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Uniform finite-horizon moments for tamed Euler approximations

Statement

Assume AC. Let $d, m \geq 1$ be finite, $T > 0$ , and let $b, σ, ξ, W$ satisfy the coefficient, Brownian, and initial-moment hypotheses of Global existence and finite-horizon moments for one-sided Lipschitz SDEs. For each integer $N \geq \max (1, ⌈ T ⌉)$ , put $h = T / N$ and define $Y_{0}, \dots, Y_{N}$ by Tamed Euler scheme for a superlinear drift. Then for every finite $p \geq 1$ , $\sup_{N \geq \max (1, ⌈ T ⌉)} \max_{0 \leq n \leq N} E ∣ Y_{n} ∣^{p} < \infty .$ The bound may depend on $p, T$ , the coefficient constants, and the initial moments, but not on $N$ or $n$ .

Facts & Assumptions

Given: AC; the autonomous coefficient and initial-moment hypotheses in the Statement; and the deterministic grid $t_{n} = n h$ .

[F1]

The tamed update is $Y_{n + 1} = Y_{n} + h b (Y_{n}) / (1 + h ∣ b (Y_{n}) ∣) + σ (Y_{n}) Δ W_{n}$ ; in particular $∣ h b (Y_{n}) / (1 + h ∣ b (Y_{n}) ∣) ∣ \leq 1$ . Tamed Euler scheme for a superlinear drift

[F2]

The sampled filtration $(F_{t_{n}})_{n = 0}^{N}$ is increasing. Filtration and filtered probability space

[F3]

Each increment $Δ W_{n}$ is independent of $F_{t_{n}}$ and has law $N_{m} (0, h I_{m})$ . $d$ -dimensional Brownian motion

[F4]

For a nonnegative submartingale $Z$ and $r > 1$ , $∥ \max_{k \leq N} Z_{k} ∥_{r} \leq r / (r - 1) ∥ Z_{N} ∥_{r}$ . Doob Lp maximal inequality

[F5]

The coordinates of $Δ W_{n}$ are independent $N (0, h)$ variables by Standard normal and normal laws. Their density gives, for $2 a h < 1$ , $E e^{a ∣ Δ W_{n} ∣^{2}} = (1 - 2 a h)^{- m / 2}$ : in one coordinate, integrate $(2 π h)^{- 1 / 2} e^{- (1 / (2 h) - a) x^{2}}$ and rescale to the normalized Gaussian density. The same density and $x = \sqrt{h} u$ show $E ∣ Δ W_{n} ∣^{r} = h^{r / 2} E ∣ G ∣^{r} < \infty$ for every finite $r$ , where $G \sim N_{m} (0, I_{m})$ .

[F6]

For predictable $G$ and $p \geq 2$ , $∥ \sup_{t \leq T} ∣ \int_{0}^{t} G_{s} d W_{s} ∣ ∥_{p} \leq C_{p, d} ∥ (\int_{0}^{T} ∥ G_{s} ∥_{F}^{2} d s)^{1 / 2} ∥_{p}$ . Lp maximal moment bound for Brownian Itô integrals

[F7]

Hölder's inequality yields the $L^{r}$ triangle inequality for $r \geq 1$ , the three-factor bound $E ∣ U V W ∣ \leq (E ∣ U ∣^{3} E ∣ V ∣^{3} E ∣ W ∣^{3})^{1 / 3}$ , and $E [1_{A} ∣ Z ∣^{p}] \leq P (A)^{1 / 2} (E ∣ Z ∣^{2 p})^{1 / 2}$ . Holder's inequality for integrals, including the endpoint cases Cauchy-Schwarz for random variables

[F8]

For a nonnegative random variable $Z$ and $a > 0$ , $P (Z \geq a) \leq E Z / a$ . Markov's inequality for random variables

[F9]

For a finite family of events, the probability of their union is at most the sum of their probabilities. The finite union bound

[A1]

AC is inherited through the Brownian, normal, filtration, and Itô interfaces; the estimates below use only countable grid constructions. The Axiom of Choice

Proof

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

1.1F1givenalgebra

The derivative bound, integrated on the segment from $0$ to $x$ , gives $∣ b (x) ∣ \leq B (1 + ∣ x ∣^{q + 1})$ for a finite $B$ . One-sided Lipschitzness with $y = 0$ gives $⟨ x, b (x) ⟩ \leq B (1 + ∣ x ∣^{2})$ , and global Lipschitzness of $σ$ gives $∥ σ (x) ∥_{F} \leq B (1 + ∣ x ∣)$ . Increase $B$ so these bounds hold with one constant. For $∣ x ∣ \geq 1$ they imply $∥ σ (x) ∥_{F}^{2} \leq B_{1} ∣ x ∣^{2}$ , $⟨ x, b (x) ⟩ \leq B_{1} ∣ x ∣^{2}$ , and $∣ b (x) ∣^{2} \leq B_{1} ∣ x ∣^{2 q + 2}$ . Thus, whenever $1 \leq ∣ x ∣ \leq N^{1 / (2 q)}$ and $h = T / N$ , $h^{2} ∣ b (x) ∣^{2} \leq B_{1} T h ∣ x ∣^{2}$ . Fix once and for all $Λ \geq \max {1, 2 B_{1} (1 + T), 1 + \sup_{∣ x ∣ \leq 1} (∣ b (x) ∣ + ∥ σ (x) ∥_{F})}$ ; continuity and compactness make this choice finite.

1.2F2F3F4F5F7given

The recursion and Brownian adaptedness show by induction that $Y_{k}$ is $F_{t_{k}}$ -measurable for every $k$ . For $0 \leq k < N$ , define $α_{k} = 1_{{∣ Y_{k} ∣ \geq 1}} \frac{⟨ Y_{k}, σ (Y_{k}) Δ W_{k} ⟩}{∣ Y_{k} ∣^{2}}, S_{0} = 0, S_{j} = \sum_{k < j} α_{k} (1 \leq j \leq N) .$ Since $α_{k}$ is $F_{t_{k + 1}}$ -measurable, $(S_{j})$ is adapted to the sampled filtration. For $0 \leq n \leq N$ , also define $D_{n} = (Λ + ∣ ξ ∣) \exp ⁣ (Λ T + \max_{0 \leq u \leq n} \sum_{k = u}^{n - 1} (Λ h + Λ ∣ Δ W_{k} ∣^{2} + α_{k})) .$ On $∣ Y_{k} ∣ \geq 1$ , write $α_{k} = v_{k} \cdot Δ W_{k}$ with $v_{k} = σ (Y_{k})^{T} Y_{k} / ∣ Y_{k} ∣^{2}$ ; global Lipschitzness of $σ$ gives $∣ v_{k} ∣ \leq ∥ σ (Y_{k}) ∥_{F} / ∣ Y_{k} ∣ \leq C$ . Conditionally on $F_{t_{k}}$ its variance is $h ∣ v_{k} ∣^{2} \leq C^{2} h$ ; on $∣ Y_{k} ∣ < 1$ , $α_{k} = 0$ . Brownian independence and the Gaussian density therefore give $E [α_{k} ∣ F_{t_{k}}] = 0$ and, by completing the square in each coordinate, $E [e^{r α_{k}} ∣ F_{t_{k}}] \leq e^{C^{2} r^{2} h / 2} (r \in R) .$ Conditional Jensen shows that $e^{Q S_{j}}$ and $e^{- Q S_{j}}$ are nonnegative submartingales for each $Q > 0$ . Iterating the preceding bound with $r = 2 Q$ gives terminal second moments at most $e^{2 C^{2} Q^{2} T}$ for both processes. Doob's $L^{2}$ inequality [F4] then gives $\sup_{N} E e^{Q \max_{j \leq N} ∣ S_{j} ∣} < \infty$ : the exponential of the maximum is bounded by the sum of the maxima of $e^{Q S_{j}}$ and $e^{- Q S_{j}}$ , whose expectations are bounded by their $L^{2}$ maxima. For any $r > 0$ , choose $Q \geq \max (1, r)$ ; since $x^{r / Q} \leq 1 + x$ for $x \geq 0$ , this also bounds $\sup_{N} E e^{r \max_{j \leq N} ∣ S_{j} ∣}$ . Finally, for $N \geq 4 r Λ T$ , [F5] and $- \log (1 - u) \leq 2 u$ on $0 \leq u \leq 1 / 2$ give $E \exp ⁣ (r Λ \sum_{k < N} ∣ Δ W_{k} ∣^{2}) = \prod_{k < N} (1 - 2 r Λ h)^{- m / 2} \leq e^{2 r m Λ T} .$ Every suffix sum $\sum_{k = u}^{j - 1} α_{k} = S_{j} - S_{u}$ is bounded in absolute value by $2 \max_{i \leq N} ∣ S_{i} ∣$ . Since $\max_{n \leq N} D_{n}^{p} \leq (Λ + ∣ ξ ∣)^{p} e^{2 p Λ T} \exp ⁣ (p Λ \sum_{k < N} ∣ Δ W_{k} ∣^{2} + 2 p \max_{j \leq N} ∣ S_{j} ∣),$ the three-factor Hölder bound [F7], the all-moment assumption on $ξ$ , the square-exponential estimate with $r = 3 p$ , and the exponential-martingale bound with $r = 6 p$ give a uniform moment for $\max_{n \leq N} D_{n}^{p}$ when $N \geq N_{0} (p) : = \max {1, ⌈ T ⌉, ⌈ 12 p Λ T ⌉}$ . The finitely many smaller $N$ will be handled separately.

1.3F1F3F6F7givenalgebra

Set $a_{n} = (E ∣ Y_{n} ∣^{p})^{1 / p}$ for $p \geq 2$ . Decompose the recursion as $Y_{n} = ξ + \sum_{k < n} h b_{h} (Y_{k}) + σ (0) W_{t_{n}} + \sum_{k < n} (σ (Y_{k}) - σ (0)) Δ W_{k} .$ The drift sum has norm at most $N$ by [F1]. The last sum is a continuous Brownian Itô integral with a predictable step integrand. By [F6], global Lipschitzness of $σ$ , and the $L^{p / 2}$ triangle inequality from [F7], its $L^{p}$ norm is at most $C_{p} {(\sum_{k < n} h ∥ σ (Y_{k}) - σ (0) ∥_{L^{p}}^{2})}^{1 / 2} \leq C_{p} {(h \sum_{k < n} a_{k}^{2})}^{1 / 2} .$ The pointwise inequality $∣ u_{1} + \dots + u_{4} ∣^{p} \leq 4^{p - 1} \sum_{i} ∣ u_{i} ∣^{p}$ and the bounded $p$ th moments of $W_{t}$ on $[0, T]$ therefore give $a_{n}^{2} \leq C_{p} (1 + ∥ ξ ∥_{L^{p}}^{2} + N^{2}) + C_{p} h \sum_{k < n} a_{k}^{2} .$ Iterating this finite recurrence yields $\max_{n \leq N} a_{n}^{2} \leq C_{p} (1 + ∥ ξ ∥_{L^{p}}^{2} + N^{2}) e^{C_{p} T}$ , hence $\max_{n \leq N} ∥ Y_{n} ∥_{L^{p}} \leq C_{p} (1 + N)$ .

2.1F1step 1.1step 1.2givenalgebraconstruct

Write $Δ W_{k} = W_{t_{k + 1}} - W_{t_{k}}$ and use $α_{k}, D_{n}$ from step 1.2. Define $Ω_{n} = {\max_{k < n} D_{k} \leq N^{1 / (2 q)}, \max_{k < n} ∣ Δ W_{k} ∣ \leq 1} .$ On $∣ Y_{k} ∣ \leq 1$ and $∣ Δ W_{k} ∣ \leq 1$ , [F1] gives $∣ Y_{k + 1} ∣ \leq 1 + h ∣ b (Y_{k}) ∣ + ∥ σ (Y_{k}) ∥_{F} \leq Λ$ . On $1 \leq ∣ Y_{k} ∣ \leq N^{1 / (2 q)}$ , expand the squared update. The drift cross term is at most $2 B_{1} h ∣ Y_{k} ∣^{2}$ ; the squared taming increment is at most $B_{1} T h ∣ Y_{k} ∣^{2}$ by step 1.1; the diffusion square is at most $B_{1} ∣ Y_{k} ∣^{2} ∣ Δ W_{k} ∣^{2}$ ; and the drift-noise cross term is at most the sum of those two square terms. The remaining cross term is exactly $2 ∣ Y_{k} ∣^{2} α_{k}$ . With the fixed $Λ$ chosen in step 1.1 large enough for these coefficients, $1 + u \leq e^{u}$ therefore gives $∣ Y_{k + 1} ∣ \leq ∣ Y_{k} ∣ \exp (Λ h + Λ ∣ Δ W_{k} ∣^{2} + α_{k}) .$ Induction on $n$ , starting either from $Y_{0} = ξ$ or from the last index with $∣ Y_{k} ∣ \leq 1$ , now gives $1_{Ω_{n}} ∣ Y_{n} ∣ \leq D_{n}$ : the first step after such an index is at most $Λ$ , and the later factors are bounded by the suffix sum in $D_{n}$ .

2.2F3F5F8F9step 1.2givenalgebra

Markov's inequality [F8] and step 1.2 imply, for any $r > 0$ , $P ⁣ (\max_{n \leq N} D_{n} > N^{1 / (2 q)}) \leq N^{- r / (2 q)} E \max_{n \leq N} D_{n}^{r} .$ Also, by the finite union bound [F9] and the $r$ th Gaussian moment, $P (\max_{k < N} ∣ Δ W_{k} ∣ > 1) \leq C_{r} N h^{r / 2} = C_{r, T} N^{1 - r / 2}$ . Choosing $r$ arbitrarily large in these bounds shows that for every $s > 0$ there is $C_{s}$ with $P (Ω_{N}^{c}) \leq C_{s} N^{- s}$ for all $N$ .

3.1A1F1F7step 2.1step 1.2step 2.2step 1.3algebra∎

For $p \geq 1$ , on $Ω_{N}$ step 2.1 bounds $E [1_{Ω_{N}} ∣ Y_{n} ∣^{p}] \leq E D_{n}^{p} \leq C_{p}$ ; here $Ω_{N} \subseteq Ω_{n}$ . On the complement, [F7] and step 1.3 give $E [1_{Ω_{N}^{c}} ∣ Y_{n} ∣^{p}] \leq P (Ω_{N}^{c})^{1 / 2} ∥ Y_{n} ∥_{L^{2 p}}^{p} \leq C_{s} N^{- s / 2} (1 + N)^{p} .$ Choose $s > 2 p$ . This is uniformly bounded. For the finitely many small $N$ excluded in step 1.2, the finite recursion and linear growth of $σ$ give finite moments by induction; taking their finite maximum completes the uniform bound. For $1 \leq p < 2$ , apply Hölder on the probability space to the $p = 2$ result. AC is exactly [A1].

Depends on

Used by

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

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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, Lemmas 3.1–3.6 and 3.9 (standard reference, not scraped)
Martin Hutzenthaler and Arnulf Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Chapters 2–3 (standard reference, not scraped)