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A continuous finite-variation martingale is constant

Statement

Let $D = (D_{t})_{0 \leq t \leq T}$ be a continuous martingale relative to a filtration $(F_{t})$ , with $D_{0} = 0$ . Suppose each path has finite total variation $V_{T} (D)$ and $E V_{T} (D) < \infty$ . Then $D$ is indistinguishably zero on $[0, T]$ .

Facts & Assumptions

Given: A finite horizon $T$ , a filtered probability space, and a continuous adapted martingale $D$ with $D_{0} = 0$ and integrable pathwise total variation.

[F1]

The all-pairs martingale identity holds at every $s \leq t$ , and the continuous-path hypothesis supplies a measurable probability-one event on which each path of $D$ is continuous. (Continuous-time adapted processes and martingales)

[F2]

A bounded $F_{s}$ -measurable factor can be taken outside conditional expectation given $F_{s}$ . (Taking out what is known)

[F3]

Dominated convergence passes limits through integrals under an integrable majorant. (Dominated convergence)

[F4]

Total variation bounds every partition sum of absolute increments. (Bounded variation and total variation on an interval)

[F5]

Every finite closed interval $[0, T]$ is compact. (Heine-Borel by bisection: every closed bounded interval $[a, b]$ is compact)

[F6]

A continuous real function on a compact subset of $R$ is uniformly continuous. (Heine-Cantor in $R$ : a continuous real function on a compact subset of $R$ is uniformly continuous)

[F7]

A nonnegative random variable with expectation zero vanishes almost surely. (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere)

[F8]

Indistinguishability means equality for every time outside one measurable probability-one event. (Process law, modification, and indistinguishability)

[A1]

AC is inherited from the conditional-expectation interface used by the continuous-time martingale definition. (The Axiom of Choice)

Proof

1.1constructalgebra

Fix $R > 0$ and define $ϕ_{R} (x) = \max (- R, \min (x, R))$ and $G_{R} (x) = {\begin{cases} x^{2} / 2, & ∣ x ∣ \leq R, \\ R ∣ x ∣ - R^{2} / 2, & ∣ x ∣ \geq R . \end{cases}$ The explicit piecewise formula gives $G_{R}^{'} = ϕ_{R}$ ; $ϕ_{R}$ is bounded by $R$ and is $1$ -Lipschitz, while $G_{R} \geq 0$ and $G_{R} (x) = 0$ exactly when $x = 0$ . Splitting an interval at $- R$ and $R$ if needed and using the Lipschitz bound gives, for all $x, y \in R$ , $∣ G_{R} (y) - G_{R} (x) - ϕ_{R} (x) (y - x) ∣ \leq \frac{1}{2} ∣ y - x ∣^{2} .$

2.1A1F1F2F3F4F5F6F7givenconstruct

Fix $t \in (0, T]$ and deterministic partitions $0 = t_{0}^{m} < \dots < t_{q_{m}}^{m} = t$ with mesh tending to zero. Put $S_{m} = \sum_{j = 0}^{q_{m} - 1} ϕ_{R} (D_{t_{j}^{m}}) (D_{t_{j + 1}^{m}} - D_{t_{j}^{m}}) .$ Each coefficient is bounded and $F_{t_{j}^{m}}$ -measurable. The martingale identity and [F2] give $E S_{m} = 0$ . The estimate of step 1.1 and the total-variation bound [F4] imply $∣ S_{m} - G_{R} (D_{t}) + G_{R} (D_{0}) ∣ \leq \frac{1}{2} \sum_{j} ∣ D_{t_{j + 1}^{m}} - D_{t_{j}^{m}} ∣^{2} \leq \frac{1}{2} \max_{j} ∣ D_{t_{j + 1}^{m}} - D_{t_{j}^{m}} ∣ V_{T} (D) .$ On the probability-one continuity event in [F1], compactness [F5] and Heine--Cantor [F6] make the maximum increment tend to zero. Thus $S_{m} \to G_{R} (D_{t})$ almost surely, since $D_{0} = 0$ . Also $∣ S_{m} ∣ \leq R V_{T} (D)$ , an integrable bound by the hypothesis. Dominated convergence [F3] yields $E G_{R} (D_{t}) = 0$ . Since $G_{R}$ is nonnegative and vanishes only at zero, [F7] gives $D_{t} = 0$ almost surely.

3.1F1F8step 2.1givenconstruct∎

The conclusion also holds at $t = 0$ because $D_{0} = 0$ . For every rational $r \in [0, T]$ , step 2.1 gives a measurable probability-one event on which $D_{r} = 0$ . Intersect these countably many events with the probability-one continuity event from [F1]. On the resulting measurable probability-one event, continuity and density of the rationals give $D_{t} = 0$ for every $t \in [0, T]$ . By [F8], $D$ is indistinguishably zero.

Source notes

Bass's Section 2.3 discusses càdlàg martingale decompositions and the jump-bracket framework. The bounded-integrand partition argument above supplies the finite-horizon finite-variation martingale proof used here.

Depends on

Used by

Uniqueness of continuous predictable brackets Lemma

Dependency tree · two levels

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Sources

Richard F. Bass, SDEs with Jumps, Section 2.3, PDF pp. 3–4 (standard reference, not scraped)