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Lp maximal moment bound for Brownian Itô integrals

Statement

Assume AC and the usual filtered Brownian conventions of Continuous Brownian Ito processes. Let $G$ be a predictable $d \times m$ matrix process on $[0, T]$ with $E {(\int_{0}^{T} ∥ G_{s} ∥_{F}^{2} d s)}^{p / 2} < \infty$ for some real $p \geq 2$ , and let $M_{t} = \int_{0}^{t} G_{s} d W_{s}$ be its continuous vector Itô integral. There is a finite constant $C_{p, d}$ , independent of $G$ and $T$ , such that $E \sup_{0 \leq t \leq T} ∣ M_{t} ∣^{p} \leq C_{p, d} E {(\int_{0}^{T} ∥ G_{s} ∥_{F}^{2} d s)}^{p / 2} .$

Facts & Assumptions

Given: AC, finite $d, m$ , a usual filtered Brownian space, a predictable matrix integrand $G$ , and the finite $p / 2$ moment of its quadratic energy.

[F1]

The multidimensional Itô formula applies to each stopped real coordinate $M^{i}$ , driven by the $m$ -dimensional Brownian motion. For $f (x) = ∣ x ∣^{p}$ , $p \geq 2$ , one has $f \in C^{2} (R)$ and $f^{''} (x) = p (p - 1) ∣ x ∣^{p - 2}$ , with the continuous value $f^{''} (0) = 2$ when $p = 2$ and $f^{''} (0) = 0$ when $p > 2$ . Multidimensional Ito formula for Brownian-driven processes

[F2]

For a martingale $M$ , $∣ M ∣$ is a submartingale; for a nonnegative submartingale $Z$ with $Z_{N} \in L^{p}$ , Doob's finite time inequality gives $∥ \max_{k \leq N} Z_{k} ∥_{p} \leq \frac{p}{p - 1} ∥ Z_{N} ∥_{p}$ . Absolute value and powers of a martingale are submartingales Doob Lp maximal inequality

[F3]

Stopping a finite-energy Brownian Itô integral at a stopping time gives the integral of the stopped predictable integrand; a stopping time satisfies ${τ \leq t} \in F_{t}$ . Stopping an Ito integral Continuous-time stopping times and stopped sigma-algebras

[F4]

A continuous Brownian Itô process has an adapted continuous version and its finite-energy stochastic integral is a martingale. Continuous Brownian Ito processes

[F5]

Hölder's inequality with conjugate exponents $p / (p - 2)$ and $p / 2$ (for $p > 2$ ), and its $p = 2$ endpoint, gives $E [S^{p - 2} Q] \leq (E S^{p})^{(p - 2) / p} (E Q^{p / 2})^{2 / p}$ for nonnegative $S, Q$ . Holder's inequality for integrals, including the endpoint cases

[F6]

On nested dyadic grids that include $T$ , continuity makes the grid maxima increase to the path supremum; monotone convergence passes the expectations to that supremum. Monotone convergence for the integral

[F7]

Fatou's lemma passes a uniform bound on nonnegative stopped path suprema to their almost-sure limit. Fatou's lemma

[A1]

AC is inherited from the Brownian and Itô-integral interfaces; no choice is made in this estimate. The Axiom of Choice

Proof

Given: The assumptions in the Statement.

1.1F2F3F4F6given

For each coordinate $i$ , set $M_{t}^{i} = \sum_{j = 1}^{m} \int_{0}^{t} G_{s}^{i j} d W_{s}^{j}$ , $Q_{i} (t) = \int_{0}^{t} \sum_{j} ∣ G_{s}^{i j} ∣^{2} d s$ , and $M_{i}^{*} (t) = \sup_{u \leq t} ∣ M_{u}^{i} ∣$ . Stop at $ρ_{R} = \inf {t : Q_{i} (t) \lor M_{i}^{*} (t) \geq R} \land T$ , for $R > 0$ . These processes are continuous and adapted, and ${ρ_{R} \leq t} = {Q_{i} (t) \lor M_{i}^{*} (t) \geq R}$ for $t < T$ , so $ρ_{R}$ is a stopping time Continuous-time stopping times and stopped sigma-algebras. The stopped process is a continuous square-integrable martingale by [F3]–[F4], its quadratic energy is bounded by $R$ , and its path supremum is bounded by $R$ . Apply [F2] to $∣ M^{i} ∣$ sampled on nested dyadic grids including $T$ . By [F6], the grid maxima increase to the path supremum and monotone convergence passes the discrete bounds to $S_{i} (R) : = E \sup_{t \leq T} ∣ M_{t \land ρ_{R}}^{i} ∣^{p} \leq q^{p} E ∣ M_{T \land ρ_{R}}^{i} ∣^{p}$ , where $q = p / (p - 1)$ .

2.1F1F5step 1.1algebra

Apply [F1] to the stopped coordinate process. The stochastic term has mean zero because its integrand is bounded by the stopping level and has finite energy. Put $U_{i} (R) = \sup_{t \leq T} ∣ M_{t \land ρ_{R}}^{i} ∣$ . The Hessian term is bounded by $\frac{p (p - 1)}{2} U_{i} (R)^{p - 2} Q_{i} (R)$ , where $Q_{i} (R) = \int_{0}^{T \land ρ_{R}} \sum_{j} ∣ G_{s}^{i j} ∣^{2} d s$ . Therefore $S_{i} (R) \leq q^{p} \frac{p (p - 1)}{2} E [U_{i} (R)^{p - 2} Q_{i} (R)]$ . For $p = 2$ , interpret $U_{i} (R)^{0} = 1$ , giving $S_{i} (R) \leq C_{2} E Q_{i} (R)$ . For $p > 2$ , [F5] gives $S_{i} (R) \leq C_{p} S_{i} (R)^{(p - 2) / p} (E Q_{i} (R)^{p / 2})^{2 / p}$ .

3.1step 2.1givenalgebra

For $p = 2$ , step 2.1 gives the coordinate estimate. For $p > 2$ , if $S_{i} (R) = 0$ it is immediate; otherwise divide the bound in step 2.1 by $S_{i} (R)^{(p - 2) / p}$ and raise to the power $p / 2$ . In both cases $S_{i} (R) \leq C_{p} E Q_{i} (R)^{p / 2} \leq C_{p} E (\int_{0}^{T} ∥ G_{s} ∥_{F}^{2} d s)^{p / 2}$ . Since $∣ M_{t} ∣ \leq \sqrt{d} \max_{i} ∣ M_{t}^{i} ∣$ , summing the coordinate estimates gives the required bound for the stopped vector integral with constant $C_{p, d}$ .

4.1A1F3F7step 3.1given∎

As $R \to \infty$ , finite pathwise energy and continuity make $ρ_{R} = T$ eventually almost surely. By [F7], Fatou's lemma passes the uniform bound of step 3.1 to $E \sup_{t \leq T} ∣ M_{t} ∣^{p}$ . The constant depends only on $p, d$ . AC is exactly the inherited assumption [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, proof of Theorem 4.4 (standard reference, not scraped)