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Strong order one-half for tamed Euler under polynomial drift growth

Statement

Assume AC. Let $d, m \geq 1$ , $T > 0$ , $b \in C^{1} (R^{d}; R^{d})$ , and $σ : R^{d} \to R^{d \times m}$ be globally Lipschitz. Suppose constants $L, C < \infty$ and an integer $q \geq 1$ satisfy $⟨ x - y, b (x) - b (y) ⟩ \leq L ∣ x - y ∣^{2}, ∥ D b (x) ∥ \leq C (1 + ∣ x ∣^{q}) .$ Let $X$ be the unique strong solution of the corresponding autonomous SDE from Global existence and finite-horizon moments for one-sided Lipschitz SDEs, with initial state $ξ$ having all finite moments. For $N \geq \max (1, ⌈ T ⌉)$ put $h = T / N$ , and let $\tilde{Y}$ be the continuous tamed Euler interpolation from Tamed Euler scheme for a superlinear drift. There is a constant $C_{T}$ , independent of $N$ , such that $E \sup_{0 \leq t \leq T} ∣ X_{t} - {\tilde{Y}}_{t} ∣^{2} \leq C_{T} h .$ Thus the tamed Euler method has strong order one-half in the path-supremum $L^{2}$ norm under the displayed hypotheses.

Facts & Assumptions

Given: AC; the autonomous SDE and coefficient hypotheses in the Statement; $h = T / N \leq 1$ ; and the tamed update and interpolation.

[F1]

The local Itô SDE definition specifies the filtered Brownian conventions and the integral equation that a local strong solution must satisfy; it does not assert existence. The global adapted continuous solution used here is supplied by [F2]. Locally Lipschitz Itô SDE on an open state domain

[F2]

Under the stated one-sided Lipschitz, polynomial derivative, and diffusion Lipschitz assumptions, $X$ is global and has finite path-supremum moments of every finite order. Global existence and finite-horizon moments for one-sided Lipschitz SDEs

[F3]

The tamed grid values satisfy $\sup_{N} \max_{n} E ∣ Y_{n} ∣^{p} < \infty$ for every finite $p \geq 1$ . Uniform finite-horizon moments for tamed Euler approximations

[F4]

The algorithm gives ${\tilde{Y}}_{t} = Y_{n} + (t - t_{n}) b (Y_{n}) / (1 + h ∣ b (Y_{n}) ∣) + σ (Y_{n}) (W_{t} - W_{t_{n}})$ on $[t_{n}, t_{n + 1}]$ . Tamed Euler scheme for a superlinear drift

[F5]

For a continuous Brownian Itô process $Z$ and $f \in C^{1, 2}$ , Itô's formula for $∣ Z ∣^{2}$ has drift $2 ⟨ Z, b_{Z} ⟩ + ∥ σ_{Z} ∥_{F}^{2}$ and martingale integrand $2 Z^{T} σ_{Z}$ . Multidimensional Ito formula for Brownian-driven processes

[F6]

For a finite-energy row-vector integrand $H$ against an $m$ -dimensional Brownian motion, $E \sup_{s \leq t} ∣ \int_{0}^{s} H_{u} d W_{u} ∣ \leq 16 \sqrt{m} E (\int_{0}^{t} ∥ H_{u} ∥_{2}^{2} d u)^{1 / 2}$ . An L1 maximal bound for finite-energy Brownian Itô integrals

[F7]

Cauchy--Schwarz for random variables bounds $E ∣ U V ∣ \leq (E ∣ U ∣^{2})^{1 / 2} (E ∣ V ∣^{2})^{1 / 2}$ . Cauchy-Schwarz for random variables

[F8]

If $v (t) \leq A + \int_{0}^{t} B v (s) d s$ with 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

[F9]

A standard normal variable has finite absolute moments of every finite order, since $\int_{0}^{\infty} x^{r} e^{- x^{2} / 2} d x < \infty$ ; scaling gives $E ∣ W_{s} - W_{t} ∣^{r} = C_{r, m} ∣ s - t ∣^{r / 2}$ . Standard normal and normal laws

[F10]

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

[F11]

For fixed finite $N$ , the interpolation has finite fourth path-supremum moment: the grid values and coefficients have finite eighth moments by [F3] and polynomial drift growth, while the Brownian path supremum has finite eighth moment by the $p = 8$ maximal Itô-integral bound with constant integrand. Apply Cauchy--Schwarz to each product of a grid diffusion coefficient and a Brownian interval supremum, then sum over the finitely many intervals in [F4]. In particular the second path-supremum moment is finite as well. Lp maximal moment bound for Brownian Itô integrals

[F12]

Since $e$ has continuous paths and $E \sup_{s \leq T} ∣ e_{s} ∣^{2} < \infty$ by [F2] and [F11], dominated convergence makes $v (t) = E \sup_{s \leq t} ∣ e_{s} ∣^{2}$ continuous. Dominated convergence

[A1]

AC is inherited from the SDE and Brownian Itô interfaces; the estimates below use only the specified Brownian increments and stopping conventions. The Axiom of Choice

Proof

Given: The assumptions and facts in the Statement and Facts.

1.1F1F2F3F4F5given

Let ${\underline{Y}}_{s} = Y_{n}$ on $[t_{n}, t_{n + 1})$ and $δ_{s} = {\tilde{Y}}_{s} - {\underline{Y}}_{s}$ . Induction in the recursion shows $Y_{n}$ is $F_{t_{n}}$ -measurable, so the step coefficient processes have predictable representatives, using $Y_{n}$ on $(t_{n}, t_{n + 1}]$ for stochastic integrands; changing values at grid times does not change the integrals. The interpolation is adapted and continuous and satisfies ${\tilde{Y}}_{t} = ξ + \int_{0}^{t} \frac{b ({\underline{Y}}_{s})}{1 + h ∣ b ({\underline{Y}}_{s}) ∣} d s + \int_{0}^{t} σ ({\underline{Y}}_{s}) d W_{s} .$ Its coefficient energies are finite by [F2]–[F3] and the polynomial growth of $b$ . Hence $e_{t} = X_{t} - {\tilde{Y}}_{t}$ is a continuous Brownian Itô process and [F5] applies.

1.2F3F4F9givenalgebra

On an interval $[t_{n}, t_{n + 1}]$ , $δ_{s} = (s - t_{n}) b_{h} (Y_{n}) + σ (Y_{n}) (W_{s} - W_{t_{n}})$ . For $r = 2, 4$ , $∣ (s - t_{n}) b_{h} (Y_{n}) ∣ \leq h ∣ b (Y_{n}) ∣$ , and the Brownian increment is independent of $F_{t_{n}}$ with $r$ th moment $C_{r, m} (s - t_{n})^{r / 2}$ by [F9]. The moment bounds [F3], polynomial growth $∣ b (x) ∣ \leq C_{1} (1 + ∣ x ∣^{q + 1})$ , and linear growth of $σ$ give $\sup_{n} E ∣ b (Y_{n}) ∣^{r} < \infty$ and $\sup_{n} E ∥ σ (Y_{n}) ∥_{F}^{r} < \infty$ . Since $h^{r} \leq h^{r / 2}$ for $h \leq 1$ , these estimates imply $\sup_{n < N} \sup_{s \in [t_{n}, t_{n + 1}]} E ∣ δ_{s} ∣^{r} \leq C_{r} h^{r / 2}, r = 2, 4.$ Also, with $b_{h} (y) = b (y) / (1 + h ∣ b (y) ∣)$ , $∣ b (y) - b_{h} (y) ∣ \leq h ∣ b (y) ∣^{2}$ , so $\sup_{n} E ∣ b (Y_{n}) - b_{h} (Y_{n}) ∣^{2} \leq C h^{2}$ by [F3].

2.1F2F3F7F10step 1.2givenalgebra

Put $G_{s} = σ (X_{s}) - σ ({\underline{Y}}_{s})$ and $A_{s} = 1 + ∣ X_{s} ∣^{q} + ∣ {\underline{Y}}_{s} ∣^{q}$ . The derivative bound and integration along the segment from $X_{s}$ to ${\underline{Y}}_{s}$ give $∣ b (X_{s}) - b ({\underline{Y}}_{s}) ∣ \leq C A_{s} ∣ X_{s} - {\underline{Y}}_{s} ∣$ . Since $X_{s} - {\underline{Y}}_{s} = e_{s} + δ_{s}$ , one-sided Lipschitzness gives $2 ⟨ e_{s}, b (X_{s}) - b ({\underline{Y}}_{s}) ⟩ = 2 ⟨ X_{s} - {\underline{Y}}_{s}, b (X_{s}) - b ({\underline{Y}}_{s}) ⟩ - 2 ⟨ δ_{s}, b (X_{s}) - b ({\underline{Y}}_{s}) ⟩ \leq C ∣ e_{s} ∣^{2} + C (1 + A_{s}^{2}) ∣ δ_{s} ∣^{2} .$ Here the second term uses $2 A_{s} ∣ δ_{s} ∣ ∣ e_{s} ∣ \leq ∣ e_{s} ∣^{2} + A_{s}^{2} ∣ δ_{s} ∣^{2}$ . The taming correction contributes at most $∣ e_{s} ∣^{2} + ∣ b ({\underline{Y}}_{s}) - b_{h} ({\underline{Y}}_{s}) ∣^{2}$ , and $∥ G_{s} ∥_{F}^{2} \leq C (∣ e_{s} ∣^{2} + ∣ δ_{s} ∣^{2})$ by global Lipschitzness of $σ$ . Thus a nonnegative $H_{s} = C (∣ e_{s} ∣^{2} + (1 + A_{s}^{2}) ∣ δ_{s} ∣^{2} + ∣ b ({\underline{Y}}_{s}) - b_{h} ({\underline{Y}}_{s}) ∣^{2})$ satisfies $2 ⟨ e_{s}, b (X_{s}) - b_{h} ({\underline{Y}}_{s}) ⟩ + ∥ G_{s} ∥_{F}^{2} \leq H_{s} .$ By [F2]–[F3], Cauchy--Schwarz [F7], and the $r = 2, 4$ increment estimates in step 1.2, Tonelli [F10] gives $E \int_{0}^{t} H_{s} d s \leq C \int_{0}^{t} v (s) d s + C h$ , where $v (t) = E \sup_{s \leq t} ∣ e_{s} ∣^{2}$ .

2.2F2F3F6F7F11step 1.2algebra

The martingale in Itô's identity is $M_{t} = 2 \int_{0}^{t} e_{s}^{T} G_{s} d W_{s}$ . For fixed $N$ , [F2], [F3], [F11], and Cauchy--Schwarz [F7] give $E \int_{0}^{T} ∣ e_{s} ∣^{2} ∥ G_{s} ∥_{F}^{2} d s < \infty$ , since $e$ has a finite fourth path-supremum moment and $G_{s}$ has uniformly finite fourth moments. Thus [F6] applies and gives $E \sup_{s \leq t} ∣ M_{s} ∣ \leq 32 \sqrt{m} E [\sup_{s \leq t} ∣ e_{s} ∣ {(\int_{0}^{t} ∥ G_{s} ∥_{F}^{2} d s)}^{1 / 2}] .$ Using Cauchy--Schwarz [F7] and $u v \leq ε u^{2} + (4 ε)^{- 1} v^{2}$ (which follows by expanding $(\sqrt{ε} u - v / (2 \sqrt{ε}))^{2} \geq 0$ ), the Lipschitz bound on $σ$ , and step 1.2, this is at most $\frac{1}{2} v (t) + C \int_{0}^{t} v (s) d s + C h$ .

3.1A1F2F3F5F8F10F11F12step 2.1step 2.2algebra∎

For each finite $N$ , [F2] and [F11] give $E \sup_{s \leq T} ∣ e_{s} ∣^{2} < \infty$ , and [F12] makes $v (t)$ continuous. Taking the supremum over times up to $t$ in Itô's identity [F5], the pointwise majorant in step 2.1 gives $\sup_{u \leq t} \int_{0}^{u} H_{s} d s \leq \int_{0}^{t} H_{s} d s$ ; now take expectations and insert steps 2.1–2.2. Moving $v (t) / 2$ to the left gives $v (t) \leq C \int_{0}^{t} v (s) d s + C h$ . Gronwall [F8] yields $v (T) \leq C_{T} h$ . AC is inherited exactly through [A1], so the stated strong order follows.

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Sources

Martin Hutzenthaler, Arnulf Jentzen and Peter E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Theorem 1.1 and proof (standard reference, not scraped)
Desmond J. Higham, Xuerong Mao and Andrew M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, §4 (standard reference, not scraped)