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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.

Depends on

Used by

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

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Sources

Desmond J. Higham, Xuerong Mao, and Andrew M. Stuart, Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations, Lemma 4.3 (standard reference, not scraped)
Simo Särkkä and Arno Solin, Applied Stochastic Differential Equations, §8.2, Algorithm 8.1 (standard reference, not scraped)