Pipeline-generated

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.

Euler–Maruyama Strong and Weak Convergence

1 · Prerequisites

2 · Summary

This page separates three Euler–Maruyama error contracts. Under global state Lipschitz, linear growth, a half-Hölder time modulus and only a finite second moment at the start, the continuous interpolation has mean-square supremum error $O (h)$ , hence root-mean-square error $O (h^{1 / 2})$ . A separate autonomous Higham–Mao–Stuart theorem gives the same rate under its one-sided-Lipschitz, polynomial-local-Lipschitz and all-moment premises. Weak order one is proved conditionally on explicit backward Kolmogorov regularity, mixed derivatives and the $p \geq 2 q + 8$ moment bound. Nested fine/coarse Brownian increments then give correction variance $O (h)$ and cost $O (h^{- 1})$ ; the bias exponent is $1 / 2$ from the strong contract and improves to $1$ only under the weak contract. No theorem here bounds learned-drift or learned-score bias.

3 · Logical flowchart

4 · Definitions, theorems and proofs

AlgorithmAlgorithm: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Autonomous Euler–Maruyama scheme and interpolation

Definition

Assume AC. Fix finite integers $d, m \geq 1$ , $T > 0$ , and $N \geq 1$ such that $h = T / N \leq 1$ . Work on a filtered probability space whose filtration is right-continuous and contains every $P$ -null subset of $F$ in $F_{0}$ . Let $W$ be an adapted standard $m$ -dimensional Brownian motion whose increments after time $s$ are independent of $F_{s}$ . Let $X_{0}$ be finite, $F_{0}$ -measurable, and independent of $σ (W_{t} - W_{0} : 0 \leq t \leq T)$ . Let $f : R^{d} \to R^{d}$ and $g : R^{d} \to R^{d \times m}$ be $C^{1}$ . Assume finite constants $μ, c, D \geq 0$ and an integer $q \geq 1$ satisfy, for all $x, y \in R^{d}$ , $⟨ x - y, f (x) - f (y) ⟩ \leq μ ∣ x - y ∣^{2}, ∥ g (x) - g (y) ∥_{F}^{2} \leq c ∣ x - y ∣^{2},$ $∣ f (x) - f (y) ∣^{2} \leq D (1 + ∣ x ∣^{q} + ∣ y ∣^{q}) ∣ x - y ∣^{2} .$ Here $∣ \cdot ∣$ is the Euclidean norm and $∥ \cdot ∥_{F}$ is the Frobenius norm.

Set $t_{k} = k h$ and construct the grid approximation recursively by $Y_{0} = X_{0}, Y_{k + 1} = Y_{k} + f (Y_{k}) h + g (Y_{k}) (W_{t_{k + 1}} - W_{t_{k}}), 0 \leq k < N .$ The output also includes the left-endpoint step process ${\overset{ˉ}{Y}}_{t} = Y_{k} (t_{k} \leq t < t_{k + 1}), {\overset{ˉ}{Y}}_{T} = Y_{N},$ and the continuous interpolation $Y_{t} = Y_{k} + f (Y_{k}) (t - t_{k}) + g (Y_{k}) (W_{t} - W_{t_{k}}) (t_{k} \leq t \leq t_{k + 1}) .$ The recursion makes the two interpolation formulas agree at every shared endpoint. The finite recursion is defined pathwise: $C^{1}$ coefficients are finite-valued, each $Y_{k}$ is finite almost surely by induction, and Brownian paths are continuous. This algorithm defines no moment or convergence claim.

LemmaStatement: AI-adaptedProof: AI-generatedOpen item page →

An L1 maximal bound for finite-energy Brownian Itô integrals

Statement

Assume AC. Work on a complete filtered probability space $(Ω, F, (F_{t})_{0 \leq t \leq T}, P)$ whose filtration is right-continuous and contains every $P$ -null subset of $F$ in $F_{0}$ . Let $m \geq 1$ be finite, let $W = (W^{1}, \dots, W^{m})$ be an adapted standard $m$ -dimensional Brownian motion $d$ -dimensional Brownian motion whose increments after time $s$ are independent of $F_{s}$ , and let $H = (H^{1}, \dots, H^{m})$ be a predictable row-vector process Progressively measurable and predictable processes with $E \int_{0}^{T} ∥ H_{s} ∥_{2}^{2} d s < \infty$ . For $M_{t} = \sum_{j = 1}^{m} \int_{0}^{t} H_{s}^{j} d W_{s}^{j},$ the continuous version satisfies $E \sup_{0 \leq t \leq T} ∣ M_{t} ∣ \leq 16 \sqrt{m} E {(\int_{0}^{T} ∥ H_{s} ∥_{2}^{2} d s)}^{1 / 2} .$

Facts & Assumptions

Given: AC; the complete usual filtered probability space, adapted Brownian motion and increment-independence hypotheses above; and the predictable finite-energy integrand.

[F1]

For a real Brownian Itô integral $N_{t} = \int_{0}^{t} K_{s} d B_{s}$ with finite energy and Brownian increments independent of the current filtration, $E \sup_{t \leq T} ∣ N_{t} ∣^{2} \leq 4 E \int_{0}^{T} K_{s}^{2} d s$ . Doob maximal bound for the Ito integral

[F2]

Stopping a finite-energy Itô integral at a stopping time agrees, up to indistinguishability, with integrating the predictable stopped integrand. Stopping an Ito integral

[F3]

The coordinates of a standard finite-dimensional Brownian motion are independent standard one-dimensional Brownian motions. $d$ -dimensional Brownian motion

[F4]

The predictable sigma-algebra is generated by rectangles $(s, u] \times C$ with $C \in F_{s}$ and by the time-zero sets; for any stopping time $ρ$ , the indicator process $1_{[0, ρ]}$ is predictable. Progressively measurable and predictable processes Continuous-time stopping times and stopped sigma-algebras Progressively measurable and predictable processes

[F5]

The finite-energy Itô integral is an isometry from predictable $L^{2}$ into mean-zero $L^{2}$ random variables. Ito isometry and linearity in predictable L2

[F6]

An elementary predictable integrand has integral equal to its defining finite sum over Brownian increments. Ito integral of an elementary predictable process

[F7]

Bounded elementary predictable integrands are dense in predictable $L^{2} ([0, T] \times Ω)$ . Density of elementary predictable processes in predictable L2

[F8]

For a nonnegative measurable function, the integral over a product space equals either iterated integral; applied to $1_{{(λ, ω) : λ < Z (ω)}}$ , this gives $E Z = \int_{0}^{\infty} P (Z > λ) d λ$ . Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F9]

A map $ρ$ is a stopping time when ${ρ \leq t} \in F_{t}$ for every $t \geq 0$ , and $A \in F_{ρ}$ when $A \cap {ρ \leq t} \in F_{t}$ for every $t \geq 0$ . Continuous-time stopping times and stopped sigma-algebras

[F10]

For a continuous adapted real process, its supremum on a compact interval equals its supremum over rational times in the interval together with the right endpoint, and is measurable in the terminal sigma-algebra. The rationals embed densely in the reals

[F11]

Dominated convergence applies to the product-measure limit below: the squared difference of the stopped-interval indicators times $H^{2}$ tends to zero almost everywhere and is bounded by the integrable function $H^{2}$ . Dominated convergence

[A1]

AC is stated because the Brownian, Itô-integral, stopping and maximal-inequality interfaces in [F1]–[F4] adopt it. The Axiom of Choice

Proof

1.1F1F4given

If $T = 0$ , both sides vanish. Henceforth $T > 0$ . For stopping-time notation only, extend the filtration by $F_{t} = F_{T}$ for $t > T$ ; it remains complete and right-continuous, and the predictable sigma-algebra on $[0, T]$ is unchanged by [F4]. All Brownian integrals and estimates below are on $[0, T]$ ; no Brownian property for this extension after $T$ is asserted or used. First take $m = 1$ and write $B = W^{1}$ . On the common null exceptional set, set $M$ and $Q$ to zero; completeness keeps them adapted and makes their paths continuous everywhere. This changes no integral class. Put $Q_{t} = \int_{0}^{t} H_{s}^{2} d s$ and $M^{*} = \sup_{t \leq T} ∣ M_{t} ∣$ . The continuous version makes $M^{*}$ measurable, and [F1] gives $E (M^{*})^{2} \leq 4 E Q_{T} < \infty$ , so $E M^{*} < \infty$ .

1.2givenF4F9F10

Fix $λ > 0$ . Let $τ$ be the first time $∣ M_{t} ∣ \geq λ$ , capped at $T$ , and put $A_{λ} = {M^{*} \geq λ}$ . Continuity and adaptation make $τ$ a stopping time by [F9]–[F10], and $A_{λ} \in F_{τ}$ : for each $t < T$ , $A_{λ} \cap {τ \leq t} = {\sup_{s \leq t} ∣ M_{s} ∣ \geq λ}$ , whose supremum is measurable by [F10]; for $t \geq T$ , the intersection is $A_{λ} \in F_{T} \subseteq F_{t}$ by the same terminal supremum fact. Let $R_{t} = Q_{t} - Q_{t \land τ}$ ; this is continuous and adapted. Let $σ$ be its first hitting time of $δ^{2} λ^{2}$ , capped at $T$ ; it is a stopping time by [F9]–[F10], where $0 < δ \leq 1 / 4$ . The set $A_{λ} \cap {τ < s}$ is the countable union, over rational $u > 0$ , of $(u, \infty) \times (A_{λ} \cap {τ \leq u})$ . Each second factor belongs to $F_{u}$ by $A_{λ} \in F_{τ}$ . Thus it is predictable; intersecting with ${s \leq σ}$ preserves predictability by [F4].

2.1F1F2F5F6F7F8F11step 1.2algebra

Define $N_{t} = \int_{0}^{t} 1_{A_{λ}} 1_{(τ, σ]} (s) H_{s} d B_{s} .$ For each $n$ , let $G_{n} = {j T 2^{- n} : 0 \leq j \leq 2^{n}}$ and round $τ, σ$ upward to their least grid points $τ_{n}, σ_{n} \in G_{n}$ . They are stopping times because ${τ_{n} \leq t} = ⋃_{q \in G_{n}, q \leq t} {τ \leq q} \in F_{t}$ , and likewise for $σ_{n}$ ; monotonicity of the ceiling map gives $τ_{n} \leq σ_{n}$ and $τ_{n} ↓ τ$ , $σ_{n} ↓ σ$ . Since $A_{λ} \in F_{τ}$ and $τ \leq τ_{n}$ , one has $A_{λ} \in F_{τ_{n}}$ : for every $t$ , $A_{λ} \cap {τ_{n} \leq t} = (A_{λ} \cap {τ \leq t}) \cap {τ_{n} \leq t} \in F_{t}$ . For an elementary predictable integrand on a refinement of this grid, the coefficient on $(t_{k}, t_{k + 1}]$ after multiplying by $1_{A_{λ}} 1_{(τ_{n}, σ_{n}]}$ is measurable at $t_{k}$ : the active event ${τ_{n} \leq t_{k} < σ_{n}}$ is in $F_{t_{k}}$ , and $A_{λ} \cap {τ_{n} \leq t_{k}}$ is in $F_{t_{k}}$ . Thus the finite-sum definition [F6] gives $\int_{0}^{t} 1_{A_{λ}} 1_{(τ_{n}, σ_{n}]} H_{s} d B_{s} = 1_{A_{λ}} (M_{t \land σ_{n}} - M_{t \land τ_{n}})$ for elementary $H$ . For general predictable $H$ , approximate in predictable $L^{2}$ by elementary $H^{k}$ using [F7]. The isometry [F5] passes the left side to its limit in $L^{2} (P)$ , and [F1] applied to $H^{k} - H$ gives $E \sup_{t \leq T} ∣ M_{t}^{k} - M_{t} ∣^{2} \to 0$ , passing the stopped evaluations on the right to their limits for this fixed grid. As $n \to \infty$ , the interval indicators converge outside the graphs of $τ$ and $σ$ , which are $d t \otimes P$ -null by Tonelli [F8]. Dominated convergence [F11] with majorant $H^{2}$ and the isometry [F5] pass the identity to $(τ, σ]$ . Continuity gives convergence of the right side to $1_{A_{λ}} (M_{t \land σ} - M_{t \land τ})$ almost surely for each fixed $t$ , proving the required stopped-integral identity entirely on $[0, T]$ . Both sides have continuous versions, so their equality at the countable rational times and continuity give equality up to indistinguishability. The accumulated energy of the integrand is at most $δ^{2} λ^{2} 1_{A_{λ}}$ on every path. Thus [F1] gives $E \sup_{t \leq T} ∣ N_{t} ∣^{2} \leq 4 δ^{2} λ^{2} P (A_{λ}) .$ On ${M^{*} > 2 λ, Q_{T}^{1 / 2} < δ λ}$ the threshold defining $σ$ is not reached before $T$ , and after the first hit of $λ$ the continuous path moves by more than $λ$ . Hence $\sup_{t \leq T} ∣ N_{t} ∣ > λ$ on that event. Markov's inequality and the preceding display yield $P (M^{*} > 2 λ, Q_{T}^{1 / 2} < δ λ) \leq 4 δ^{2} P (M^{*} \geq λ) .$

3.1F8step 1.1step 2.1algebra

Split the event ${M^{*} > 2 λ}$ according to whether $Q_{T}^{1 / 2} \geq δ λ$ . The bound in step 2.1 gives $P (M^{*} > 2 λ) \leq P (Q_{T}^{1 / 2} \geq δ λ) + 4 δ^{2} P (M^{*} \geq λ) .$ For any nonnegative $Z$ , the integrals of $1_{{λ < Z}}$ and $1_{{λ \leq Z}}$ over $λ > 0$ agree pathwise, since they differ at at most one point. Integrating this inequality over $λ > 0$ and applying Tonelli [F8] to $1_{{(λ, ω) : λ < M^{*} (ω)}}$ and $1_{{(λ, ω) : λ < Q_{T} (ω)^{1 / 2}}}$ gives $E M^{*} \leq \frac{2}{δ} E Q_{T}^{1 / 2} + 8 δ^{2} E M^{*} .$ With $δ = 1 / 4$ , the last coefficient is $1 / 2$ , so $E M^{*} \leq 16 E Q_{T}^{1 / 2}$ .

4.1A1F3step 3.1algebra∎

For general finite $m$ , write $M = \sum_{j = 1}^{m} M^{j}$ with $M_{t}^{j} = \int_{0}^{t} H_{s}^{j} d W_{s}^{j}$ . Apply the scalar result of step 3.1 to each coordinate and then the finite-sum Cauchy–Schwarz inequality pathwise: $E \sup_{t \leq T} ∣ M_{t} ∣ \leq 16 \sum_{j = 1}^{m} E {(\int_{0}^{T} ∣ H_{s}^{j} ∣^{2} d s)}^{1 / 2} \leq 16 \sqrt{m} E {(\int_{0}^{T} \sum_{j = 1}^{m} ∣ H_{s}^{j} ∣^{2} d s)}^{1 / 2} .$ This is the claimed estimate. AC enters only through [A1] and the cited Brownian and Itô interfaces.

Source notes

The proof is a local good-lambda derivation from the published $L^{2}$ Doob maximal bound and a finite-grid proof of the stopped-integral identity on $[0, T]$ . It supplies the $L^{1}$ maximal estimate used in the H-M-S energy proof; no general BDG theorem is imported without proof.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

All-moment Euler interpolation increment bound under autonomous hypotheses

Statement

Assume the Axiom of Choice, let $d, m \geq 1$ be finite, and fix $T > 0$ . Work on a complete filtered probability space whose filtration is right-continuous and contains every $P$ -null subset of $F$ in $F_{0}$ . Let $W$ be an adapted standard $m$ -dimensional Brownian motion whose increments after time $s$ are independent of $F_{s}$ . Let $X_{0}$ be finite and $F_{0}$ -measurable, and independent of $σ (W_{t} - W_{0} : 0 \leq t \leq T)$ . Use the Euclidean state norm and Frobenius matrix norm. Let $f : R^{d} \to R^{d}$ and $g : R^{d} \to R^{d \times m}$ be autonomous $C^{1}$ coefficients. Assume finite constants $μ, c, D \geq 0$ and an integer $q \geq 1$ satisfy, for all $x, y \in R^{d}$ , $⟨ x - y, f (x) - f (y) ⟩ \leq μ ∣ x - y ∣^{2}, ∥ g (x) - g (y) ∥_{F}^{2} \leq c ∣ x - y ∣^{2},$ $∣ f (x) - f (y) ∣^{2} \leq D (1 + ∣ x ∣^{q} + ∣ y ∣^{q}) ∣ x - y ∣^{2} .$ For each $0 < h \leq 1$ with $N = T / h$ a positive integer, let $Y^{h}$ be the continuous Euler–Maruyama interpolation and ${\overset{ˉ}{Y}}^{h}$ its left-endpoint step process from Autonomous Euler–Maruyama scheme and interpolation, using this same $W$ and $X_{0}$ . Assume that for every finite $p \geq 1$ , $\sup_{0 < h \leq 1, T / h \in N} (E \sup_{t \leq T} ∣ Y_{t}^{h} ∣^{p} + E \sup_{t \leq T} ∣ {\overset{ˉ}{Y}}_{t}^{h} ∣^{p}) < \infty .$ Then for every even integer $r \geq 2$ there is a finite constant $C_{r}$ , independent of $h$ , such that $\sup_{0 \leq t \leq T} E ∣ Y_{t}^{h} - {\overset{ˉ}{Y}}_{t}^{h} ∣^{r} \leq C_{r} h^{r / 2} .$ The premise is an all-moment premise for this autonomous coefficient class; it does not follow from the finite- $L^{2}$ moment bound alone.

Facts & Assumptions

Given: AC; finite dimensions $d, m$ ; $T > 0$ ; the stated complete right-continuous filtered space, adapted Brownian motion and initial value; the $C^{1}$ , one-sided-Lipschitz, diffusion-Lipschitz and polynomial drift inequalities in the Statement; $0 < h \leq 1$ with $T / h \in N$ ; and the uniform all-moment bounds stated there.

[F1]

On $[t_{k}, t_{k + 1}]$ , the continuous Euler interpolation is $Y_{t} = Y_{k} + f (Y_{k}) (t - t_{k}) + g (Y_{k}) (W_{t} - W_{t_{k}})$ , while ${\overset{ˉ}{Y}}_{t} = Y_{k}$ on $[t_{k}, t_{k + 1})$ and $Y_{T} = {\overset{ˉ}{Y}}_{T} = Y_{N}$ . Autonomous Euler–Maruyama scheme and interpolation

[F2]

If a Brownian coordinate has increment law $N (0, ∣ t - s ∣)$ , then for every integer $n \geq 1$ its $2 n$ th absolute moment is $(2 n - 1)!! ∣ t - s ∣^{n} < \infty$ . Gaussian even moments for Brownian increments

[F3]

A standard finite-dimensional Brownian motion has independent coordinates, each a standard one-dimensional Brownian motion; its vector increments have law $N_{m} (0, (t - s) I_{m})$ . $d$ -dimensional Brownian motion

[A1]

AC is declared because the deterministic Brownian-increment and Gaussian-moment interfaces used in [F2] and [F3] assume it. The Axiom of Choice

Proof

1.1F1algebra

Fix $t \in [t_{k}, t_{k + 1})$ and put $r_{t} = Y_{t}^{h} - {\overset{ˉ}{Y}}_{t}^{h}$ and $δ = t - t_{k}$ . By [F1], $r_{t} = δ f (Y_{k}^{h}) + g (Y_{k}^{h}) (W_{t} - W_{t_{k}}) .$ For even $r \geq 2$ , the finite-dimensional power inequality gives $∣ r_{t} ∣^{r} \leq 2^{r - 1} (δ^{r} ∣ f (Y_{k}^{h}) ∣^{r} + ∥ g (Y_{k}^{h}) ∥_{F}^{r} ∣ W_{t} - W_{t_{k}} ∣^{r}) .$

2.1givenalgebra

Taking $y = 0$ in the coefficient bounds gives $∣ f (x) ∣^{2} \leq 2 ∣ f (0) ∣^{2} + 2 D (1 + ∣ x ∣^{q}) ∣ x ∣^{2}, ∥ g (x) ∥_{F}^{2} \leq 2 ∥ g (0) ∥_{F}^{2} + 2 c ∣ x ∣^{2} .$ Thus $∣ f (x) ∣^{r} \leq C_{r} (1 + ∣ x ∣^{(q + 2) r / 2})$ and $∥ g (x) ∥_{F}^{r} \leq C_{r} (1 + ∣ x ∣^{r})$ . The uniform all-moment premise therefore bounds both coefficient moments in step 1.1 uniformly over $h$ and $t$ .

2.2F2F3given

Each coordinate of $W_{t} - W_{t_{k}}$ has the Gaussian law in [F3], so [F2] and the finite-dimensional inequality $∣ z ∣^{r} \leq C_{r, m} \sum_{j = 1}^{m} ∣ z_{j} ∣^{r}$ give $E ∣ W_{t} - W_{t_{k}} ∣^{r} \leq C_{r, m} δ^{r / 2}$ . Since $Y_{k}^{h}$ is $F_{t_{k}}$ -measurable and Brownian increments are independent of $F_{t_{k}}$ , conditioning in step 1.1 separates the coefficient factor from this Gaussian moment.

3.1A1F1step 1.1step 2.1step 2.2∎

Using $0 \leq δ \leq h \leq 1$ , so $δ^{r} \leq δ^{r / 2}$ , steps 1.1, 2.1 and 2.2 yield $E ∣ Y_{t}^{h} - {\overset{ˉ}{Y}}_{t}^{h} ∣^{r} \leq C_{r} h^{r / 2}$ uniformly for $t < T$ . At grid points the increment is zero by [F1], including $t = T$ . The constants depend on $r, T, d, m, c, D, q, f (0), g (0)$ and the stated moment bounds, but not on $h$ . AC is inherited only through [A1] and the declared Gaussian interfaces; no arbitrary-index choice is used in the estimate.

Source notes

Higham–Mao–Stuart, Lemma 4.3 (printed p. 1055), proves this interpolation increment estimate under its autonomous coefficient and all-moment hypotheses. Särkkä–Solin §8.2, Algorithm 8.1 (printed p. 132), specifies the Euler–Maruyama recursion; the estimate above is proved here under the stated all-moment premise.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Euler–Maruyama strong rate under Higham–Mao–Stuart hypotheses

Statement

Assume AC. Let $d, m \geq 1$ be finite, fix $T > 0$ , and work on a complete filtered probability space whose filtration is right-continuous and contains every $P$ -null subset of $F$ in $F_{0}$ . Let $W = (W_{t})_{0 \leq t \leq T}$ be an adapted standard $m$ -dimensional Brownian motion whose increments are independent of the current filtration. Let $X_{0}$ be finite and $F_{0}$ -measurable, independent of $σ (W_{t} - W_{0} : 0 \leq t \leq T)$ . Use Euclidean state norms and the Frobenius matrix norm. Consider the autonomous SDE $X_{t} = X_{0} + \int_{0}^{t} f (X_{s}) d s + \int_{0}^{t} g (X_{s}) d W_{s}, X_{t} \in R^{d},$ where $f : R^{d} \to R^{d}$ and $g : R^{d} \to R^{d \times m}$ are $C^{1}$ . Assume that constants $μ, c, D \geq 0$ and an integer $q \geq 1$ satisfy, for all $x, y \in R^{d}$ , $⟨ x - y, f (x) - f (y) ⟩ \leq μ ∣ x - y ∣^{2}, ∥ g (x) - g (y) ∥_{F}^{2} \leq c ∣ x - y ∣^{2},$ and $∣ f (x) - f (y) ∣^{2} \leq D (1 + ∣ x ∣^{q} + ∣ y ∣^{q}) ∣ x - y ∣^{2} .$ For each $0 < h \leq 1$ with $N = T / h \in N$ , let $Y^{h}$ be the continuous Euler–Maruyama interpolation and ${\overset{ˉ}{Y}}^{h}$ its left-endpoint step process from Autonomous Euler–Maruyama scheme and interpolation. Assume that an exact strong solution $X$ and these exact, step and continuous Euler processes have uniformly bounded moments of every finite order on $[0, T]$ : for every $p \geq 1$ , $E \sup_{t \leq T} ∣ X_{t} ∣^{p} + \sup_{0 < h \leq 1, T / h \in N} (E \sup_{t \leq T} ∣ Y_{t}^{h} ∣^{p} + E \sup_{t \leq T} ∣ {\overset{ˉ}{Y}}_{t}^{h} ∣^{p}) < \infty .$ Then a constant $C$ , independent of $h$ , satisfies $E \sup_{0 \leq t \leq T} ∣ X_{t} - Y_{t}^{h} ∣^{2} \leq C h .$ In particular the grid-point root-mean-square error is $O (h^{1 / 2})$ . The all-moment premise is part of this source contract; $X_{0} \in L^{2}$ alone does not imply it.

Facts & Assumptions

Given: AC; finite $d, m$ ; $T > 0$ ; the stated Brownian and coefficient hypotheses; an exact strong solution $X$ ; and the uniform all-moment bounds in the Statement.

[F1]

The autonomous Euler algorithm returns the grid recursion, left-endpoint step process, and continuous interpolation using $Y_{t} = Y_{k} + f (Y_{k}) (t - t_{k}) + g (Y_{k}) (W_{t} - W_{t_{k}})$ ; endpoint values agree with the recursion. Autonomous Euler–Maruyama scheme and interpolation

[F2]

For every even integer $r \geq 2$ , the autonomous Euler interpolation increment satisfies $\sup_{t \leq T} E ∣ Y_{t} - {\overset{ˉ}{Y}}_{t} ∣^{r} \leq C_{r} h^{r / 2}$ under the all-moment and coefficient premises stated in that lemma. All-moment Euler interpolation increment bound under autonomous hypotheses

[F3]

For a finite-dimensional continuous Brownian Itô process and $F \in C^{1, 2}$ , Itô's formula has drift $\partial_{t} F + b \cdot \nabla F + \frac{1}{2} (σ σ^{T}) : D^{2} F$ and stochastic term $\nabla F σ d W$ . Multidimensional Ito formula for Brownian-driven processes

[F4]

For a finite-energy $m$ -dimensional Brownian Itô integral $N_{t} = \sum_{j} \int_{0}^{t} K_{s}^{j} d W_{s}^{j}$ , $E \sup_{t \leq T} ∣ N_{t} ∣ \leq 16 \sqrt{m} E (\int_{0}^{T} \sum_{j} ∣ K_{s}^{j} ∣^{2} d s)^{1 / 2}$ . An L1 maximal bound for finite-energy Brownian Itô integrals

[F5]

If $U, V \in L^{2}$ , then $E ∣ U V ∣ \leq (E U^{2})^{1 / 2} (E V^{2})^{1 / 2}$ . Cauchy-Schwarz for random variables

[F6]

If $v (t) \leq A + \int_{0}^{t} B v (s) d s$ for nonnegative continuous $v$ and constants $A, B \geq 0$ , then $v (t) \leq A e^{B t}$ . Gronwall's integral inequality with variable and constant coefficients

[F7]

A finite-dimensional continuous Brownian Itô process is an adapted continuous solution of its coordinatewise drift-plus-Itô-integral equation on the given usual filtered space. Continuous Brownian Ito processes

[F8]

Tonelli's theorem interchanges expectation and time integration for the nonnegative measurable drift and diffusion-error terms used below. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[A1]

AC is stated because the Brownian, Itô-process, multidimensional Itô and maximal-integral interfaces in [F3], [F4], and [F7] adopt it. The Axiom of Choice

Proof

1.1F1F3F7given

Induction in the recursion shows that $Y_{k}^{h}$ is $F_{t_{k}}$ -measurable: this holds for $Y_{0} = X_{0}$ , and each update uses $F_{t_{k}}$ -measurable coefficients and a Brownian increment measurable at $t_{k + 1}$ . The interpolation is therefore adapted and continuous. For integration use the predictable step representative ${\hat{Y}}_{0}^{h} = Y_{0}$ and ${\hat{Y}}_{s}^{h} = Y_{k}^{h}$ on $(t_{k}, t_{k + 1}]$ ; its coefficient processes $f ({\hat{Y}}^{h})$ and $g ({\hat{Y}}^{h})$ are predictable because each $Y_{k}^{h}$ is $F_{t_{k}}$ -measurable and $f, g$ are continuous. It agrees with ${\overset{ˉ}{Y}}^{h}$ outside the finite set of grid times. The recursion and interpolation give the same integral equation with these predictable coefficients. Setting $y = 0$ in the coefficient inequalities yields $∣ f (x) ∣^{2} \leq 2 ∣ f (0) ∣^{2} + 2 D (1 + ∣ x ∣^{q}) ∣ x ∣^{2}$ and $∥ g (x) ∥_{F}^{2} \leq 2 ∥ g (0) ∥_{F}^{2} + 2 c ∣ x ∣^{2}$ . The all-moment premise thus gives finite expected time-integral energy for these Euler coefficients; it gives the same for $f (X)$ and $g (X)$ . The definition [F7] then makes the Euler interpolation and exact solution continuous Brownian Itô processes, and its linear-combination clause makes their difference $e$ one as well. In Itô's formula for $∣ e_{t} ∣^{2}$ , the drift is $2 ⟨ e_{t}, f (X_{t}) - f ({\hat{Y}}_{t}^{h}) ⟩ + ∥ G_{t} ∥_{F}^{2}$ and the martingale term is $M_{t} = 2 \sum_{j = 1}^{m} \int_{0}^{t} ⟨ e_{s}, G_{s, \cdot j} ⟩ d W_{s}^{j}$ , where $G_{s} = g (X_{s}) - g ({\hat{Y}}_{s}^{h})$ . Since $e$ is continuous adapted and $G$ is predictable, each martingale integrand is predictable. Its finite expected energy follows from $\sum_{j} ∣ 2 ⟨ e_{s}, G_{s, \cdot j} ⟩ ∣^{2} \leq 4 ∣ e_{s} ∣^{2} ∥ G_{s} ∥_{F}^{2}$ and $E \int_{0}^{T} ∣ e_{s} ∣^{2} ∥ G_{s} ∥_{F}^{2} d s \leq T (E \sup_{s \leq T} ∣ e_{s} ∣^{4})^{1 / 2} (E \sup_{s \leq T} ∥ G_{s} ∥_{F}^{4})^{1 / 2} < \infty$ by the all-moment and coefficient-growth premises.

1.2F2F5F8givenalgebra

Split the drift difference at $Y_{t}^{h}$ . The one-sided Lipschitz bound and $2 a b \leq a^{2} + b^{2}$ give $2 ⟨ e, f (X) - f (\hat{Y}) ⟩ \leq (2 μ + 1) ∣ e ∣^{2} + ∣ f (Y) - f (\hat{Y}) ∣^{2}$ almost everywhere in time. The polynomial local-Lipschitz condition bounds the final term by $D (1 + ∣ Y ∣^{q} + ∣ \hat{Y} ∣^{q}) ∣ Y - \hat{Y} ∣^{2}$ . Also, splitting $g (X) - g (\hat{Y})$ at $g (Y)$ and using its global Lipschitz bound gives $∥ G ∥_{F}^{2} \leq 2 c ∣ e ∣^{2} + 2 c ∣ Y - \hat{Y} ∣^{2}$ . By [F2] with $r = 2, 4$ , the all-moment premise, and [F5], $E ∣ Y_{s} - {\hat{Y}}_{s} ∣^{2} \leq C h$ and $E [(1 + ∣ Y_{s} ∣^{q} + ∣ {\hat{Y}}_{s} ∣^{q}) ∣ Y_{s} - {\hat{Y}}_{s} ∣^{2}] \leq C h$ for almost every $s$ , uniformly in $h$ . Tonelli [F8] then bounds the expected integrated drift by $C \int_{0}^{t} E ∣ e_{s} ∣^{2} d s + C h$ .

1.3F2F4F5F8givenalgebra

Apply [F4] to the vector integrand $K_{s}^{j} = 2 ⟨ e_{s}, G_{s, \cdot j} ⟩$ . Since $\sum_{j} ∣ K_{s}^{j} ∣^{2} \leq 4 ∣ e_{s} ∣^{2} ∥ G_{s} ∥_{F}^{2}$ , $E \sup_{s \leq t} ∣ M_{s} ∣ \leq 32 \sqrt{m} E [\sup_{s \leq t} ∣ e_{s} ∣ {(\int_{0}^{t} ∥ G_{s} ∥_{F}^{2} d s)}^{1 / 2}] \leq 32 \sqrt{m} {(v (t) E \int_{0}^{t} ∥ G_{s} ∥_{F}^{2} d s)}^{1 / 2},$ where $v (t) = E \sup_{s \leq t} ∣ e_{s} ∣^{2}$ . From [F2], Tonelli [F8], and the Lipschitz estimate for $g$ , $E \int_{0}^{t} ∥ G_{s} ∥_{F}^{2} d s \leq C \int_{0}^{t} E ∣ e_{s} ∣^{2} d s + C h .$ Young's inequality therefore bounds the martingale term by $\frac{1}{2} v (t) + C \int_{0}^{t} v (s) d s + C h$ .

2.1F6step 1.1step 1.2step 1.3algebra

Take the supremum over times up to $t$ in Itô's identity from step 1.1, take expectations, and insert the drift estimate from step 1.2 and martingale estimate from step 1.3. Moving $\frac{1}{2} v (t)$ to the left gives $v (t) \leq C \int_{0}^{t} v (s) d s + C h .$ The all-moment assumption makes $v$ finite; pathwise continuity of $e$ and dominated convergence make $v$ continuous. Applying [F6] with the nondecreasing constant term $C h$ yields $v (T) \leq C^{'} h$ . The constants depend on $T, d, m, μ, c, D, q$ and the all-moment bounds, but not on $h$ .

3.1A1step 2.1∎

Since $Y_{T}^{h} = Y_{N}^{h}$ , the grid endpoint is included in the supremum bound. Taking square roots gives the stated root-mean-square order. AC is inherited exactly through [A1]; no choice principle is used in the numerical estimates.

Remarks

This is the autonomous all-moment Higham–Mao–Stuart route. It does not replace the separate finite- $L^{2}$ theorem, does not extend to nonautonomous coefficients, and does not derive its all-moment premise from $X_{0} \in L^{2}$ . The local proof uses [F4], whose $L^{1}$ maximal estimate is proved from the published $L^{2}$ Doob inequality and stopping identity; no unproved Burkholder–Davis–Gundy supplier is imported.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Euler–Maruyama strong rate for square-integrable initial data

Statement

Assume the Axiom of Choice. Let $d, m \geq 1$ be finite, fix $T > 0$ , and work on the complete usual filtered probability space of Itô SDE data and strong solution. Let $0 < h \leq 1$ with $N = T / h \in N$ . Use Euclidean state norms and the Frobenius matrix norm. Let $W$ be an adapted standard $m$ -dimensional Brownian motion whose increments are independent of the current filtration. Let $b : [0, T] \times R^{d} \to R^{d}$ and $σ : [0, T] \times R^{d} \to R^{d \times m}$ be jointly Borel, and suppose finite nonnegative constants $L, K$ satisfy, for all $s, t \in [0, T]$ and $x, y \in R^{d}$ , $∣ b (t, x) - b (t, y) ∣ + ∥ σ (t, x) - σ (t, y) ∥_{F} \leq L ∣ x - y ∣,$ $∣ b (t, x) ∣^{2} + ∥ σ (t, x) ∥_{F}^{2} \leq K^{2} (1 + ∣ x ∣^{2}),$ and $∣ b (t, x) - b (s, x) ∣ + ∥ σ (t, x) - σ (s, x) ∥_{F} \leq K (1 + ∣ x ∣) ∣ t - s ∣^{1 / 2} .$ Let $X_{0}$ be $F_{0}$ -measurable, independent of $σ (W_{t} - W_{0} : 0 \leq t \leq T)$ , with $E ∣ X_{0} ∣^{2} < \infty$ . Let $X$ be the exact continuous strong solution with $E \sup_{t \leq T} ∣ X_{t} ∣^{2} < \infty,$ and let $Y^{h}, {\overset{ˉ}{Y}}^{h}$ be the continuous Euler interpolation and left-endpoint step process of Euler–Maruyama step process and continuous interpolation. The uniform pointwise moment and interpolation-increment bounds of Euler–Maruyama uniform moments and interpolation increments then imply that a constant $C$ , independent of $h$ , satisfies $E \sup_{0 \leq t \leq T} ∣ X_{t} - Y_{t}^{h} ∣^{2} \leq C h .$ Thus the grid-point root-mean-square error is $O (h^{1 / 2})$ . The constant may depend on $T, d, m, L, K$ and the exact-solution supremum-moment bound supplied by the P-SDE-01 existence theorem.

Facts & Assumptions

Given: AC; finite $d, m$ ; the stated horizon, Brownian, coefficient and initial-value hypotheses; the exact strong solution $X$ and its finite supremum second moment; and the Euler recursion/interpolation.

[F1]

The continuous Euler interpolation is $Y_{t} = Y_{k} + b (t_{k}, Y_{k}) (t - t_{k}) + σ (t_{k}, Y_{k}) (W_{t} - W_{t_{k}})$ on $[t_{k}, t_{k + 1}]$ , with $Y_{t_{k}} = Y_{k}$ and left-endpoint step process ${\overset{ˉ}{Y}}_{t} = Y_{k}$ on $[t_{k}, t_{k + 1})$ . Euler–Maruyama step process and continuous interpolation

[F2]

Under the stated global Lipschitz, linear-growth, half-Hölder time, Brownian-increment and initial-value conditions, $\sup_{t \leq T} E ∣ {\overset{ˉ}{Y}}_{t} ∣^{2}$ and $\sup_{t \leq T} E ∣ Y_{t} ∣^{2}$ are bounded uniformly in $h$ , and $\sup_{t \leq T} E ∣ Y_{t} - {\overset{ˉ}{Y}}_{t} ∣^{2} \leq C h$ . Euler–Maruyama uniform moments and interpolation increments

[F3]

For each scalar Brownian integral $M_{t} = \int_{0}^{t} H_{s} d B_{s}$ with finite energy, $E \sup_{t \leq T} ∣ M_{t} ∣^{2} \leq 4 E \int_{0}^{T} H_{s}^{2} d s$ . Doob maximal bound for the Ito integral

[F4]

For a finite-dimensional continuous Brownian Itô process, its component processes have the coordinatewise drift and Brownian-integral representation stated in the definition. Continuous Brownian Ito processes

[F5]

Stopping a finite-energy Brownian Itô integral at a stopping time agrees with integrating its predictable stopped integrand. Stopping an Ito integral

[F6]

[F7]

The stated squared growth bound implies $∣ b (t, x) ∣ + ∥ σ (t, x) ∥_{F} \leq \sqrt{2} K (1 + ∣ x ∣)$ ; with the global Lipschitz, usual-filtration and initial-value conditions, the P-SDE-01 theorem supplies a unique exact strong solution and $E \sup_{t \leq T} ∣ X_{t} ∣^{2} < \infty$ . Itô SDE data and strong solution Global strong well-posedness for Lipschitz Itô SDEs

[F8]

A map $ρ$ is a stopping time when ${ρ \leq t} \in F_{t}$ for every $t \geq 0$ . Continuous-time stopping times and stopped sigma-algebras

[F9]

For a continuous adapted process, the supremum on $[0, t]$ equals the supremum over rational times in $[0, t]$ together with $t$ , and is $F_{t}$ -measurable. The rationals embed densely in the reals

[F10]

Cauchy–Schwarz for time integrals gives $∣ \int_{0}^{t} u_{s} d s ∣^{2} \leq t \int_{0}^{t} ∣ u_{s} ∣^{2} d s$ . Holder's inequality for integrals, including the endpoint cases

[F11]

If $Z_{R} ↑ Z$ are nonnegative measurable random variables, then $E Z_{R} ↑ E Z$ . Monotone convergence for the integral

[F12]

Tonelli's theorem exchanges time integration and expectation for nonnegative measurable integrands. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[A1]

AC is stated because the SDE, Brownian, Itô-integral and maximal-inequality interfaces in [F3]–[F7] adopt it. The Axiom of Choice

Proof

[F1] The continuous Euler interpolation is $Y_{t} = Y_{k} + b (t_{k}, Y_{k}) (t - t_{k}) + σ (t_{k}, Y_{k}) (W_{t} - W_{t_{k}})$ on $[t_{k}, t_{k + 1}]$ , with $Y_{t_{k}} = Y_{k}$ and left-endpoint step process ${\overset{ˉ}{Y}}_{t} = Y_{k}$ on $[t_{k}, t_{k + 1})$ . Euler–Maruyama step process and continuous interpolation

[F2] Under the stated global Lipschitz, linear-growth, half-Hölder time, Brownian-increment and initial-value conditions, $\sup_{t \leq T} E ∣ {\overset{ˉ}{Y}}_{t} ∣^{2}$ and $\sup_{t \leq T} E ∣ Y_{t} ∣^{2}$ are bounded uniformly in $h$ , and $\sup_{t \leq T} E ∣ Y_{t} - {\overset{ˉ}{Y}}_{t} ∣^{2} \leq C h$ . Euler–Maruyama uniform moments and interpolation increments

[F3] For each scalar Brownian integral $M_{t} = \int_{0}^{t} H_{s} d B_{s}$ with finite energy, $E \sup_{t \leq T} ∣ M_{t} ∣^{2} \leq 4 E \int_{0}^{T} H_{s}^{2} d s$ . Doob maximal bound for the Ito integral

[F4] For a finite-dimensional continuous Brownian Itô process, its component processes have the coordinatewise drift and Brownian-integral representation stated in the definition. Continuous Brownian Ito processes

[F5] Stopping a finite-energy Brownian Itô integral at a stopping time agrees with integrating its predictable stopped integrand. Stopping an Ito integral

[F6] If $v (t) \leq A + \int_{0}^{t} B v (s) d s$ for nonnegative continuous $v$ and constants $A, B \geq 0$ , then $v (t) \leq A e^{B t}$ . Gronwall's integral inequality with variable and constant coefficients

[F7] The stated squared growth bound implies $∣ b (t, x) ∣ + ∥ σ (t, x) ∥_{F} \leq \sqrt{2} K (1 + ∣ x ∣)$ ; with the global Lipschitz, usual-filtration and initial-value conditions, the P-SDE-01 theorem supplies a unique exact strong solution and $E \sup_{t \leq T} ∣ X_{t} ∣^{2} < \infty$ . Itô SDE data and strong solution Global strong well-posedness for Lipschitz Itô SDEs

[F8] A map $ρ$ is a stopping time when ${ρ \leq t} \in F_{t}$ for every $t \geq 0$ . Continuous-time stopping times and stopped sigma-algebras

[F9] For a continuous adapted process, the supremum on $[0, t]$ equals the supremum over rational times in $[0, t]$ together with $t$ , and is $F_{t}$ -measurable. The rationals embed densely in the reals

[F10] Cauchy–Schwarz for time integrals gives $∣ \int_{0}^{t} u_{s} d s ∣^{2} \leq t \int_{0}^{t} ∣ u_{s} ∣^{2} d s$ . Holder's inequality for integrals, including the endpoint cases

[F11] If $Z_{R} ↑ Z$ are nonnegative measurable random variables, then $E Z_{R} ↑ E Z$ . Monotone convergence for the integral

[F12] Tonelli's theorem exchanges time integration and expectation for nonnegative measurable integrands. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[A1] AC is stated because the SDE, Brownian, Itô-integral and maximal-inequality interfaces in [F3]–[F7] adopt it. The Axiom of Choice

1.1F1F2F4F5F7F8F9F12given

By [F4] and [F7], the exact process has its finite-coordinate Itô representation; [F1] supplies the Euler interpolation and its predictable representative ${\overset{ˉ}{Y}}^{-}$ with left-endpoint map $η_{h}^{-}$ . Use that representative in the Euler integral form: $X_{t} = X_{0} + \int_{0}^{t} b (s, X_{s}) d s + \int_{0}^{t} σ (s, X_{s}) d W_{s},$ $Y_{t}^{h} = X_{0} + \int_{0}^{t} b (η_{h}^{-} (s), {\overset{ˉ}{Y}}_{s}^{-}) d s + \int_{0}^{t} σ (η_{h}^{-} (s), {\overset{ˉ}{Y}}_{s}^{-}) d W_{s} .$ The grid-valued coefficients are predictable on $(t_{k}, t_{k + 1}]$ ; the grid coefficient energy is a finite sum bounded by [F2], while Tonelli [F12] and the stated $E \sup ∣ X ∣^{2}$ bound give finite expected energy for the exact coefficients. Thus these are the required integral representations. In subsequent estimates, $\overset{ˉ}{Y}$ and $η_{h}$ may replace their predictable versions because they differ only at finitely many grid times, which do not affect the integrals. For $R > 0$ define $τ_{R} = \inf {t \leq T : ∣ e_{t} ∣ \geq R} \land T$ , with $\inf \emptyset = \infty$ . For $t < T$ , ${τ_{R} \leq t} = {\sup_{s \leq t} ∣ e_{s} ∣ \geq R}$ , which is in $F_{t}$ by continuity and [F9]; for $t \geq T$ the event is all of $Ω$ . Thus [F8] makes $τ_{R}$ a stopping time. Subtracting the displays and using [F5] gives the stopped error as the sum of the two stopped coefficient integrals.

1.2F1F2F10F12givenalgebra

Put $δ b_{s} = b (s, X_{s}) - b (η_{h} (s), {\overset{ˉ}{Y}}_{s}), δ σ_{s} = σ (s, X_{s}) - σ (η_{h} (s), {\overset{ˉ}{Y}}_{s}) .$ The state Lipschitz bounds, time modulus, and $∣ X_{s} - {\overset{ˉ}{Y}}_{s} ∣ \leq ∣ e_{s} ∣ + ∣ Y_{s}^{h} - {\overset{ˉ}{Y}}_{s} ∣$ give $∣ δ b_{s} ∣^{2} + ∥ δ σ_{s} ∥_{F}^{2} \leq C (∣ e_{s} ∣^{2} + ∣ Y_{s}^{h} - {\overset{ˉ}{Y}}_{s} ∣^{2} + h (1 + ∣ {\overset{ˉ}{Y}}_{s} ∣^{2})) .$ Consequently, with $v_{R} (t) = E \sup_{s \leq t} ∣ e_{s \land τ_{R}} ∣^{2}$ , Tonelli [F12] and [F2] bound the expected drift-integrand error and the expected squared diffusion-integrand error, after stopping, by $C \int_{0}^{t} v_{R} (s) d s + C h .$

2.1F2F3F10F12step 1.2algebra

Cauchy–Schwarz for time integrals bounds the squared supremum of the stopped drift integral by $T \int_{0}^{t} E [1_{{s \leq τ_{R}}} ∣ δ b_{s} ∣^{2}] d s$ . For the vector stochastic integral, write each state coordinate as a sum of the $m$ scalar Brownian integrals and use $∣ \sum_{j = 1}^{m} z_{j} ∣^{2} \leq m \sum_{j} ∣ z_{j} ∣^{2}$ . Applying [F3] to each scalar term yields the bound $4 m E \int_{0}^{t} 1_{{s \leq τ_{R}}} ∥ δ σ_{s} ∥_{F}^{2} d s$ . Using Tonelli [F12] and step 1.2 in both bounds gives $v_{R} (t) \leq C \int_{0}^{t} v_{R} (s) d s + C h,$ where $C$ is independent of $R$ and $h$ .

3.1A1F6F11step 2.1given∎

The stopped error is bounded by $R$ , so $v_{R}$ is finite and continuous; applying [F6] to step 2.1 gives $v_{R} (T) \leq C_{T} h$ uniformly in $R$ . Since $e$ is continuous on the compact interval, $τ_{R} = T$ for all sufficiently large $R$ on each sample path. Taking $R = n \in N$ , the stopped suprema increase to $\sup_{s \leq T} ∣ e_{s} ∣^{2}$ ; monotone convergence [F11] yields $E \sup_{t \leq T} ∣ e_{t} ∣^{2} \leq C_{T} h$ . Taking square roots gives the grid-point root-mean-square rate. AC enters only through [A1], and no arbitrary-index choice is used in the estimates.

Remarks

The proof is the finite- $L^{2}$ interpolation-defect route. It uses the pointwise-in-time moment and increment estimates of the scheme supplier; its stopping argument does not assume a path-supremum moment bound for the Euler interpolation. This rate is proved separately from the autonomous Higham–Mao–Stuart all-moment theorem.

TheoremStatement: AI-adaptedProof: AI-generatedjudge pass (gpt-6-luna)Open item page →

Euler–Maruyama weak order one under backward regularity

Statement

Assume the Axiom of Choice. Let $d, m \geq 1$ be finite, fix $T > 0$ , and work on the complete usual filtered probability space of Itô SDE data and strong solution. Take $0 < h \leq 1$ with $N = T / h \in N$ . Let $W$ be an adapted standard $m$ -dimensional Brownian motion with increments independent of the current filtration. Use Euclidean state norms and the Frobenius matrix norm, and write $A : B = tr (A^{T} B)$ . Let $b : [0, T] \times R^{d} \to R^{d}$ and $σ : [0, T] \times R^{d} \to R^{d \times m}$ be jointly Borel, globally Lipschitz in state, of uniform linear growth, and satisfy the half-Hölder time modulus in Euler–Maruyama recursion. Let $X_{0}$ be $F_{0}$ -measurable, independent of $σ (W_{t} - W_{0} : 0 \leq t \leq T)$ , with $E ∣ X_{0} ∣^{p} < \infty$ for some $p \geq 2 q + 8$ , where $q \geq 0$ is finite; when $q = 0$ , use the convention $∣ x ∣^{q} = 1$ . Let $X$ be the exact strong solution and $Y^{h}, {\overset{ˉ}{Y}}^{h}$ the continuous Euler interpolation and step process of Euler–Maruyama step process and continuous interpolation.

For a bounded Borel payoff $φ : R^{d} \to R$ , define $u (t, x) = E [φ (X_{T}^{t, x})],$ where $X^{t, x}$ is the unique shifted strong solution started from $x$ at time $t$ , supplied by Global strong well-posedness for Lipschitz Itô SDEs. Set $L_{t} v (x) = b (t, x) \cdot \nabla v (x) + \frac{1}{2} (σ (t, x) σ (t, x)^{T}) : D_{x}^{2} v (x) .$ Assume $u (T, x) = φ (x)$ and $(\partial_{t} + L_{t}) u = 0$ on $[0, T) \times R^{d}$ . A multi-index is $α = (α_{1}, \dots, α_{d})$ with nonnegative integer entries and $∣ α ∣ = \sum_{i} α_{i}$ . Assume every mixed derivative $\partial_{t}^{j} D_{x}^{α} u$ with $0 \leq j \leq 2$ and $∣ α ∣ \leq 4$ exists, is continuous, and satisfies $∣ \partial_{t}^{j} D_{x}^{α} u (t, x) ∣ \leq C_{u} (1 + ∣ x ∣^{q})$ for a common constant $C_{u}$ . Finally, assume the exact and Euler processes have uniform pointwise moments of order $p$ on $[0, T]$ ; the Euler bounds are provided by Euler–Maruyama uniform moments and interpolation increments and the exact bound by the P-SDE-01 moment contract. Then $∣ E φ (X_{T}) - E φ (Y_{N}^{h}) ∣ \leq C h,$ where $C$ is independent of $h$ . The theorem is conditional on the stated backward regularity; it does not derive that regularity from coefficient or payoff smoothness.

Facts & Assumptions

Given: AC; finite $d, m$ ; the fixed horizon and grid; the Brownian, coefficient, initial-value, exact-solution, backward-equation, mixed-derivative and moment hypotheses in the Statement.

[F1]

The Euler recursion uses $Y_{k + 1} = Y_{k} + b (t_{k}, Y_{k}) h + σ (t_{k}, Y_{k}) (W_{t_{k + 1}} - W_{t_{k}})$ , and its continuous interpolation uses those same left-endpoint coefficients. Euler–Maruyama recursion Euler–Maruyama step process and continuous interpolation

[F2]

For every finite $p \geq 2$ allowed by the initial moment, the Euler step, continuous interpolation and left-endpoint process have pointwise moments bounded uniformly over $h$ ; their interpolation increment satisfies $E ∣ Y_{t} - {\overset{ˉ}{Y}}_{t} ∣^{2} \leq C h$ . Euler–Maruyama uniform moments and interpolation increments

[F3]

Brownian increments have law $N_{m} (0, (t - s) I_{m})$ and are independent of the current filtration. $d$ -dimensional Brownian motion

[F4]

For each even integer $r \geq 2$ , a scalar Brownian increment has $r$ th absolute moment $(r - 1)!! ∣ t - s ∣^{r / 2} < \infty$ . Gaussian even moments for Brownian increments

[F5]

[F6]

A continuous Brownian Itô process has the coordinatewise drift-plus-Itô-integral representation used to interpret the exact SDE solution; every finite-energy piece of its localized Brownian integral is a square-integrable mean-zero martingale. Continuous Brownian Ito processes

[F7]

Under the global Lipschitz and linear-growth assumptions and $X_{0} \in L^{2}$ , the exact strong solution exists uniquely and has a finite path-supremum second moment. For each deterministic $t < T$ and $x \in R^{d}$ , the same theorem applies on the shifted interval $[0, T - t]$ with coefficients $b (t + s, \cdot), σ (t + s, \cdot)$ , shifted filtration $F_{t + s}$ , Brownian motion $W_{t + s} - W_{t}$ , and deterministic initial value $x$ ; these data retain the theorem's hypotheses, so $X^{t, x}$ is well-defined. Global strong well-posedness for Lipschitz Itô SDEs

[F8]

For every real $p \geq 2$ with $E ∣ X_{0} ∣^{p} < \infty$ , the exact solution has $\sup_{t \leq T} E ∣ X_{t} ∣^{p} < \infty$ . Finite-horizon moments for Lipschitz Itô SDEs

[F9]

A standard normal has density $e^{- z^{2} / 2} / \sqrt{2 π}$ and the centered Gaussian law of a Brownian increment of length $h$ is symmetric with covariance $h I_{m}$ . Its real nonnegative absolute-moment bounds are derived in step 1.3 from [F4]. Standard normal and normal laws $d$ -dimensional Brownian motion

[F10]

Tonelli's theorem allows the expected energy of the exact stochastic-integral coefficient to be bounded by the time integral of its pointwise moments. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[A1]

AC is declared because the SDE, Brownian, normal-law, Gaussian-moment and multidimensional Itô interfaces in [F3]–[F9] adopt it. The Axiom of Choice

Proof

Given: The hypotheses in the Statement or Example and the facts above.

1.1A1F5F6F7F8F10given

By [F6], the exact solution has the continuous Itô decomposition; apply [F5] to $u (t, X_{t})$ . The backward equation cancels its drift. Moreover, $∣ \nabla u (t, x) ∣ \leq C (1 + ∣ x ∣^{q})$ and linear growth of $σ$ give $E \int_{0}^{T} ∥ \nabla u (s, X_{s}) σ (s, X_{s}) ∥_{F}^{2} d s < \infty$ by Tonelli [F10], because $p \geq 2 q + 8$ and [F8] supplies the exact process moments. Thus the stochastic integral is a square-integrable mean-zero martingale by [F6]. With $u (T, \cdot) = φ$ , this proves $E φ (X_{T}) = E u (0, X_{0}) .$

1.2givenalgebra

Fix $t < T$ , $x \in R^{d}$ , and $0 < h \leq \min (1, T - t)$ ; set $Δ W \sim N_{m} (0, h I_{m})$ and $δ = b (t, x) h + σ (t, x) Δ W$ . Taylor's formula along the spatial segment gives $u (t, x + δ) = \sum_{k = 0}^{3} \frac{1}{k!} D_{x}^{k} u (t, x) [δ^{\otimes k}] + R_{4},$ $u_{t} (t, x + δ) = \sum_{k = 0}^{2} \frac{1}{k!} D_{x}^{k} u_{t} (t, x) [δ^{\otimes k}] + R_{3}^{t},$ where $D_{x}^{k} u [z^{\otimes k}]$ means the $k$ th spatial derivative applied to $z$ in all $k$ slots. The integral remainders are $R_{4} = \frac{1}{3!} \int_{0}^{1} (1 - r)^{3} D_{x}^{4} u (t, x + r δ) [δ^{\otimes 4}] d r,$ $R_{3}^{t} = \frac{1}{2!} \int_{0}^{1} (1 - r)^{2} D_{x}^{3} u_{t} (t, x + r δ) [δ^{\otimes 3}] d r .$ Time Taylor expansion gives $u (t + h, x + δ) = u (t, x + δ) + h u_{t} (t, x + δ) + R_{t t}, R_{t t} = h^{2} \int_{0}^{1} (1 - r) u_{t t} (t + r h, x + δ) d r .$ The derivative bounds in the Statement and $∣ x + r δ ∣^{q} \leq C_{q} (∣ x ∣^{q} + ∣ δ ∣^{q})$ imply $∣ R_{4} ∣ \leq C (1 + ∣ x ∣^{q} + ∣ δ ∣^{q}) ∣ δ ∣^{4}, ∣ R_{3}^{t} ∣ \leq C (1 + ∣ x ∣^{q} + ∣ δ ∣^{q}) ∣ δ ∣^{3}, ∣ R_{t t} ∣ \leq C h^{2} (1 + ∣ x + δ ∣^{q}) .$ If $q = 0$ , interpret $1 + ∣ x ∣^{q}$ as the constant $2$ .

1.3F3F4F9choosealgebra

For each finite real $s > 0$ , choose an integer $n \geq 1$ with $2 n \geq s$ . If $Z$ is scalar standard normal, $∣ Z ∣^{s} \leq 1 + ∣ Z ∣^{2 n}$ , so [F4] gives $E ∣ Z ∣^{s} \leq 1 + (2 n - 1)!! < \infty$ . For a standard $m$ -dimensional Gaussian vector $G$ , $∣ G ∣ \leq \sqrt{m} \max_{j} ∣ G_{j} ∣$ gives $∣ G ∣^{s} \leq m^{s / 2} \sum_{j = 1}^{m} ∣ G_{j} ∣^{s}$ , and hence $E ∣ G ∣^{s} \leq m^{1 + s / 2} {1 + (2 n - 1)!!}$ . By the Gaussian scaling convention in [F3] and [F9], $Δ W$ has law $\sqrt{h} G$ , so $E ∣ Δ W ∣^{s} \leq C_{s, m} h^{s / 2}$ . This also covers $h = 0$ when $s > 0$ ; for $s = 0$ adopt $∣ x ∣^{0} = 1$ , and its expectation is one. No initial-state moment stronger than the Statement is used in this Gaussian calculation.

2.1F3F4F9step 1.2step 1.3algebra

Put $a (t, x) = σ (t, x) σ (t, x)^{T}$ . The coefficient growth bounds and [F3]–[F4], [F9] give $E δ = b (t, x) h, E (δ δ^{T}) = a (t, x) h + h^{2} b (t, x) b (t, x)^{T},$ $∥ E (δ^{\otimes 3}) ∥ \leq C h^{2} (1 + ∣ x ∣^{3}), E ∣ δ ∣^{s} \leq C_{s} h^{s / 2} (1 + ∣ x ∣^{s})$ for each finite exponent $s > 0$ used in the Taylor remainders. Indeed, $∣ b (t, x) ∣ + ∥ σ (t, x) ∥_{F} \leq C (1 + ∣ x ∣)$ and $∣ δ ∣ \leq h ∣ b (t, x) ∣ + ∥ σ (t, x) ∥_{F} ∣ Δ W ∣$ ; [F9] and $h \leq 1$ give the stated bound. When $q = 0$ , its $∣ δ ∣^{q}$ factor is the constant $1$ . For the third tensor moment, write $δ = h b + Z$ with $Z = σ (t, x) Δ W$ . The centered Gaussian $Z$ has covariance $h a (t, x)$ and zero third moment, so expanding the tensor cube leaves three terms with one $h b$ and two $Z$ factors, plus $(h b)^{\otimes 3}$ ; each has expectation $O (h^{2} (1 + ∣ x ∣^{3}))$ . Substituting these moments in step 1.2, the first-order terms combine to $h (\partial_{t} u + b \cdot \nabla u + \frac{1}{2} a : D_{x}^{2} u) (t, x) = 0$ . Every other Taylor term is bounded by $C h^{2} (1 + ∣ x ∣^{q + 4})$ : the fourth spatial remainder uses $E ∣ δ ∣^{4} (1 + ∣ x ∣^{q} + ∣ δ ∣^{q})$ , the time-space terms use $E δ$ and $E δ δ^{T}$ , and the $u_{t}$ third-order remainder is $O (h^{5 / 2}) (1 + ∣ x ∣^{q + 3}) \leq C h^{2} (1 + ∣ x ∣^{q + 4})$ . Therefore $∣ E u (t + h, x + δ) - u (t, x) ∣ \leq C h^{2} (1 + ∣ x ∣^{q + 4}) .$

3.1F1F2F3step 2.1

Condition on $F_{t_{k}}$ . By [F1], $Y_{k + 1}^{h}$ is the one-step update in step 2.1 with $t = t_{k}$ and $x = Y_{k}^{h}$ ; [F3] makes the Brownian increment independent of $F_{t_{k}}$ . Thus $∣ E [u (t_{k + 1}, Y_{k + 1}^{h}) - u (t_{k}, Y_{k}^{h}) ∣ F_{t_{k}}] ∣ \leq C h^{2} (1 + ∣ Y_{k}^{h} ∣^{q + 4}) .$ Since $p \geq 2 q + 8 \geq q + 4$ , [F2] and the exact pointwise moment premise make the expectation of the right side at most $C^{'} h^{2}$ , uniformly in $k, h$ .

4.1A1step 1.1step 3.1algebra∎

Sum the conditional one-step errors over $k = 0, \dots, N - 1$ . The sum telescopes to $∣ E u (T, Y_{N}^{h}) - E u (0, Y_{0}^{h}) ∣ \leq N C^{'} h^{2} = C^{'} T h .$ Here $u (T, Y_{N}^{h}) = φ (Y_{N}^{h})$ and $Y_{0}^{h} = X_{0}$ ; step 1.1 gives $E u (0, X_{0}) = E φ (X_{T})$ . Combining the equalities proves the desired weak error bound. AC enters only through [A1].

Remarks

The explicit mixed derivative premise includes $\partial_{t} D_{x}^{3} u$ , which is used in the Taylor remainder for $u_{t}$ . The moment exponent $p \geq 2 q + 8$ is a conservative bound for the $q + 4$ polynomial remainder and the martingale integrability in step 1.1. The proof assumes the backward regularity and does not prove it from $b, σ$ or $φ$ smoothness.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Shared-Brownian Euler level correction rates

Statement

Assume AC and the finite- $L^{2}$ coefficient and initial-value hypotheses of Euler–Maruyama strong rate for square-integrable initial data. Fix an integer refinement factor $M \geq 2$ and a fine step $h$ with $0 < h \leq 1 / M$ , $T / h \in N$ , and $T / (M h) \in N$ . Construct the fine and coarse Euler paths with the same initial value and Brownian motion. The coarse Brownian increment over each coarse interval is, by definition, the sum of the $M$ fine Brownian increments in that interval. Let $X_{T}$ be the common exact solution, and let $φ : R^{d} \to R$ be globally Lipschitz with constant $L_{φ}$ . Then constants independent of $h$ satisfy $E ∣ Y_{T}^{h} - Y_{T}^{M h} ∣^{2} \leq C_{M} h,$ $Var (φ (Y_{T}^{h}) - φ (Y_{T}^{M h})) \leq C_{M} L_{φ}^{2} h, ∣ E φ (X_{T}) - E φ (Y_{T}^{h}) ∣ \leq C L_{φ} h^{1 / 2} .$ Thus the Lipschitz-payoff Euler level pair has correction variance exponent $β = 1$ and strong-error bias exponent $α = 1 / 2$ . If, in addition, $φ$ is bounded and its backward value function satisfies the hypotheses of Euler–Maruyama weak order one under backward regularity, then the bias bound improves to $∣ E φ (X_{T}) - E φ (Y_{T}^{h}) ∣ \leq C h$ , so $α = 1$ while $β = 1$ remains. With fixed dimension, fixed $M$ , and unit-cost coefficient and payoff evaluations, one coupled fine/coarse sample has work $O (h^{- 1})$ .

Facts & Assumptions

Given: AC; the stated finite- $L^{2}$ SDE assumptions; the fixed refinement factor and nested grids; a common exact solution and Brownian path; and the Lipschitz payoff (plus, for the weak branch, the bounded-payoff backward contract).

[F1]

Under the finite- $L^{2}$ coefficient, time-modulus, Brownian and initial-value hypotheses, the continuous Euler path satisfies $E \sup_{t \leq T} ∣ X_{t} - Y_{t}^{h} ∣^{2} \leq C h$ , uniformly in $0 < h \leq 1$ with $T / h \in N$ . Euler–Maruyama strong rate for square-integrable initial data

[F2]

For the weak-bias branch, retain finite $d, m$ , the Euclidean/Frobenius norm convention and $A : B = tr (A^{T} B)$ , and the same finite-horizon, complete usual filtered Brownian and grid setup as in the Statement. Assume jointly Borel coefficients that are globally Lipschitz in state, have uniform linear growth, and satisfy the half-Hölder time modulus. Require an $F_{0}$ -measurable initial value independent of future Brownian increments with $E ∣ X_{0} ∣^{p} < \infty$ for finite real $q \geq 0$ and finite $p \geq 2 q + 8$ (using $∣ x ∣^{0} = 1$ ), and uniform pointwise order- $p$ moments of the exact solution, Euler interpolation, and Euler step process on $[0, T]$ . The payoff is bounded Borel, and its backward value function $u (t, x) = E [φ (X_{T}^{t, x})]$ , using the unique shifted strong solutions supplied by the global well-posedness theorem, satisfies $u (T, x) = φ (x)$ , $(\partial_{t} + L_{t}) u = 0$ for $L_{t} v = b (t, \cdot) \cdot \nabla v + \frac{1}{2} (σ σ^{T}) (t, \cdot) : D_{x}^{2} v$ , and continuous mixed derivatives $\partial_{t}^{j} D_{x}^{α} u$ for $0 \leq j \leq 2$ , $∣ α ∣ \leq 4$ , bounded by $C_{u} (1 + ∣ x ∣^{q})$ . Under this full contract of Euler–Maruyama weak order one under backward regularity, $∣ E φ (X_{T}) - E φ (Y_{T}^{h}) ∣ \leq C h$ .

[A1]

The refinement factor $M$ is fixed as $h \to 0$ ; both mesh sizes obey $0 < h \leq 1$ and have $T / h$ and $T / (M h)$ integral. The coarse path uses the sum of each corresponding block of $M$ fine Brownian increments, so both Euler paths use the same Brownian path and initial value.

[A2]

$∣ φ (x) - φ (y) ∣ \leq L_{φ} ∣ x - y ∣$ for all states. In the weak branch, the same $φ$ is bounded and satisfies the additional backward regularity contract [F2].

[A3]

AC is included because [F1] carries that assumption from the Doob and Brownian Itô interfaces. The Axiom of Choice

Proof

[F1] Under the finite- $L^{2}$ coefficient, time-modulus, Brownian and initial-value hypotheses, the continuous Euler path satisfies $E \sup_{t \leq T} ∣ X_{t} - Y_{t}^{h} ∣^{2} \leq C h$ , uniformly in $0 < h \leq 1$ with $T / h \in N$ . Euler–Maruyama strong rate for square-integrable initial data

[F2] For the weak-bias branch, retain finite $d, m$ , the Euclidean/Frobenius norm convention and $A : B = tr (A^{T} B)$ , and the same finite-horizon, complete usual filtered Brownian and grid setup as in the Statement. Assume jointly Borel coefficients that are globally Lipschitz in state, have uniform linear growth, and satisfy the half-Hölder time modulus. Require an $F_{0}$ -measurable initial value independent of future Brownian increments with $E ∣ X_{0} ∣^{p} < \infty$ for finite real $q \geq 0$ and finite $p \geq 2 q + 8$ (using $∣ x ∣^{0} = 1$ ), and uniform pointwise order- $p$ moments of the exact solution, Euler interpolation, and Euler step process on $[0, T]$ . The payoff is bounded Borel, and its backward value function $u (t, x) = E [φ (X_{T}^{t, x})]$ , using the unique shifted strong solutions supplied by the global well-posedness theorem, satisfies $u (T, x) = φ (x)$ , $(\partial_{t} + L_{t}) u = 0$ for $L_{t} v = b (t, \cdot) \cdot \nabla v + \frac{1}{2} (σ σ^{T}) (t, \cdot) : D_{x}^{2} v$ , and continuous mixed derivatives $\partial_{t}^{j} D_{x}^{α} u$ for $0 \leq j \leq 2$ , $∣ α ∣ \leq 4$ , bounded by $C_{u} (1 + ∣ x ∣^{q})$ . Under this full contract of Euler–Maruyama weak order one under backward regularity, $∣ E φ (X_{T}) - E φ (Y_{T}^{h}) ∣ \leq C h$ .

[A1] The refinement factor $M$ is fixed as $h \to 0$ ; both mesh sizes obey $0 < h \leq 1$ and have $T / h$ and $T / (M h)$ integral. The coarse path uses the sum of each corresponding block of $M$ fine Brownian increments, so both Euler paths use the same Brownian path and initial value.

[A2] $∣ φ (x) - φ (y) ∣ \leq L_{φ} ∣ x - y ∣$ for all states. In the weak branch, the same $φ$ is bounded and satisfies the additional backward regularity contract [F2].

[A3] AC is included because [F1] carries that assumption from the Doob and Brownian Itô interfaces. The Axiom of Choice

1.1F1A1algebra

By [A1], the fine and coarse paths are driven by nested increments of this common Brownian path. Since $h \leq 1 / M$ , both meshes lie in the range of [F1]. Therefore $∣ Y_{T}^{h} - Y_{T}^{M h} ∣^{2} \leq 2 ∣ Y_{T}^{h} - X_{T} ∣^{2} + 2 ∣ Y_{T}^{M h} - X_{T} ∣^{2} .$ Taking expectations and applying [F1] at step sizes $h$ and $M h$ gives $E ∣ Y_{T}^{h} - Y_{T}^{M h} ∣^{2} \leq 2 C h + 2 C M h = 2 C (1 + M) h .$ This is the correction second-moment bound.

1.2A2F1algebra

The same Lipschitz bound and Cauchy–Schwarz give $∣ E φ (X_{T}) - E φ (Y_{T}^{h}) ∣ \leq L_{φ} E ∣ X_{T} - Y_{T}^{h} ∣ \leq L_{φ} (E ∣ X_{T} - Y_{T}^{h} ∣^{2})^{1 / 2} \leq C^{1 / 2} L_{φ} h^{1 / 2} .$ Thus $β = 1$ and the strong route gives $α = 1 / 2$ .

2.1A2step 1.1algebra

Let $D = φ (Y_{T}^{h}) - φ (Y_{T}^{M h})$ . Since variance is at most second moment and [A2] gives $∣ D ∣ \leq L_{φ} ∣ Y_{T}^{h} - Y_{T}^{M h} ∣$ , step 1.1 yields $Var (D) \leq E D^{2} \leq 2 C (1 + M) L_{φ}^{2} h .$

3.1F2step 2.1

If the additional bounded-payoff backward contract holds, [F2] applied to the same fine Euler scheme gives bias at most $C h$ . It changes only the bias exponent to $α = 1$ ; the correction-variance bound of step 2.1 is unchanged.

4.1A1A3step 3.1∎

A fine path takes $T / h$ Euler steps and the coarse path takes $T / (M h)$ steps. The coarse increments are sums of already generated fine increments, so they add no Brownian draws; for fixed $M, d$ and unit-cost coefficient/payoff evaluations, total path work is proportional to $T / h + T / (M h) = O (h^{- 1})$ . AC is inherited only through [A3].

Remarks

This is the solver-rate interface for multilevel examples; it makes no claim about learned-drift or learned-score bias. The variance-rate and work statements do not import the abstract MLMC complexity theorem.

5 · Examples, counterexamples and false statements

None yet.

Euler–Maruyama Strong and Weak Convergence

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Autonomous Euler–Maruyama scheme and interpolation

Definition

An L1 maximal bound for finite-energy Brownian Itô integrals

Statement

Facts & Assumptions

Proof

Source notes

All-moment Euler interpolation increment bound under autonomous hypotheses

Statement

Facts & Assumptions

Proof

Source notes

Euler–Maruyama strong rate under Higham–Mao–Stuart hypotheses

Statement

Facts & Assumptions

Proof

Remarks

Euler–Maruyama strong rate for square-integrable initial data

Statement

Facts & Assumptions

Proof

Remarks

Euler–Maruyama weak order one under backward regularity

Statement

Facts & Assumptions

Proof

Remarks

Shared-Brownian Euler level correction rates

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, counterexamples and false statements

Sources