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Finite-horizon L2 maximal bound for martingales with right-continuous paths and left limits

Statement

Assume the Axiom of Choice. Let $T \geq 0$ , let $(Ω, F, P)$ be a probability space, and let $(F_{t})_{0 \leq t \leq T}$ be an increasing family of sub- $σ$ -algebras of $F$ . Let $(M_{t})_{0 \leq t \leq T}$ be a family of real random variables such that each $M_{t}$ is $F_{t}$ -measurable, $E [M_{t}^{2}] < \infty$ , $M_{0} = 0$ , and $E [M_{t} ∣ F_{s}] = M_{s}$ almost surely for every $0 \leq s \leq t \leq T$ . Suppose it has càdlàg paths almost surely, meaning there is a measurable event $Ω_{0} \in F$ with $P (Ω_{0}) = 1$ such that, for every $ω \in Ω_{0}$ , the path $t \mapsto M_{t} (ω)$ is right-continuous on $[0, T)$ and has a finite left limit at every $t \in (0, T]$ . Define

D_{T} = ⋃_{n \geq 0} {k T / 2^{n} : 0 \leq k \leq 2^{n}}, S_{T} = \sup_{t \in D_{T}} ∣ M_{t} ∣ .

Then $S_{T}$ is measurable and agrees with the pathwise supremum $\sup_{0 \leq t \leq T} ∣ M_{t} ∣$ on $Ω_{0}$ , and

E [S_{T}^{2}] \leq 4 E [∣ M_{T} ∣^{2}] .

Statement references

Continuous-time adapted processes and martingales

Facts & Assumptions

Given: A finite horizon $T \geq 0$ , the filtered probability space, square-integrable all-pairs martingale, and measurable probability-one path-regularity event specified in the Statement, and the Axiom of Choice.

[A1]

Assume Choice for the conditional-expectation interface in the continuous-time martingale definition (The Axiom of Choice).

[F1]

Continuous-time adapted processes and martingales: $E [M_{T} ∣ F_{t}] = M_{t}$ almost surely for every deterministic $t \leq T$ .

[F2]

Basic algebra and order properties of conditional expectation: $∣ E [Z ∣ G] ∣ \leq E [∣ Z ∣ ∣ G]$ almost surely for integrable real $Z$ .

[F3]

Cauchy-Schwarz for random variables: $E [∣ U V ∣] \leq (E U^{2})^{1 / 2} (E V^{2})^{1 / 2}$ for square-integrable real random variables.

[F4]

Layer-cake formulas for random variables: for nonnegative measurable $Z$ , $E Z = \int_{0}^{\infty} P (Z > r) d r$ , including infinite values.

[F5]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: a nonnegative measurable function on the product of the time half-line and the probability space may be integrated in either order.

[F6]

Monotone convergence for the integral: expectations of increasing nonnegative sequences converge to the expectation of their pointwise limit.

[F7]

Conditional expectation given a sigma algebra: a version $U$ of $E [Z ∣ G]$ is integrable and $G$ -measurable, and $E [U 1_{C}] = E [Z 1_{C}]$ for every $C \in G$ .

Proof

1.1A1F1F2F7construct

The result is immediate when $T = 0$ . Suppose $T > 0$ , fix $n \geq 0$ , and let $t_{k} = k T / 2^{n}$ for $0 \leq k \leq 2^{n}$ . Put $S_{n} = \max_{0 \leq k \leq 2^{n}} ∣ M_{t_{k}} ∣$ , which is square-integrable because it is bounded by the finite sum $\sum_{k} ∣ M_{t_{k}} ∣$ . For $r > 0$ , let $τ$ be the first grid index $k$ with $∣ M_{t_{k}} ∣ \geq r$ , or leave it undefined if no grid value reaches $r$ . Each event $C_{k} = {τ = k}$ belongs to $F_{t_{k}}$ . By [F1] and [F2], $∣ M_{t_{k}} ∣ = ∣ E [M_{T} ∣ F_{t_{k}}] ∣ \leq E [∣ M_{T} ∣ ∣ F_{t_{k}}]$ almost surely. Thus [F7] gives $E [∣ M_{T} ∣ 1_{C_{k}}] = E [E [∣ M_{T} ∣ ∣ F_{t_{k}}] 1_{C_{k}}] \geq E [∣ M_{t_{k}} ∣ 1_{C_{k}}] \geq r P (C_{k})$ . Summing over the finite grid yields $r P (S_{n} \geq r) \leq E [∣ M_{T} ∣ 1_{{S_{n} \geq r}}] .$

2.1F3F4F5step 1.1algebra

Apply [F4] to $S_{n}^{2}$ , use $P (S_{n} > r) \leq P (S_{n} \geq r)$ and integrate the bound from step 1.1. [F5] for the nonnegative measurable function $(r, ω) \mapsto ∣ M_{T} (ω) ∣ 1_{{S_{n} (ω) \geq r}}$ gives $E S_{n}^{2} = \int_{0}^{\infty} 2 r P (S_{n} > r) d r \leq 2 E [∣ M_{T} ∣ S_{n}] .$ By [F3], the right side is at most $2 (E ∣ M_{T} ∣^{2})^{1 / 2} (E S_{n}^{2})^{1 / 2}$ . If $E S_{n}^{2} = 0$ the desired finite-grid bound is immediate; otherwise divide by its square root and square to obtain $E S_{n}^{2} \leq 4 E ∣ M_{T} ∣^{2}$ .

3.1F6step 2.1algebra∎

The dyadic grids are nested, so $S_{n} ↑ S_{*} : = \sup_{t \in D_{T}} ∣ M_{t} ∣ = S_{T}$ . For every $ω \in Ω_{0}$ and each $t < T$ , there are dyadic grid times decreasing to $t$ , and right-continuity makes their process values converge to $M_{t} (ω)$ ; time $T$ is in every grid. Hence $S_{*}$ equals the pathwise supremum on $[0, T]$ almost surely. Apply [F6] to $S_{n}^{2}$ and take limits in the finite-grid estimate from step 2.1 to obtain the stated inequality.

Finite-horizon L2 maximal bound for martingales with right-continuous paths and left limits

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources