mean $- r_{n}$ and centered terms $M_{n + 1}, Q_{n + 1}$ .

3.1A1L4step 2.2given∎

Initially $e_{0} = 0$ . For each $n \leq N$ , apply [L4] to the recurrence restricted to the first $n$ steps; its constants are the same because $n h \leq T$ . Thus $\max_{0 \leq n \leq N} E ∣ e_{n} ∣^{2} \leq C_{T} h^{2}$ , and taking square roots gives the stated grid root-mean-square order one. When $m = 1$ , the only commutativity equation is $L_{1} σ_{1} = L_{1} σ_{1}$ ; if all diffusion fields vanish, the stochastic terms and their bounds vanish and the same proof reduces to the deterministic recurrence. The argument uses AC only through [A1].

Depends on

Used by

Dependency tree · two levels

28 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

Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, Sections 10.3 and 10.6, printed pp. 345-351 and 356-365 (standard reference, not scraped)
Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review 43 (2001), Section 6 (standard reference, not scraped)
M. Tretyakov, Lecture Notes for a Mini-Course on Numerical Methods for SDEs, Section 6.4 (standard reference, not scraped)