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Strong order one-half for drift-implicit Euler–Maruyama

Statement

Assume AC. Let $d, m \geq 1$ be finite, $T > 0$ , and consider the autonomous SDE with $b \in C^{1} (R^{d}; R^{d})$ and globally Lipschitz $σ : R^{d} \to R^{d \times m}$ . Suppose that for constants $μ \geq 0$ , $C < \infty$ , and an integer $q \geq 1$ , $⟨ x - y, b (x) - b (y) ⟩ \leq - μ ∣ x - y ∣^{2}, ∥ D b (x) ∥ \leq C (1 + ∣ x ∣^{q}) .$ Let $ξ$ have all finite moments and let $X$ be the corresponding global strong solution. For $N \geq \max (1, ⌈ T ⌉)$ , put $h = T / N$ and let $Y_{n}$ be the exact-resolvent drift-implicit Euler–Maruyama values from Drift-implicit Euler–Maruyama with explicit solve-failure status. There is a constant $C_{T}$ independent of $N$ such that $E \max_{0 \leq n \leq N} ∣ X_{t_{n}} - Y_{n} ∣^{2} \leq C_{T} h .$ Moreover, for two such numerical sequences driven by the same Brownian increments but with initial values $ξ, \tilde{ξ}$ , $E ∣ Y_{n} - {\tilde{Y}}_{n} ∣^{2} \leq e^{L_{σ}^{2} t_{n}} E ∣ ξ - \tilde{ξ} ∣^{2},$ where $L_{σ}$ is a global Lipschitz constant of $σ$ . The second estimate is finite-time mean-square stability; it does not claim strict contraction when diffusion is present.

Facts & Assumptions

Given: AC; the dissipative SDE and moment hypotheses in the Statement; $h = T / N \leq 1$ ; and the drift-implicit scheme.

[F1]

The backward Euler resolvent exists on all of $R^{d}$ and obeys $∣ R_{h} (u) - R_{h} (v) ∣ \leq (1 + h μ)^{- 1} ∣ u - v ∣$ . Backward Euler has a globally defined nonexpansive resolvent (contractive when strongly dissipative)

[F2]

The exact SDE solution 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 exact SDE satisfies the local Itô integral equation from the definition. For a scalar predictable finite-energy integrand $H$ and each Brownian coordinate $W^{j}$ , $E ∣ \int_{a}^{b} H_{s} d W_{s}^{j} ∣^{2} = E \int_{a}^{b} H_{s}^{2} d s$ ; the matrix bound needed below is derived coordinatewise in step 1.3. Locally Lipschitz Itô SDE on an open state domain Ito isometry and linearity in predictable L2

[F4]

The Brownian Itô integral satisfies the $L^{p}$ maximal estimate in Lp maximal moment bound for Brownian Itô integrals and the $L^{1}$ maximal estimate $E \sup ∣ \int H d W ∣ \leq 16 \sqrt{m} E (\int ∥ H ∥_{F}^{2})^{1 / 2}$ in An L1 maximal bound for finite-energy Brownian Itô integrals.

[F5]

The polynomial derivative bound implies $∣ b (x) - b (y) ∣ \leq C^{'} (1 + ∣ x ∣^{q} + ∣ y ∣^{q}) ∣ x - y ∣$ by integrating $D b$ along the segment from $x$ to $y$ . This is a consequence of the Statement's derivative hypothesis.

[F6]

Hölder's inequality gives the $L^{p}$ bounds for time integrals and products used in the local-increment estimates. Holder's inequality for integrals, including the endpoint cases

[F7]

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

[F8]

Euclidean Cauchy--Schwarz gives $∣ ⟨ x, y ⟩ ∣ \leq ∣ x ∣ ∣ y ∣$ . Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation

[F9]

For each Brownian coordinate $W^{j}$ , a scalar finite-energy Itô integral has a continuous square-integrable martingale version. Each coordinate of a matrix integral is a finite sum of these scalar martingales, hence is a square-integrable martingale by linearity of conditional expectation. Its grid increments therefore have conditional mean zero given the preceding sigma-algebra. The Ito integral process has a continuous martingale version

[F10]

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

[A1]

AC is inherited through the Brownian, resolvent, and Itô interfaces. The Axiom of Choice

Proof

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

1.1A1F1F2F6givenalgebra

First fix $N$ . Since $R_{h} (- h b (0)) = 0$ , [F1] gives $∣ R_{h} (u) ∣ \leq ∣ u ∣ + h ∣ b (0) ∣$ . Its continuity, linear growth of $σ$ , and the independent Gaussian increments show by finite induction that $Y_{n}$ is $F_{t_{n}}$ -measurable and has every finite moment; the same holds for the comparison sequence. Together with [F2] and Hölder [F6], these moments give finite expected square energy for all stochastic integrands below, including products of a grid error and a diffusion difference. These preliminary bounds may depend on $N$ . For two updates with difference $e_{n} = Y_{n} - {\tilde{Y}}_{n}$ , [F1] and Brownian independence give $\begin{aligned} E (∣ e_{n + 1} ∣^{2} ∣ F_{t_{n}}) & \leq (1 + h μ)^{- 2} E (∣ e_{n} + (σ (Y_{n}) - σ ({\tilde{Y}}_{n})) Δ W_{n} ∣^{2} ∣ F_{t_{n}}) \\ \leq (1 + h L_{σ}^{2}) ∣ e_{n} ∣^{2} . \end{aligned}$ The cross term vanishes because $e_{n}$ and the diffusion difference are $F_{t_{n}}$ -measurable and $Δ W_{n}$ is independent with mean zero; its covariance is $h I_{m}$ . Iterating and using $(1 + h L_{σ}^{2})^{n} \leq e^{L_{σ}^{2} t_{n}}$ proves the second estimate.

1.2F1F3F9givenalgebra

Let $X_{n} = X_{t_{n}}$ and define $d_{n + 1} = \int_{t_{n}}^{t_{n + 1}} (b (X_{s}) - b (X_{n + 1})) d s, m_{n + 1} = \int_{t_{n}}^{t_{n + 1}} (σ (X_{s}) - σ (X_{n})) d W_{s} .$ The SDE integral equation gives $X_{n + 1} - h b (X_{n + 1}) = X_{n} + σ (X_{n}) Δ W_{n} + d_{n + 1} + m_{n + 1}$ . With $R_{h} = (I - h b)^{- 1}$ , the numerical update and [F1] therefore imply, for $e_{n} = Y_{n} - X_{n}$ , $∣ e_{n + 1} ∣ \leq ∣ e_{n} + ξ_{n + 1} - d_{n + 1} ∣, ξ_{n + 1} = \int_{t_{n}}^{t_{n + 1}} (σ (Y_{n}) - σ (X_{s})) d W_{s} .$ The identity for $ξ_{n + 1}$ follows by adding and subtracting $σ (X_{n}) Δ W_{n}$ . Its conditional mean given $F_{t_{n}}$ is zero by [F9].

2.1F2F3F4F5F6F8F10step 1.1givenalgebra

From [F2], linear growth of $σ$ , and the SDE equation, for $r = 2, 4$ and $s \in [t_{n}, t_{n + 1}]$ , $E ∣ X_{s} - X_{n} ∣^{r} + E ∣ X_{s} - X_{n + 1} ∣^{r} \leq C_{r} h^{r / 2} .$ Indeed the drift increment over either intervening interval has $r$ th moment at most $h^{r - 1} \int_{t_{n}}^{t_{n + 1}} E ∣ b (X_{u}) ∣^{r} d u \leq C h^{r}$ by Hölder and [F10], while [F4] bounds the stochastic increment by $C_{r} E (\int_{t_{n}}^{t_{n + 1}} ∥ σ (X_{u}) ∥_{F}^{2} d u)^{r / 2} \leq C_{r} h^{r / 2}$ . The polynomial local-Lipschitz estimate [F5], all moments in [F2], and Hölder [F6] then give $E ∣ d_{n + 1} ∣^{2} \leq h \int_{t_{n}}^{t_{n + 1}} E ∣ b (X_{s}) - b (X_{n + 1}) ∣^{2} d s \leq C h^{3} .$ For any finite-energy matrix $G$ , writing each coordinate as a sum over the $m$ Brownian coordinates, the scalar isometry [F3] and $∣ \sum_{j} a_{j} ∣^{2} \leq m \sum_{j} ∣ a_{j} ∣^{2}$ give $E {∣ \int_{a}^{b} G_{s} d W_{s} ∣}^{2} \leq m \sum_{i, j} E {∣ \int_{a}^{b} G_{s}^{i j} d W_{s}^{j} ∣}^{2} = m E \int_{a}^{b} ∥ G_{s} ∥_{F}^{2} d s .$ Apply this to $G_{s} = σ (Y_{n}) - σ (X_{s})$ and use global Lipschitzness to obtain $E ∣ ξ_{n + 1} ∣^{2} \leq C h E ∣ e_{n} ∣^{2} + C h^{2} .$ The constants here are independent of $N$ .

3.1F8F9step 1.2step 2.1algebra

Squaring the error inequality in step 1.2 and applying $2 ∣ e_{n} ∣ ∣ d_{n + 1} ∣ \leq h ∣ e_{n} ∣^{2} + h^{- 1} ∣ d_{n + 1} ∣^{2}$ and $2 ∣ ξ_{n + 1} ∣ ∣ d_{n + 1} ∣ \leq ∣ ξ_{n + 1} ∣^{2} + ∣ d_{n + 1} ∣^{2}$ gives $∣ e_{n + 1} ∣^{2} \leq (1 + h) ∣ e_{n} ∣^{2} + 2 ⟨ e_{n}, ξ_{n + 1} ⟩ + 2 ∣ ξ_{n + 1} ∣^{2} + (2 + h^{- 1}) ∣ d_{n + 1} ∣^{2} .$ By [F9], the increments $ξ_{n + 1}$ are martingale differences. Summing their unconditional second-moment bound from step 2.1 gives $\sum_{n < N} E ∣ ξ_{n + 1} ∣^{2} \leq C h \sum_{n < N} E ∣ e_{n} ∣^{2} + C h$ ; step 2.1 also gives $\sum_{n < N} (2 + h^{- 1}) E ∣ d_{n + 1} ∣^{2} \leq C h$ .

3.2F2F4F8F9F10step 1.1step 2.1givenalgebra

Put $v_{j} = E \max_{k \leq j} ∣ e_{k} ∣^{2}$ for $0 \leq j \leq N$ , finite by step 1.1. The partial sums $M_{j} = 2 \sum_{n < j} ⟨ e_{n}, ξ_{n + 1} ⟩$ form a martingale by [F9]. Written as one Brownian Itô integral, its integrand on $(t_{n}, t_{n + 1}]$ is $2 e_{n}^{T} (σ (Y_{n}) - σ (X_{s}))$ , whose expected square energy is finite by the fourth moments in step 1.1 and [F2]. For $1 \leq j \leq N$ , the $L^{1}$ maximal bound [F4] therefore gives $E \max_{l \leq j} ∣ M_{l} ∣ \leq 32 \sqrt{m} E [\max_{n < j} ∣ e_{n} ∣ {(\sum_{n < j} \int_{t_{n}}^{t_{n + 1}} ∥ σ (Y_{n}) - σ (X_{s}) ∥_{F}^{2} d s)}^{1 / 2}] .$ Applying $u v \leq ε u^{2} + (4 ε)^{- 1} v^{2}$ (obtained by expanding $(\sqrt{ε} u - v / (2 \sqrt{ε}))^{2} \geq 0$ ) factors out the error maximum and yields, for any $ε > 0,$ $E \max_{l \leq j} ∣ M_{l} ∣ \leq ε v_{j - 1} + C_{ε} E \sum_{n < j} \int_{t_{n}}^{t_{n + 1}} ∥ σ (Y_{n}) - σ (X_{s}) ∥_{F}^{2} d s .$ By Lipschitzness, $∥ σ (Y_{n}) - σ (X_{s}) ∥_{F}^{2} \leq 2 L_{σ}^{2} (∣ e_{n} ∣^{2} + ∣ X_{s} - X_{n} ∣^{2})$ . Step 2.1 and Tonelli [F10] then give $E \max_{l \leq j} ∣ M_{l} ∣ \leq ε v_{j - 1} + C_{ε} h \sum_{n < j} E ∣ e_{n} ∣^{2} + C_{ε} h,$ since $\sum_{n < j} \int_{t_{n}}^{t_{n + 1}} E ∣ X_{s} - X_{n} ∣^{2} d s \leq C h$ .

4.1A1F7step 1.1step 2.1step 3.1step 3.2algebra∎

Recall $v_{j}$ from step 3.2; $e_{0} = 0$ because both methods start from $ξ$ . Subtract $∣ e_{n} ∣^{2}$ from both sides of step 3.1 and sum from $n = 0$ to $j - 1$ . Taking the maximum over partial endpoints makes the martingale term $\max_{l \leq j} ∣ M_{l} ∣$ ; the remaining nonnegative terms have expectation at most $C h \sum_{k < j} v_{k} + C h$ by the local bounds in steps 2.1--2.2. Use step 3.2 with $ε = 1 / 2$ ; since $v_{j - 1} \leq v_{j}$ , absorb the martingale contribution to obtain $v_{j} \leq C h \sum_{k < j} v_{k} + C h .$ Choose $C_{0} \geq C$ so this implies $v_{j} \leq C_{0} w_{j}$ for $w_{j} = h + h \sum_{k < j} v_{k}$ . Then $w_{j + 1} = w_{j} + h v_{j} \leq (1 + C_{0} h) w_{j}$ and $w_{0} = h$ , so $v_{N} \leq C_{0} h (1 + C_{0} h)^{N} \leq C_{0} e^{C_{0} T} h$ . This proves the first assertion; the stability assertion was proved in step 1.1. AC is exactly [A1].

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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, Theorem 5.3 and proof (standard reference, not scraped)
Martin Hutzenthaler, Arnulf Jentzen and Peter E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, §1 (standard reference, not scraped)