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Doob maximal bound for the Ito integral

Statement

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands. Let $H$ be a predictable process on $[0, T]$ , where $T \geq 0$ , with $E \int_{0}^{T} H_{s}^{2} d s < \infty$ . Extend $H$ by zero after $T$ when applying the global continuous-version theorem, and let $M_{t} = \int_{0}^{t} H_{s} d B_{s}$ be the continuous version of The Ito integral process has a continuous martingale version. Then $E \sup_{0 \leq t \leq T} ∣ M_{t} ∣^{2} \leq 4 E \int_{0}^{T} H_{s}^{2} d s .$ The left-hand side is finite and does not depend on the chosen version, since any two continuous versions are indistinguishable. Every continuous version of the integral process satisfies the same bound. The path supremum in this expectation means its measurable version $S_{D} = \sup_{q \in D} ∣ M_{q} ∣$ on the countable scaled dyadic grid $D$ (including both endpoints). Continuity identifies it with the path supremum on a measurable event of probability one. Indistinguishability here uses that full-event convention, as in the cited continuous-version theorem, without assuming completeness of the filtration.

Facts & Assumptions

Given: AC, the standing hypothesis (H), a finite-energy predictable $H$ on $[0, T]$ , and the continuous version $M$ with $M_{t} = \int_{0}^{t} H d B$ a.s. and $E M_{t}^{2} = E \int_{0}^{t} H^{2} d s$ .

[F1]

$M$ is an adapted continuous square-integrable martingale with $E M_{t}^{2} = E \int_{0}^{t} H^{2} d s$ for every $t \leq T$ , and $M_{T}$ is the integral class at $T$ . The Ito integral process has a continuous martingale version Ito isometry and linearity in predictable L2

[F2]

For each $m \geq 1$ the sampled family $Y_{k} : = M_{k T / 2^{m}}$ , $0 \leq k \leq 2^{m}$ , is a discrete martingale for the filtration $(F_{k T / 2^{m}})_{k}$ , by the tower property; hence Doob's $L^{2}$ inequality gives $E \max_{k \leq 2^{m}} Y_{k}^{2} \leq 4 E Y_{2^{m}}^{2}$ . Tower property of conditional expectation Doob Lp maximal inequality

[F3]

By the continuous-version theorem there is an event of probability one on which the path of $M$ is continuous on $[0, T]$ . On that event its supremum equals the supremum over the dyadic grid union $D : = ⋃_{m} {k T / 2^{m}}$ , which in turn is the increasing limit $\lim_{m} \max_{k \leq 2^{m}} ∣ M_{k T / 2^{m}} ∣$ because the grids are nested. The rationals embed densely in the reals The Ito integral process has a continuous martingale version

[F4]

For $0 \leq Z_{m} ↑ Z$ , $E Z_{m} ↑ E Z$ , so the expectation of the supremum is the limit of the expectations of the finite-grid maxima. Monotone convergence for the integral

[F5]

Two continuous versions of the same integral process are indistinguishable, so their path suprema agree almost surely. The Ito integral process has a continuous martingale version Process law, modification, and indistinguishability

[F6]

AC is declared for the ambient interfaces. The Axiom of Choice

Proof

technique · direct

1.1

If $T = 0$ , $M_{0} = 0$ almost surely and both sides vanish. For $T > 0$ and fixed $m \geq 1$ the finite-grid quantity $\max_{k \leq 2^{m}} ∣ M_{k T / 2^{m}} ∣^{2}$ is integrable, and [F2] gives $E \max_{k \leq 2^{m}} ∣ M_{k T / 2^{m}} ∣^{2} \leq 4 E M_{T}^{2} = 4 E \int_{0}^{T} H^{2} d s$ .

F1F2

1.2

As $m$ increases the grids ${k T / 2^{m} : 0 \leq k \leq 2^{m}}$ are nested, so the unsquared maxima increase pointwise to the measurable random variable $S_{D} : = \sup_{q \in D} ∣ M_{q} ∣$ . By [F3], $S_{D} = \sup_{t \in [0, T]} ∣ M_{t} ∣$ almost surely.

2.1

Monotone convergence [F4] applies pointwise to $Z_{m} : = \max_{k \leq 2^{m}} ∣ M_{k T / 2^{m}} ∣^{2} ↑ S_{D}^{2}$ . Hence, using the almost-sure equality in step 1.2, $E \sup_{t \leq T} ∣ M_{t} ∣^{2} = E S_{D}^{2} = \lim_{m} E Z_{m} \leq 4 E \int_{0}^{T} H^{2} d s$ , and the bound is finite because the right-hand side is finite.

F1F4step 1.1step 1.2

3.1

For any other continuous version $N$ , intersect the fixed-time equality events ${N_{q} = M_{q}}$ over the countable grid $D$ and the two measurable continuity events. On the resulting measurable probability-one event the paths agree at every time by continuity, and their grid suprema agree. Thus the measurable supremum for $N$ has the same expectation and satisfies the bound, even if $N$ is not adapted. AC enters through the declared ambient interfaces [F6].

F5F6step 2.1∎

Source notes

Lawler, Proposition 3.2.4, uses discrete maximal estimates on refining grids to prove a uniform-convergence criterion. The expectation bound here follows directly from the library discrete Doob inequality with p=2 and monotone convergence; no fourth moment is assumed.

Depends on

Used by

Brownian-filtration martingale representation Theorem
Localized Ito integral Theorem
Multidimensional Ito formula for Brownian-driven processes Theorem
One-dimensional Ito formula Theorem
Quadratic variation of an Ito integral Theorem
Space-time harmonic functions yield Brownian local martingales up to exit lifetime Theorem

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Sources

Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Proposition 3.2.4 (standard reference, not scraped)