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Milstein stochastic taylor and iterated integrals

1 · Prerequisites

2 · Summary

This page develops the scalar Milstein method from its iterated It^o integral and proves strong order one using conditional local-defect bounds and a stochastic local-to-global recurrence. It also proves the commutative-noise reduction to Brownian increments, states where noncommuting noise needs separately sampled areas, and establishes weak order two for stochastic Heun under additive noise and bounded backward-solution derivatives. The new Milstein MLMC variance proof supplies the rate assumptions for the existing complexity theorem.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Milstein updates, iterated Brownian integrals, and commutative noise

Definition

Assume AC and the Brownian filtered-space conventions in Continuous Brownian Ito processes. Let $d, m \geq 1$ be finite integers, let $W = (W^{1}, \dots, W^{m})$ be the driving standard Brownian motion, and consider the autonomous It^o SDE $d X_{t} = b (X_{t}) d t + \sum_{j = 1}^{m} σ_{j} (X_{t}) d W_{t}^{j},$ where $b : R^{d} \to R^{d}$ and each $σ_{j} : R^{d} \to R^{d}$ is differentiable. For a differentiable vector-valued function $f$ , $D f (x)$ is its Jacobian and $L_{i} f (x) : = D f (x) σ_{i} (x)$ ; this definition is applied componentwise when $f$ is matrix-valued.

On a deterministic grid $t_{n} = n h$ , define $Δ W_{n}^{j} = W_{t_{n + 1}}^{j} - W_{t_{n}}^{j}$ and the iterated It^o integral $I_{i j, n} : = \int_{t_{n}}^{t_{n + 1}} (W_{s}^{i} - W_{t_{n}}^{i}) d W_{s}^{j}, 1 \leq i, j \leq m .$ The integrand is predictable and square-integrable, so this is the $L^{2}$ It^o integral of Ito integral for square-integrable predictable processes; its finite energy is $E \int_{t_{n}}^{t_{n + 1}} (W_{s}^{i} - W_{t_{n}}^{i})^{2} d s = h^{2} / 2$ by Ito isometry and linearity in predictable L2.

The general Milstein Taylor update is $Y_{n + 1} = Y_{n} + b (Y_{n}) h + \sum_{j = 1}^{m} σ_{j} (Y_{n}) Δ W_{n}^{j} + \sum_{i, j = 1}^{m} L_{i} σ_{j} (Y_{n}) I_{i j, n} .$ The diffusion fields have commutative noise when $L_{i} σ_{j} (x) = L_{j} σ_{i} (x)$ for every $x$ and every $i, j$ . Under that condition, the equivalent update using only Brownian increments is $Y_{n + 1} = Y_{n} + b (Y_{n}) h + \sum_{j = 1}^{m} σ_{j} (Y_{n}) Δ W_{n}^{j} + \frac{1}{2} \sum_{i, j = 1}^{m} L_{i} σ_{j} (Y_{n}) (Δ W_{n}^{i} Δ W_{n}^{j} - δ_{i j} h) .$ For scalar noise ( $d = m = 1$ ), this reads $Y_{n + 1} = Y_{n} + b (Y_{n}) h + σ (Y_{n}) Δ W_{n} + \frac{1}{2} σ (Y_{n}) σ^{'} (Y_{n}) ((Δ W_{n})^{2} - h) .$

For every $1 \leq i, j \leq m$ , define the Brownian area component $A_{i j, n} : = (I_{i j, n} - I_{j i, n}) / 2$ and the symmetric part $S_{i j, n} : = (I_{i j, n} + I_{j i, n}) / 2$ . In particular, $A_{i i, n} = 0$ and $S_{i i, n} = I_{i i, n}$ . Then $A_{j i, n} = - A_{i j, n}$ , $S_{j i, n} = S_{i j, n}$ , and reindexing the sums gives $\sum_{i, j} L_{i} σ_{j} I_{i j, n} = \frac{1}{2} \sum_{i, j} (L_{i} σ_{j} + L_{j} σ_{i}) S_{i j, n} + \frac{1}{2} \sum_{i, j} (L_{i} σ_{j} - L_{j} σ_{i}) A_{i j, n} .$ The second sum vanishes under commutative noise; otherwise the general update includes the area components. A noncommuting Milstein implementation must sample or approximate the needed areas and account for their approximation error and computational cost; this definition makes no claim that an exact area sample is free or that a particular area sampler has a specified error rate.

For this pair, $C_{b}^{k} (R^{d}; R^{q})$ means that every component and every partial derivative of orders at most $k$ exists, is continuous, and is bounded. This notation is componentwise for vector and matrix valued functions.

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

Product identity and conditional moments for iterated Brownian integrals

Statement

Let the finite-dimensional Brownian It^o setting and the integrals $I_{i j, n}$ be as in Milstein updates, iterated Brownian integrals, and commutative noise. For every grid interval of length $h > 0$ , $I_{i j, n} + I_{j i, n} = Δ W_{n}^{i} Δ W_{n}^{j} - δ_{i j} h .$ In the scalar case, writing $I_{n} = I_{11, n}$ and $Δ W_{n} = Δ W_{n}^{1}$ , this gives $I_{n} = \frac{1}{2} ((Δ W_{n})^{2} - h), E [I_{n} ∣ F_{t_{n}}] = 0, E [I_{n}^{2} ∣ F_{t_{n}}] = \frac{h^{2}}{2}, E [Δ W_{n} I_{n} ∣ F_{t_{n}}] = 0.$ Consequently, under the commutativity condition in the Definition, the double-integral term in the general Milstein Taylor increment equals $\sum_{i, j} L_{i} σ_{j} (Y_{n}) I_{i j, n} = \frac{1}{2} \sum_{i, j} L_{i} σ_{j} (Y_{n}) (Δ W_{n}^{i} Δ W_{n}^{j} - δ_{i j} h) .$

Facts & Assumptions

Given: AC, a filtered probability space with an adapted standard $m$ -dimensional Brownian motion whose increments after $t_{n}$ are independent of $F_{t_{n}}$ , and the grid interval $[t_{n}, t_{n} + h]$ .

[L1]

Milstein updates, iterated Brownian integrals, and commutative noise: $I_{i j, n} = \int_{t_{n}}^{t_{n} + h} (W_{s}^{i} - W_{t_{n}}^{i}) d W_{s}^{j}$ , and the commutative condition is $L_{i} σ_{j} = L_{j} σ_{i}$ .

[F1]

Multidimensional Ito formula for Brownian-driven processes: for a Brownian It^o process, the second-order term in It^o's formula is one half of the Hessian paired with the instantaneous covariance matrix; the theorem assumes AC and the usual-filtration/integrability conventions stated in its body.

[F2]

Continuous Brownian Ito processes: Brownian increments are independent of the current sigma algebra and have their stated centered normal laws.

[F3]

Standard normal and normal laws: a standard normal has density $(2 π)^{- 1 / 2} e^{- x^{2} / 2}$ ; its odd moments vanish and its even moments follow by integration by parts.

[F4]

Conditional expectation given a sigma algebra: conditional expectation is the measurable integrable version characterized by its integrals on events in the conditioning sigma algebra.

[A1]

The Axiom of Choice: AC is assumed by the cited Brownian and It^o interfaces; no further choice is used here.

Proof

Given: The assumptions above.

1.1F1L1A1

Apply [F1] to $f (x) = x_{i} x_{j}$ on $[t_{n}, t_{n} + h]$ . It gives $W_{t_{n} + h}^{i} W_{t_{n} + h}^{j} - W_{t_{n}}^{i} W_{t_{n}}^{j} = \int_{t_{n}}^{t_{n} + h} W_{s}^{i} d W_{s}^{j} + \int_{t_{n}}^{t_{n} + h} W_{s}^{j} d W_{s}^{i} + δ_{i j} h$ . Expand each integrand as its value at $t_{n}$ plus its Brownian increment; the two constant terms cancel from the two sides, leaving $Δ W_{n}^{i} Δ W_{n}^{j} = I_{i j, n} + I_{j i, n} + δ_{i j} h$ .

2.1F2F3step 1.1

Setting $i = j = 1$ in step 1.1 yields $I_{n} = ((Δ W_{n})^{2} - h) / 2$ . By [F2], $Δ W_{n}$ is independent of $F_{t_{n}}$ and has law $N (0, h)$ . Scaling the density in [F3] and integrating by parts give $E [Δ W_{n} ∣ F_{t_{n}}] = 0$ , $E [(Δ W_{n})^{2} ∣ F_{t_{n}}] = h$ , $E [(Δ W_{n})^{3} ∣ F_{t_{n}}] = 0$ , and $E [(Δ W_{n})^{4} ∣ F_{t_{n}}] = 3 h^{2}$ .

3.1F2F3F4step 2.1

Substituting the four conditional moments from step 2.1 into $I_{n} = ((Δ W_{n})^{2} - h) / 2$ gives $E [I_{n} ∣ F_{t_{n}}] = (h - h) / 2 = 0$ , $E [I_{n}^{2} ∣ F_{t_{n}}] = (3 h^{2} - 2 h^{2} + h^{2}) / 4 = h^{2} / 2$ , and $E [Δ W_{n} I_{n} ∣ F_{t_{n}}] = (0 - h \cdot 0) / 2 = 0$ . Each equality is an equality of conditional-expectation versions by the event-integral definition [F4].

4.1L1step 1.1∎

Under commutativity, pair the summand with indices $(i, j)$ with the summand $(j, i)$ . The coefficients agree by [L1], so the sum equals $\frac{1}{2} \sum_{i, j} L_{i} σ_{j} (Y_{n}) (I_{i j, n} + I_{j i, n})$ . Substituting step 1.1 gives the displayed Brownian-increment formula. The argument includes $i = j$ , where the summand is one half of $L_{i} σ_{i} (Y_{n}) ((Δ W_{n}^{i})^{2} - h)$ .

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

Event-weighted Ito isometry after a conditioning time

Statement

Assume AC and the Brownian filtration conventions in Continuous Brownian Ito processes. Let $t \geq 0$ , $h > 0$ , let $A \in F_{t}$ , and let $H$ be a real predictable process with finite energy on $[t, t + h]$ and $H_{s} = 0$ for $s \notin (t, t + h]$ . For any coordinate $W^{j}$ of the driving finite-dimensional Brownian motion, $E ⁣ [1_{A} {∣ \int_{t}^{t + h} H_{s} d W_{s}^{j} ∣}^{2}] = E ⁣ [1_{A} \int_{t}^{t + h} ∣ H_{s} ∣^{2} d s] .$

Facts & Assumptions

Given: AC, the Brownian filtered probability space, deterministic $t, h$ , $A \in F_{t}$ , and the predictable square-integrable integrand $H$ supported on $(t, t + h]$ .

[F1]

Continuous Brownian Ito processes: each Brownian coordinate has increments independent of the current filtration, and finite-energy Ito integrals are defined on predictable integrands.

[F2]

Elementary predictable Brownian integrands: an elementary predictable integrand is constant on finitely many intervals $(s_{k}, s_{k + 1}]$ with bounded coefficient measurable at $s_{k}$ .

[F3]

Ito integral of an elementary predictable process: on a common partition the elementary integral is the finite sum of coefficient times Brownian increment, and it is linear in the integrand.

[F4]

Ito integral for square-integrable predictable processes: a finite-energy predictable integrand is approximated in $L^{2} (d s d P)$ by elementary predictable integrands, and its Ito integral is their $L^{2} (P)$ limit.

[F5]

Ito isometry and linearity in predictable L2: for every finite-energy predictable integrand $G$ , $E ∣ \int G d W^{j} ∣^{2} = E \int ∣ G ∣^{2} d s$ , and the extension is linear in $L^{2}$ .

[F6]

Continuous-time filtrations and all-pairs martingales: a continuous-time filtration satisfies $F_{s} \subseteq F_{r}$ whenever $0 \leq s \leq r$ .

[A1]

The Axiom of Choice: AC is part of the Brownian and Ito-integral interfaces [F1], [F4], and [F5].

Proof

Given: The assumptions above.

1.1F2F4given

Choose elementary predictable $H^{n} \to H$ in $L^{2} (d s d P)$ using [F4], and replace each $H^{n}$ by ${\tilde{H}}^{n} = 1_{(t, t + h]} H^{n}$ . Refining its finite partition at $t$ and $t + h$ makes ${\tilde{H}}^{n}$ elementary; because $H$ vanishes off this interval, ${\tilde{H}}^{n} \to H$ in $L^{2} (d s d P)$ .

2.1F1F2F3F5F6step 1.1

Refine the partition of ${\tilde{H}}^{n}$ so that every nonzero cell $(s_{k}, s_{k + 1}]$ has $s_{k} \geq t$ . Its coefficient $ξ_{k}$ is $F_{s_{k}}$ -measurable, and [F6] with $A \in F_{t}$ makes $1_{A} ξ_{k}$ also $F_{s_{k}}$ -measurable. Thus $1_{A} {\tilde{H}}^{n}$ is elementary predictable for the Brownian coordinate in [F1], and the finite-sum definition [F3] gives $\int 1_{A} {\tilde{H}}^{n} d W^{j} = 1_{A} \int {\tilde{H}}^{n} d W^{j}$ . Applying [F5] to $1_{A} {\tilde{H}}^{n}$ yields $E ⁣ [1_{A} {∣ \int {\tilde{H}}^{n} d W^{j} ∣}^{2}] = E ⁣ [1_{A} \int ∣ {\tilde{H}}^{n} ∣^{2} d s] .$

3.1F4F5step 2.1

Multiplication by $1_{A}$ is a contraction in both $L^{2} (P)$ and $L^{2} (d s d P)$ . Hence [F4] and [F5] give $\int {\tilde{H}}^{n} d W^{j} \to \int H d W^{j}$ and $\int 1_{A} {\tilde{H}}^{n} d W^{j} \to \int 1_{A} H d W^{j}$ in $L^{2} (P)$ , while $1_{A} {\tilde{H}}^{n} \to 1_{A} H$ in $L^{2} (d s d P)$ . Passing the finite-sum identity of step 2.1 to these limits gives $\int 1_{A} H d W^{j} = 1_{A} \int H d W^{j}$ in $L^{2} (P)$ .

4.1F5step 3.1algebra

By [F5] and step 3.1, $E ⁣ [1_{A} {∣ \int H d W^{j} ∣}^{2}] = E {∣ \int 1_{A} H d W^{j} ∣}^{2} = E \int ∣ 1_{A} H ∣^{2} d s = E ⁣ [1_{A} \int ∣ H ∣^{2} d s] .$ All terms are finite because $H$ has finite energy, so the asserted identity follows.

5.1A1step 4.1∎

The argument applies to each coordinate separately and uses only finite partitions, $L^{2}$ limits, and the supplied Ito isometry. Its only inherited choice assumption is [A1].

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

Second and fourth moments of bounded-coefficient SDE increments

Statement

Let $X$ solve the autonomous SDE in Milstein updates, iterated Brownian integrals, and commutative noise on a filtered probability space with an $m$ -dimensional Brownian motion relative to the filtration. Assume $b, σ_{1}, \dots, σ_{m} \in C_{b}^{3} (R^{d})$ , with the $C_{b}^{3}$ convention of the Definition. There is a constant $C$ depending only on the coefficient bounds and $d, m$ such that for every $t \geq 0$ , $0 \leq r \leq 1$ , and $A \in F_{t}$ , $E ⁣ [1_{A} ∣ X_{t + r} - X_{t} ∣^{2}] \leq C r P (A), E ⁣ [1_{A} ∣ X_{t + r} - X_{t} ∣^{4}] \leq C r^{2} P (A) .$ These event-weighted inequalities are the conditional second- and fourth-moment bounds for the increment.

Facts & Assumptions

Given: AC, the Brownian It^o process and SDE coefficient assumptions in the Statement, $t \geq 0$ , $0 \leq r \leq 1$ , and an event $A \in F_{t}$ .

[L1]

Milstein updates, iterated Brownian integrals, and commutative noise: the state space is finite-dimensional, the drift and diffusion vector fields are componentwise $C_{b}^{3}$ , and their sup norms and derivative sup norms are finite.

[F1]

Continuous Brownian Ito processes: the usual filtration and Brownian hypotheses give a continuous adapted representative with a drift-plus-It^o-integral representation; Brownian increments after $t$ are independent of $F_{t}$ , and the process class is closed under stopping with drift and diffusion multiplied by the stopped indicator.

[F2]

Event-weighted Ito isometry after a conditioning time: for $A \in F_{t}$ , each scalar It^o integral over $(t, t + r]$ satisfies $E [1_{A} ∣ \int H, d W^{j} ∣^{2}] = E [1_{A} \int ∣ H ∣^{2} d s]$ .

[F3]

Multidimensional Ito formula for Brownian-driven processes: for $f \in C^{1, 2}$ , It^o's drift is $D f b + \frac{1}{2} \sum_{j} D^{2} f [σ_{j}, σ_{j}]$ and the martingale term is $\sum_{j} D f σ_{j} d W^{j}$ .

[F4]

The Ito integral process has a continuous martingale version: finite-energy It^o integral processes are square-integrable martingales, so their stopped increments have mean zero after multiplication by an event known at the start time.

[F5]

Fatou's lemma: for nonnegative random variables converging almost surely, the expectation of the limit is at most the lower limit of the expectations.

[A1]

The Axiom of Choice: AC is inherited from [F1]--[F3]; the moment estimates introduce no further choice.

Proof

Given: The assumptions above.

1.1L1F1F2A1

Put $Y_{s} = X_{s} - X_{t}$ , and let $B$ bound $∣ b ∣$ and $S^{2}$ bound $\sum_{j} ∣ σ_{j} ∣^{2}$ . The integral equation, $∣ \int_{t}^{t + r} b (X_{s}) d s ∣^{2} \leq B^{2} r^{2}$ , the finite-sum inequality $∣ \sum_{j} v_{j} ∣^{2} \leq m \sum_{j} ∣ v_{j} ∣^{2}$ , and [F2] give $E [1_{A} ∣ Y_{t + r} ∣^{2}] \leq 2 B^{2} r^{2} P (A) + 2 S^{2} r P (A) \leq C r P (A),$ since $r \leq 1$ . The event-weighted identity applies to each integrand $σ_{j} (X_{s}) 1_{(t, t + r]} (s)$ , whose squared norm is bounded by $∥ σ_{j} ∥_{\infty}^{2}$ .

2.1F1F2F3F4step 1.1

For $R > 0$ , use the continuous adapted version from [F1], modifying it on its exceptional null set if necessary; usual completeness makes this an adapted null modification and leaves all moment and SDE assertions unchanged. Let $θ_{R} = \inf {u \geq 0 : ∣ Y_{t + u} ∣ \geq R}$ . This is a stopping time: for fixed $u$ , ${θ_{R} \leq u} = {\sup_{0 \leq v \leq u} ∣ Y_{t + v} ∣ \geq R}$ , and continuity makes the supremum equal to the supremum over rational $v \in [0, u]$ together with $u$ , hence $F_{t + u}$ -measurable. Put $Y_{u}^{R} = X_{t + (u \land θ_{R})} - X_{t} = Y_{t + (u \land θ_{R})}$ , $q_{R} (u) = E [1_{A} ∣ Y_{u}^{R} ∣^{4}]$ , and $p_{R} (u) = E [1_{A} ∣ Y_{u}^{R} ∣^{2}]$ . By the stopped representation in [F1], the drift and diffusion integrands are multiplied by $1_{s - t \leq θ_{R}}$ . Its drift square is at most $B^{2} u^{2}$ , and applying [F2] coordinatewise with the finite-sum bound gives $p_{R} (u) \leq 2 B^{2} u^{2} P (A) + 2 S^{2} u P (A) \leq C u P (A)$ , by the same bound as step 1.1. Apply [F3] to $f (y) = ∣ y ∣^{4}$ and the stopped process. The Hessian formula gives drift $4 ∣ Y^{R} ∣^{2} ⟨ Y^{R}, b (X) ⟩ + 2 ∣ Y^{R} ∣^{2} \sum_{j} ∣ σ_{j} (X) ∣^{2} + 4 \sum_{j} ⟨ Y^{R}, σ_{j} (X) ⟩^{2}$ before the stopping time. Bounded coefficients and $4 B ∣ Y^{R} ∣^{3} \leq C (∣ Y^{R} ∣^{4} + ∣ Y^{R} ∣^{2})$ bound its absolute value by $C (∣ Y^{R} ∣^{4} + ∣ Y^{R} ∣^{2})$ . The stopped stochastic integrand is bounded, so [F4] makes its expectation after multiplication by $1_{A}$ zero. Consequently $q_{R} (u) \leq C \int_{0}^{u} q_{R} (v) d v + C u^{2} P (A), 0 \leq u \leq r .$ The second term follows from $p_{R} (v) \leq C v P (A)$ ; all constants are independent of $R$ .

3.1F5step 2.1algebra

Define $Q_{R} (u) = \int_{0}^{u} q_{R} (v) d v$ . The inequality in step 2.1 implies $Q_{R}^{'} (u) \leq C Q_{R} (u) + C u^{2} P (A)$ for almost every $u$ , with $Q_{R} (0) = 0$ . Multiplying by $e^{- C u}$ and integrating gives $Q_{R} (u) \leq C^{'} u^{3} P (A)$ for $u \leq 1$ . Returning to step 2.1 gives $q_{R} (u) \leq C^{''} u^{2} P (A)$ . As $R \to \infty$ , path continuity implies $Y_{u}^{R} \to Y_{t + u}$ almost surely; [F5] therefore yields the asserted fourth-moment bound.

4.1A1step 1.1step 3.1∎

Steps 1.1 and 3.1 are exactly the asserted event-weighted second- and fourth-moment estimates, with constants independent of $t, r, A$ . The stopped argument and finite-dimensional It^o calculation add no axiom beyond [A1].

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

Conditional local defect bounds for the Milstein Taylor increment

Statement

Let $X$ solve the finite-dimensional autonomous SDE in Milstein updates, iterated Brownian integrals, and commutative noise with $b, σ_{1}, \dots, σ_{m} \in C_{b}^{3} (R^{d})$ , using the $C_{b}^{3}$ convention in that Definition. Let $0 < h \leq 1$ and let $[t, t + h]$ be a deterministic interval. Define $L_{0} f : = D f b + \frac{1}{2} \sum_{k = 1}^{m} D^{2} f [σ_{k}, σ_{k}]$ and $L_{i} f : = D f σ_{i}$ for $i \geq 1$ . Then there is a remainder $R_{t, h} \in L^{2} (R^{d})$ such that $X_{t + h} = X_{t} + b (X_{t}) h + \sum_{j = 1}^{m} σ_{j} (X_{t}) Δ W^{j} + \sum_{i, j = 1}^{m} L_{i} σ_{j} (X_{t}) I_{i j} + R_{t, h},$ where $Δ W^{j} = W_{t + h}^{j} - W_{t}^{j}$ and $I_{i j} = \int_{t}^{t + h} (W_{s}^{i} - W_{t}^{i}) d W_{s}^{j}$ . Moreover, for the conditioning sigma algebra $F_{t}$ there is a conditional-expectation version $r_{t, h} = E [R_{t, h} ∣ F_{t}]$ with $∣ r_{t, h} ∣ \leq C h^{2}, E [∣ R_{t, h} ∣^{2} ∣ F_{t}] \leq C h^{3} almost surely,$ where $C$ depends only on $d, m$ and the coefficient bounds, not on $t, h$ or $X_{t}$ .

Facts & Assumptions

Given: AC, a Brownian It^o solution on a usual filtered probability space, the coefficient bounds in the Statement, and $0 < h \leq 1$ .

[L1]

Milstein updates, iterated Brownian integrals, and commutative noise: $I_{i j} = \int_{t}^{t + h} (W_{s}^{i} - W_{t}^{i}) d W_{s}^{j}$ , $L_{i} f = D f σ_{i}$ , and $C_{b}^{3}$ means all derivatives through order three, including the function itself, are bounded and continuous.

[L2]

Second and fourth moments of bounded-coefficient SDE increments: for $s \in [t, t + h]$ and every $A \in F_{t}$ , $E [1_{A} ∣ X_{s} - X_{t} ∣^{2}] \leq C (s - t) P (A)$ .

[F1]

Continuous Brownian Ito processes: $X$ has the drift-plus-It^o-integral decomposition, Brownian increments after $t$ are independent of $F_{t}$ , and finite-energy It^o integrals are martingales.

[F2]

Multidimensional Ito formula for Brownian-driven processes: for $f \in C^{1, 2}$ , $d f (X) = L_{0} f (X) d t + \sum_{i} L_{i} f (X) d W^{i}$ .

[F3]

Event-weighted Ito isometry after a conditioning time: for $A \in F_{t}$ , each scalar It^o integral over $(t, t + h]$ satisfies $E [1_{A} ∣ \int H, d W^{j} ∣^{2}] = E [1_{A} \int ∣ H ∣^{2} d s]$ .

[F4]

The Ito integral process has a continuous martingale version: a finite-energy It^o integral increment has conditional mean zero at its start time.

[F5]

Conditional expectation given a sigma algebra: a conditional expectation version has the same integrals as the original variable on every event in the conditioning sigma algebra.

[F6]

Conditional expectation exists by radon nikodym and Basic algebra and order properties of conditional expectation: under AC the conditional expectation versions used below exist, are linear, and preserve expectation and order.

[A1]

The Axiom of Choice: AC is used only through the declared Brownian, It^o, and conditional-expectation interfaces [F1]--[F6].

Proof

Given: The assumptions above.

1.1L1F1F2A1

Apply [F2] to $f = σ_{j}$ on $[t, s]$ . With the operators in [L1], $σ_{j} (X_{s}) = σ_{j} (X_{t}) + \int_{t}^{s} (L_{0} σ_{j}) (X_{r}) d r + \sum_{i} \int_{t}^{s} (L_{i} σ_{j}) (X_{r}) d W_{r}^{i} .$ Integrating this identity against $d W_{s}^{j}$ , summing over $j$ , and substituting the SDE decomposition of $X_{t + h} - X_{t}$ isolates the constant coefficient terms. Thus the remainder is exactly $R_{t, h} = R_{b} + R_{0} + R_{1},$ where $R_{b} = \int_{t}^{t + h} (b (X_{s}) - b (X_{t})) d s$ , $R_{0} = \sum_{j} \int_{t}^{t + h} (\int_{t}^{s} (L_{0} σ_{j}) (X_{r}) d r) d W_{s}^{j}$ , and $R_{1} = \sum_{i, j} \int_{t}^{t + h} (\int_{t}^{s} ((L_{i} σ_{j}) (X_{r}) - (L_{i} σ_{j}) (X_{t})) d W_{r}^{i}) d W_{s}^{j}$ . The subtraction in $R_{1}$ leaves precisely $\sum_{i, j} (L_{i} σ_{j}) (X_{t}) I_{i j}$ .

1.2L2F2F4F5F6A1

Since $L_{0} b$ is bounded, apply [F2] to $b (X_{s})$ and take event integrals over $A \in F_{t}$ . The stochastic increments have zero event integral by [F4]; componentwise this gives $∣ E [1_{A} R_{b}] ∣ \leq ∥ L_{0} b ∥_{\infty} h^{2} P (A) / 2$ . By [F5]--[F6], the vector conditional expectation $r_{b} = E [R_{b} ∣ F_{t}]$ , formed coordinatewise, exists. If any coordinate exceeded its bound on an $F_{t}$ event, the defining event-integral identity [F5] would contradict the bound just proved, so $∣ r_{b} ∣ \leq C h^{2}$ after increasing $C$ by a dimension factor. Also, the pathwise inequality $∣ \int_{t}^{t + h} g (s) d s ∣^{2} \leq h \int_{t}^{t + h} ∣ g (s) ∣^{2} d s$ follows by applying $0 \leq \int_{t}^{t + h} (∣ g ∣ - a)^{2} d s$ with $a = h^{- 1} \int_{t}^{t + h} ∣ g ∣ d s$ ; [L2] therefore gives $E [1_{A} ∣ R_{b} ∣^{2}] \leq h ∥ D b ∥_{\infty}^{2} \int_{t}^{t + h} C (s - t) P (A) d s \leq C h^{3} P (A)$ .

1.3F3F4A1

Each $L_{0} σ_{j}$ is bounded. Therefore the inner integral in $R_{0}$ has magnitude at most $C (s - t)$ . Its outer It^o integrals have zero conditional mean by [F4]. Apply the scalar event-weighted identity [F3] to each component of each outer vector integral and use $∣ \sum_{j} v_{j} ∣^{2} \leq m \sum_{j} ∣ v_{j} ∣^{2}$ ; this gives $E [1_{A} ∣ R_{0} ∣^{2}] \leq C \int_{t}^{t + h} (s - t)^{2} d s P (A) \leq C h^{3} P (A)$ .

1.4L1L2F3F4A1

Each $L_{i} σ_{j}$ is globally Lipschitz: its derivative is a sum of products of first and second derivatives of the $σ_{k}$ , bounded by [L1]. The inner integral $J_{i j} (s) = \int_{t}^{s} ((L_{i} σ_{j}) (X_{r}) - (L_{i} σ_{j}) (X_{t})) d W_{r}^{i}$ has zero conditional mean and, by [F3] coordinatewise and [L2], satisfies $E [1_{A} ∣ J_{i j} (s) ∣^{2}] \leq C \int_{t}^{s} E [1_{A} ∣ X_{r} - X_{t} ∣^{2}] d r \leq C (s - t)^{2} P (A)$ . Each outer integral has zero conditional mean by [F4]; applying the scalar identity [F3] componentwise and $∣ \sum_{i j} v_{i j} ∣^{2} \leq m^{2} \sum_{i j} ∣ v_{i j} ∣^{2}$ gives $E [1_{A} ∣ R_{1} ∣^{2}] \leq C h^{3} P (A)$ .

2.1F5F6step 1.2step 1.3step 1.4

The sum $R_{0} + R_{1}$ has conditional mean zero, while $R_{b}$ has conditional mean $r_{b}$ from step 1.2; set $r_{t, h} = r_{b}$ . The inequality $∣ x + y + z ∣^{2} \leq 3 (∣ x ∣^{2} + ∣ y ∣^{2} + ∣ z ∣^{2})$ and steps 1.2--1.4 show $E [1_{A} ∣ R_{t, h} ∣^{2}] \leq C h^{3} P (A)$ for every $A \in F_{t}$ . By the defining event-integral property [F5] and order property [F6], this is exactly $E [∣ R_{t, h} ∣^{2} ∣ F_{t}] \leq C h^{3}$ almost surely.

3.1A1step 1.1step 1.2step 2.1∎

Steps 1.2 and 2.1 give the conditional mean and second-moment bounds in the Statement, and step 1.1 gives the exact Taylor expansion. Every constant is uniform in the starting state because the coefficients and derivatives are bounded. AC enters only through [A1].

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Discrete stochastic local-to-global RMS stability bound

Statement

Let $N \geq 1$ , $0 < h \leq 1$ , and $N h \leq T$ . Let $(F_{n})_{n = 0}^{N}$ be a filtration as in Filtration and filtered probability space. For every $n$ , let $e_{n}, u_{n}, r_{n}$ be square-integrable $R^{d}$ -valued, $F_{n}$ -measurable random vectors; let $M_{n + 1}, Q_{n + 1}$ be square-integrable and $F_{n + 1}$ -measurable. Say $Z$ is orthogonal to $F_{n}$ if $E ⟨ H, Z ⟩ = 0$ for every square-integrable $F_{n}$ -measurable vector $H$ . Suppose $e_{0} = 0, e_{n + 1} = e_{n} + h u_{n} + r_{n} + M_{n + 1} + Q_{n + 1},$ $∣ u_{n} ∣ \leq L ∣ e_{n} ∣, ∣ r_{n} ∣ \leq C_{0} h^{2},$ and $M_{n + 1}, Q_{n + 1}$ are orthogonal to $F_{n}$ , with $E ∣ M_{n + 1} ∣^{2} \leq C_{1} h E ∣ e_{n} ∣^{2}, E ∣ Q_{n + 1} ∣^{2} \leq C_{1} h^{3} .$ Then $E ∣ e_{N} ∣^{2} \leq C_{T} h^{2}$ , where $C_{T}$ depends only on $T, L, C_{0}, C_{1}$ and not on $h$ or $N$ . This is a local-to-global estimate for an order-one grid RMS error when the drift error is Lipschitz and the centered one-step defect has second moment $O (h^{3})$ .

Facts & Assumptions

Given: The filtered probability space, the square-integrable vectors, the recurrence, orthogonality, and numerical bounds in the Statement.

[F1]

Filtration and filtered probability space: $F_{n} \subseteq F_{n + 1}$ and each $F_{n}$ is a sigma algebra on the underlying probability space.

Proof

Given: The assumptions above.

1.1givenF1algebra

Since $∣ e_{n} + h u_{n} ∣ \leq (1 + L h) ∣ e_{n} ∣$ and $∣ r_{n} ∣ \leq C_{0} h^{2}$ , the elementary inequality $2 a b \leq h a^{2} + h^{- 1} b^{2}$ and $h \leq 1$ give $E ∣ e_{n} + h u_{n} + r_{n} ∣^{2} \leq (1 + C h) E ∣ e_{n} ∣^{2} + C h^{3} .$ Indeed, the square of $e_{n} + h u_{n}$ contributes at most $(1 + L h)^{2} ∣ e_{n} ∣^{2} \leq (1 + C h) ∣ e_{n} ∣^{2}$ , its cross term with $r_{n}$ is bounded by $C h^{2} ∣ e_{n} ∣ \leq C h ∣ e_{n} ∣^{2} + C h^{3}$ , and $∣ r_{n} ∣^{2} \leq C h^{4} \leq C h^{3}$ .

2.1givenstep 1.1algebra

Orthogonality makes the cross expectation between $e_{n} + h u_{n} + r_{n}$ and $M_{n + 1} + Q_{n + 1}$ zero. To bound the remaining cross term, put $a = ∣ M_{n + 1} ∣$ and $b = ∣ Q_{n + 1} ∣$ ; nonnegativity of $E (a - λ b)^{2}$ for every real $λ$ implies $E [a b] \leq (E a^{2})^{1 / 2} (E b^{2})^{1 / 2}$ . Hence $2 ∣ E ⟨ M_{n + 1}, Q_{n + 1} ⟩ ∣ \leq 2 (E ∣ M_{n + 1} ∣^{2})^{1 / 2} (E ∣ Q_{n + 1} ∣^{2})^{1 / 2} \leq C h^{2} (E ∣ e_{n} ∣^{2})^{1 / 2} \leq C h E ∣ e_{n} ∣^{2} + C h^{3} .$ The final inequality is $2 a b \leq a^{2} + b^{2}$ with factors proportional to $h^{1 / 2} (E ∣ e_{n} ∣^{2})^{1 / 2}$ and $h^{3 / 2}$ . Therefore the recurrence and step 1.1 imply $E ∣ e_{n + 1} ∣^{2} \leq (1 + C h) E ∣ e_{n} ∣^{2} + C h^{3} .$

3.1givenstep 2.1algebra∎

Iterating the inequality in step 2.1 from $e_{0} = 0$ gives $E ∣ e_{N} ∣^{2} \leq C h^{3} \sum_{j = 0}^{N - 1} (1 + C h)^{j} \leq C h^{3} N e^{C N h} \leq C_{T} h^{2},$ because $(1 + C h)^{j} \leq e^{C j h} \leq e^{C T}$ and $N h \leq T$ . Taking square roots proves the stated RMS estimate. This recurrence argument uses only finite algebraic estimates and the given orthogonality conditions.

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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].

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Explicit stochastic Heun update for additive noise

Definition

Assume AC and the Brownian filtered-space conventions in Continuous Brownian Ito processes. Consider the scalar additive-noise It^o SDE $d X_{t} = b (X_{t}) d t + σ d W_{t},$ where $σ \in R$ is constant. Given a step size $h > 0$ , a grid value $Y_{n}$ , and the Brownian increment $Δ W_{n} = W_{t_{n + 1}} - W_{t_{n}}$ , the explicit stochastic Heun update is the two-stage rule $Z_{n} = Y_{n} + b (Y_{n}) h + σ Δ W_{n}, Y_{n + 1} = Y_{n} + \frac{h}{2} (b (Y_{n}) + b (Z_{n})) + σ Δ W_{n} .$ The predictor $Z_{n}$ uses the same increment as the corrector. One step evaluates $b$ at $Y_{n}$ and $Z_{n}$ and uses one Gaussian increment. The weak order-two guarantee authored in this pair is only for this constant-diffusion scheme; the displayed update omits terms required for the corresponding multiplicative-noise guarantee.

In this definition and its consumers, $C_{b}^{k} (R; R)$ means that the function and all its derivatives through order $k$ are continuous and bounded.

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One-step moment matching for additive-noise stochastic Heun

Statement

Let $b \in C_{b}^{6} (R; R)$ , let $σ \in R$ be constant, and define the generator $L f = b f^{'} + \frac{1}{2} σ^{2} f^{''}$ . Let $u : [0, T] \times R \to R$ satisfy $u_{t} + L u = 0$ . Assume all mixed derivatives $\partial_{t}^{r} \partial_{x}^{j} u$ for $0 \leq r \leq 3$ and $0 \leq j \leq 6$ are continuous and bounded. For $0 < h \leq 1$ , $t + h \leq T$ , fixed $x \in R$ , and $w \sim N (0, h)$ , take one Heun step from Explicit stochastic Heun update for additive noise and let $D = Y_{n + 1} - x$ . Then a constant $C$ , uniform in $t, x, h$ , satisfies $∣ E [u (t + h, x + D)] - u (t, x) ∣ \leq C h^{3} .$

Facts & Assumptions

Given: The additive-noise Heun update, $b \in C_{b}^{6}$ , the constant $σ$ , the backward equation $u_{t} + L u = 0$ , its bounded derivative assumptions, $w \sim N (0, h)$ , and $0 < h \leq 1$ .

[L1]

Explicit stochastic Heun update for additive noise: for increment $w$ , $Z = x + b (x) h + σ w$ and $D = σ w + \frac{h}{2} (b (x) + b (Z))$ ; $C_{b}^{6}$ means bounded continuous derivatives through order six.

[F1]

Standard normal and normal laws: $w = \sqrt{h} Z_{0}$ with $Z_{0}$ standard normal of density $(2 π)^{- 1 / 2} e^{- z^{2} / 2}$ ; its odd moments vanish and its even moments are obtained by integration by parts.

[A1]

The Axiom of Choice: AC is inherited through the Brownian/normal-law interface in [L1] and [F1]; the one-step Taylor calculation itself uses no further choice.

Proof

Given: The assumptions above.

1.1L1F1A1

Put $B = b (x)$ , $B_{j} = b^{(j)} (x)$ , $δ = B h + σ w$ , and $D = σ w + B h + \frac{h}{2} (b (x + δ) - B)$ . Taylor's formula through degree three gives $b (x + δ) = B + B_{1} δ + \frac{1}{2} B_{2} δ^{2} + \frac{1}{6} B_{3} δ^{3} + ρ_{4}$ , with $∣ ρ_{4} ∣ \leq ∥ b^{(4)} ∥_{\infty} ∣ δ ∣^{4} / 24$ . From [F1], $E w = 0$ , $E w^{2} = h$ , $E w^{3} = 0$ , $E w^{4} = 3 h^{2}$ , $E w^{5} = 0$ , and $E ∣ w ∣^{6} = 15 h^{3}$ . Substituting $δ = B h + σ w$ and collecting terms of order at most $h^{2}$ gives $\begin{aligned} E D & = B h + (\frac{1}{2} B B_{1} + \frac{1}{4} σ^{2} B_{2}) h^{2} + O (h^{3}), \\ E D^{2} & = σ^{2} h + (B^{2} + σ^{2} B_{1}) h^{2} + O (h^{3}), \\ E D^{3} & = 3 B σ^{2} h^{2} + O (h^{3}), \\ E D^{4} & = 3 σ^{4} h^{2} + O (h^{3}), E D^{5} = O (h^{3}) . \end{aligned}$ For the second moment, the $h^{2}$ terms are $(B h)^{2}$ and $2 (σ w) (\frac{1}{2} B_{1} σ h w)$ ; for the third they are $3 (σ w)^{2} (B h)$ ; for the fourth it is $E (σ w)^{4}$ . All other terms either contain an odd Gaussian moment or have expected magnitude $O (h^{3})$ by $E ∣ δ ∣^{4} = O (h^{2})$ and the displayed remainder bound. Since $b$ is bounded, $∣ D ∣ \leq ∣ σ ∣ ∣ w ∣ + h ∥ b ∥_{\infty}$ , so $E ∣ D ∣^{6} \leq C h^{3}$ .

2.1L1F1step 1.1

Taylor-expand $u (t + h, x + D)$ in its spatial argument through degree five. The sixth-derivative bound and step 1.1 yield $E u (t + h, x + D) = u + u_{x} E D + \frac{1}{2} u_{x x} E D^{2} + \frac{1}{6} u_{x x x} E D^{3} + \frac{1}{24} u_{x x x x} E D^{4} + \frac{1}{120} u_{x x x x x} E D^{5} + O (h^{3}),$ where the displayed derivatives are evaluated at $(t + h, x)$ . Substituting the moment formulas, the coefficients through order $h^{2}$ are $h L u$ and $\frac{1}{2} h^{2} L^{2} u$ , because $L^{2} u = (B B_{1} + \frac{1}{2} σ^{2} B_{2}) u_{x} + (B^{2} + σ^{2} B_{1}) u_{x x} + B σ^{2} u_{x x x} + \frac{1}{4} σ^{4} u_{x x x x} .$ Thus $E u (t + h, x + D) = u (t + h, x) + h L u (t + h, x) + \frac{1}{2} h^{2} L^{2} u (t + h, x) + O (h^{3})$ .

3.1givenstep 2.1algebra

Since $u_{t} = - L u$ and $L$ is independent of $t$ , differentiating the backward equation gives $u_{t t} = L^{2} u$ . Taylor's formula in time at fixed $x$ , with bounded $u_{t t t}$ , therefore gives $u (t, x) = u (t + h, x) + h L u (t + h, x) + \frac{1}{2} h^{2} L^{2} u (t + h, x) + O (h^{3}) .$ The coefficient and derivative bounds in the Statement make the constants in the preceding expansions uniform in $t, x, h$ .

4.1step 2.1step 3.1∎

The expansions in steps 2.1 and 3.1 have identical terms through order $h^{2}$ ; subtracting them leaves a remainder bounded by $C h^{3}$ , which is the asserted one-step weak defect.

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Weak order two for additive-noise stochastic Heun

Statement

Assume AC and the Brownian filtration conventions in Continuous Brownian Ito processes. Consider $d X_{t} = b (X_{t}) d t + σ d W_{t}$ on $[0, T]$ , where $b \in C_{b}^{6} (R; R)$ , $σ \in R$ is constant, and $X_{0} = x_{0} \in R$ is deterministic. For $t \leq T$ and $x \in R$ , let $X^{t, x}$ denote the solution of the same SDE started from $x$ at time $t$ . Let $ϕ : R \to R$ have a bounded classical backward solution $u (t, x) = E [ϕ (X_{T}^{t, x})]$ with $u (T, x) = ϕ (x)$ , $u_{t} + L u = 0$ , $L = b \partial_{x} + \frac{1}{2} σ^{2} \partial_{x x}$ , and all mixed derivatives $\partial_{t}^{r} \partial_{x}^{j} u$ for $0 \leq r \leq 3$ and $0 \leq j \leq 6$ continuous and bounded. On a uniform grid $h = T / N \leq 1$ , initialize $Y_{0} = x_{0}$ and apply the additive-noise stochastic Heun update in Explicit stochastic Heun update for additive noise. There is a constant $C_{T}$ , independent of $h$ , such that $∣ E ϕ (Y_{N}) - E ϕ (X_{T}) ∣ \leq C_{T} h^{2} .$ This is weak order two, meaning the expectation error is bounded by a constant times $h^{2}$ for the stated payoff class. The claim is specific to constant additive diffusion; no multiplicative-noise order-two result is asserted.

Facts & Assumptions

Given: AC; the scalar additive-noise SDE, deterministic initial value $x_{0}$ , drift and constant diffusion assumptions; the backward solution and derivative bounds; and the uniform grid of the Statement.

[L1]

Explicit stochastic Heun update for additive noise: $Z = Y + b (Y) h + σ Δ W$ and $Y^{+} = Y + \frac{h}{2} (b (Y) + b (Z)) + σ Δ W$ .

[L2]

One-step moment matching for additive-noise stochastic Heun: for every starting time/state, the expectation of $u (t + h, Y^{+}) - u (t, Y)$ over one Brownian increment has absolute value at most $C h^{3}$ , uniformly in the starting state.

[F1]

Continuous Brownian Ito processes: each next Brownian increment is independent of the current filtration and has its stated normal law.

[A1]

The Axiom of Choice: AC is inherited through the Brownian interface [F1]; the telescoping argument uses no further choice.

Proof

Given: The assumptions above.

1.1F1L1L2given

For each $n$ , the numerical value $Y_{n}$ is $F_{t_{n}}$ -measurable by induction from the update [L1]. The next increment $Δ W_{n}$ is independent of $F_{t_{n}}$ by [F1]. Therefore the joint law of $(Y_{n}, Δ W_{n})$ is the product of the law of $Y_{n}$ and $N (0, h)$ ; integrating the uniform one-step bound [L2] against the law of $Y_{n}$ gives $∣ E u (t_{n + 1}, Y_{n + 1}) - E u (t_{n}, Y_{n}) ∣ \leq C h^{3} .$

2.1step 1.1algebra

Sum the inequalities in step 1.1 over $n = 0, \dots, N - 1$ . The intermediate expectations telescope, so $∣ E u (T, Y_{N}) - E u (0, Y_{0}) ∣ \leq N C h^{3} = C_{T} h^{2},$ because $N h = T$ .

3.1A1step 2.1given∎

The terminal condition gives $u (T, Y_{N}) = ϕ (Y_{N})$ . Since $Y_{0} = x_{0}$ and $u (0, x_{0})$ is the expectation of $ϕ$ for the exact SDE started at $(0, x_{0})$ , $E u (0, Y_{0}) = u (0, x_{0}) = E ϕ (X_{T})$ . Substituting these identities into step 2.1 yields the claimed weak error. The argument uses only the admitted additive-noise backward solution and makes no inference for state-dependent diffusion.

TheoremStatement: Literature-sourcedProof: AI-generatedOpen item page →

Milstein MLMC correction variance and inverse-epsilon-squared cost

Statement

Assume AC and the scalar or commutative-noise SDE hypotheses of Strong order one for scalar and commutative-noise Milstein schemes. Let $T > 0$ , use dyadic step sizes $h_{ℓ} = T 2^{- ℓ}$ , and let $ψ : R^{d} \to R$ be Lipschitz. At each level form the Milstein approximation $Y_{ℓ} (T)$ ; couple adjacent levels by using the same Brownian path, so each coarse increment is the sum of its two fine increments. Put $P_{ℓ} = ψ (Y_{ℓ} (T))$ and let $Δ_{ℓ} = P_{ℓ}^{f} - P_{ℓ - 1}^{c}$ for $ℓ \geq 1$ , as in Geometric MLMC hierarchy and coupled corrections. For fixed finite $d, m$ and bounded-cost coefficient and derivative evaluations, constants independent of $ℓ$ satisfy $∣ E P_{ℓ} - E ψ (X_{T}) ∣ \leq C_{b} h_{ℓ}, Var (Δ_{ℓ}) \leq C_{v} h_{ℓ}^{2}, C_{ℓ} \leq C_{c} h_{ℓ}^{- 1} .$ The exact statement and proof of MLMC cost complexity under bias, variance, and work rates then give an MLMC estimator with root-mean-square error at most $ϵ$ and total work $O (ϵ^{- 2})$ for all sufficiently small $ϵ > 0$ . The variance statement is for Lipschitz terminal payoffs and the stated nested coupling; no Euler correction-rate theorem or area-sampling cost is substituted.

Facts & Assumptions

Given: The SDE and Milstein hypotheses, a fixed finite horizon $T$ , a Lipschitz payoff $ψ$ , the dyadic hierarchy, nested Brownian coupling, and bounded work per Milstein step.

[L1]

Milstein updates, iterated Brownian integrals, and commutative noise: under commutative noise the Milstein update uses Brownian increments and coefficient derivatives, with no iterated-area samples.

[L2]

Strong order one for scalar and commutative-noise Milstein schemes: on a shared Brownian path, $E ∣ Y_{h} (T) - X_{T} ∣^{2} \leq C h^{2}$ for all sufficiently small grid steps $h$ .

[F1]

Geometric MLMC hierarchy and coupled corrections: dyadic levels use $h_{ℓ} = T 2^{- ℓ}$ , a fine/coarse coupling, and correction $Δ_{ℓ} = P_{ℓ}^{f} - P_{ℓ - 1}^{c}$ .

[F2]

MLMC cost complexity under bias, variance, and work rates: if bias is $O (h^{α})$ with $α \geq 1 / 2$ , correction variance is $O (h^{β})$ with $β > 0$ , and one correction sample costs $O (h^{- 1})$ , then the MLMC work is $O (ϵ^{- 2})$ when $β > 1$ .

[F3]

Cauchy-Schwarz for random variables: $E ∣ U V ∣ \leq (E ∣ U ∣^{2})^{1 / 2} (E ∣ V ∣^{2})^{1 / 2}$ for square-integrable real variables; applied to norms it gives the $L^{2}$ triangle bound and $E ∣ U ∣ \leq ∥ U ∥_{2}$ .

[A1]

The Axiom of Choice: AC is inherited through the Brownian and strong Milstein interfaces; no additional choice is used in the coupling or rate argument.

Proof

Given: The hypotheses above.

1.1L1L2given

For all sufficiently fine levels, [L2] gives $∥ Y_{ℓ} (T) - X_{T} ∥_{2} \leq C h_{ℓ}$ , where $∥ Z ∥_{2} = (E ∣ Z ∣^{2})^{1 / 2}$ . There are only finitely many coarser dyadic levels; each has finitely many steps and finite second moments because the coefficients and derivatives are bounded. Enlarging $C$ therefore makes the same error bound valid for every level. The same argument gives $E ∣ ψ (Y_{ℓ} (T)) ∣^{2} < \infty$ , since $∣ ψ (x) ∣ \leq ∣ ψ (0) ∣ + Lip (ψ) ∣ x ∣$ .

1.2L1F1givenalgebra

A level- $ℓ$ fine path has $T / h_{ℓ} = 2^{ℓ}$ steps and its coarse companion has $2^{ℓ - 1}$ steps. Each commutative Milstein step uses a fixed finite number of coefficient/derivative evaluations and Gaussian vector samples for fixed $d, m$ ; [L1] requires no Lévy-area sampler. Thus one correction sample has work at most $C (2^{ℓ} + 2^{ℓ - 1}) \leq C_{c} h_{ℓ}^{- 1}$ , including the finitely many coarse initial levels after increasing $C_{c}$ .

2.1givenL2F1F3step 1.1algebra

The fine and coarse approximations in $Δ_{ℓ}$ use the same Brownian path and hence the same exact terminal value $X_{T}$ . By [F3], $∥ U + V ∥_{2} \leq ∥ U ∥_{2} + ∥ V ∥_{2}$ ; with $h_{ℓ - 1} = 2 h_{ℓ}$ this gives $∥ Y_{ℓ}^{f} (T) - Y_{ℓ - 1}^{c} (T) ∥_{2} \leq ∥ Y_{ℓ}^{f} (T) - X_{T} ∥_{2} + ∥ Y_{ℓ - 1}^{c} (T) - X_{T} ∥_{2} \leq C (h_{ℓ} + h_{ℓ - 1}) \leq C^{'} h_{ℓ} .$ For each $ℓ \geq 1$ , the Lipschitz property yields $Var (Δ_{ℓ}) \leq E ∣ Δ_{ℓ} ∣^{2} \leq Lip (ψ)^{2} E ∣ Y_{ℓ}^{f} (T) - Y_{ℓ - 1}^{c} (T) ∣^{2} \leq C_{v} h_{ℓ}^{2}$ . At level zero, $Δ_{0} = P_{0}$ by [F1] and has finite variance by step 1.1; increasing $C_{v}$ to also dominate $Var (P_{0}) / h_{0}^{2}$ makes the same envelope hold there.

2.2L2F1F3step 1.1algebra

Again by the Lipschitz property and [F3], $∣ E P_{ℓ} - E ψ (X_{T}) ∣ \leq E ∣ ψ (Y_{ℓ} (T)) - ψ (X_{T}) ∣ \leq Lip (ψ) ∥ Y_{ℓ} (T) - X_{T} ∥_{2} \leq C_{b} h_{ℓ} .$ The finite-level second-moment statement in step 1.1 verifies the square-integrability premise of [F1].

3.1F2L2A1step 2.1step 2.2step 1.2∎

The estimates in steps 2.1, 2.2, and 1.2 verify [F2] with $α = 1$ and $β = 2$ . Since $2 > 1$ and $1 \geq 1 / 2$ , applying [F2] proves the stated $O (ϵ^{- 2})$ MLMC work bound. The local variance argument uses only the strong order-one result [L2] for nested commutative-Milstein paths; AC is inherited as stated in [A1].

5 · Examples, counterexamples and false statements

None yet.

Milstein stochastic taylor and iterated integrals

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Milstein updates, iterated Brownian integrals, and commutative noise

Definition

Product identity and conditional moments for iterated Brownian integrals

Statement

Facts & Assumptions

Proof

Event-weighted Ito isometry after a conditioning time

Statement

Facts & Assumptions

Proof

Second and fourth moments of bounded-coefficient SDE increments

Statement

Facts & Assumptions

Proof

Conditional local defect bounds for the Milstein Taylor increment

Statement

Facts & Assumptions

Proof

Discrete stochastic local-to-global RMS stability bound

Statement

Facts & Assumptions

Proof

Strong order one for scalar and commutative-noise Milstein schemes

Statement

Facts & Assumptions

Proof

Explicit stochastic Heun update for additive noise

Definition

One-step moment matching for additive-noise stochastic Heun

Statement

Facts & Assumptions

Proof

Weak order two for additive-noise stochastic Heun

Statement

Facts & Assumptions

Proof

Milstein MLMC correction variance and inverse-epsilon-squared cost

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources