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Stopping an Ito integral

Statement

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands. Let $H$ be a locally square-integrable predictable process Locally square-integrable predictable Brownian integrands with localized integral $H \cdot B$ in the progressively measurable version of Localized Ito integral, with its measurable-full-event convention for indistinguishability. The filtration satisfies the usual conditions and local energy is finite almost surely on every finite horizon, as required by the local-integrability definition. Let $τ$ be a stopping time Continuous-time stopping times and stopped sigma-algebras. Then the process $s \mapsto (H \cdot B)_{s \land τ}$ and the localized integral of the process $1_{[0, τ]} H$ are indistinguishable: $(H \cdot B)_{t \land τ} = \int_{0}^{t} 1_{[0, τ]} (s) H_{s} d B_{s} for every t \geq 0, up to indistinguishability .$ In particular, for $H$ of finite energy this reduces to the stopping identity of Localized Ito integral, and for $τ \equiv \infty$ it is the definition of the localized integral.

Facts & Assumptions

Given: AC, the standing hypothesis (H), a locally square-integrable predictable $H$ with energy $A$ and canonical stopping times $τ_{n}$ , $n \geq 1$ , its localized integral $H \cdot B$ , and a stopping time $τ$ .

[F1]

$1_{[0, τ]}$ is predictable for every stopping time, and $1_{[0, τ]} H$ is predictable and locally square-integrable with energy $\int_{0}^{t} 1_{[0, τ]} H^{2} d s \leq A_{t} < \infty$ almost surely for every finite $t$ ; its localized integral exists and is a continuous local martingale. Progressively measurable and predictable processes Localized Ito integral

[F2]

The canonical times $τ_{n}$ satisfy $τ_{n} \leq n$ , $τ_{n} ↑ \infty$ a.s., $1_{(0, τ_{n}]} H$ has finite energy $E A_{t \land τ_{n}} \leq n$ , and $(H \cdot B)^{τ_{n}}$ is the finite-energy integral of $H 1_{(0, τ_{n}]}$ ; the finite-energy stopping identity gives $(G \cdot B)_{t}^{σ} = \int_{0}^{t} G 1_{(0, σ]} d B$ for finite-energy $G$ and any stopping time $σ$ . Locally square-integrable predictable Brownian integrands Localized Ito integral

[F3]

If $(ρ_{k})$ is a nondecreasing sequence of stopping times with $ρ_{k} ↑ \infty$ a.s., each $H 1_{(0, ρ_{k}]}$ has finite energy, and a continuous adapted $N$ with $N_{0} = 0$ has $N^{ρ_{k}}$ indistinguishable from the finite-energy integral of $H 1_{(0, ρ_{k}]}$ for every $k$ , then $N$ is indistinguishable from the localized integral of $H$ . Localized Ito integral

[F4]

The chosen localized integral is progressive. The measurable stopped-evaluation argument of Localized Ito integral, proof step 1.1, shows that its stopped values are adapted; stopping also preserves continuity on the same full event. Thus $N : = (H \cdot B)^{τ}$ is an adapted continuous process with $N_{0} = 0$ . The proof below uses the original sequence $(τ_{n}) ↑ \infty$ to localize $N$ ; the bounded sequence $(τ \land τ_{n})$ is not asserted to be a localizing sequence. Continuous-time adapted processes and martingales Localized Ito integral

[F5]

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

Proof

technique · direct

1.1F1F4given

Put $G : = 1_{[0, τ]} H$ and $N : = (H \cdot B)^{τ}$ , so that $N$ is adapted with continuous paths and $N_{0} = (H \cdot B)_{0} = 0$ by [F4]; by [F1] the localized integral $G \cdot B$ exists and is a continuous local martingale. No local-martingale property of $N$ is assumed at this stage.

1.2F2given

For every $k$ the stopped integral $(H \cdot B)^{τ_{k}}$ is the finite-energy integral of $H 1_{(0, τ_{k}]}$ by [F2], and the finite-energy stopping identity applied with the stopping time $τ$ gives $(H \cdot B)^{τ \land τ_{k}} = (H 1_{(0, τ_{k}]} \cdot B)^{τ} = \int_{0}^{t} H 1_{(0, τ_{k}]} 1_{(0, τ]} d B$ .

2.1F2step 1.2

The integrand identity $H 1_{(0, τ_{k}]} 1_{(0, τ]} = G 1_{(0, τ \land τ_{k}]} = G 1_{(0, τ_{k}]}$ holds for every $(s, ω)$ with $s > 0$ , the three expressions differing at most at $s = 0$ , a $(d t \otimes P)$ -null set; hence $\int_{0}^{t} H 1_{(0, τ_{k}]} 1_{(0, τ]} d B$ equals the finite-energy integral of $G 1_{(0, τ_{k}]}$ , and the canonical sequence $ρ_{k} : = τ_{k}$ consists of stopping times, is nondecreasing with $ρ_{k} ↑ \infty$ almost surely, (indexed by $k \geq 1$ , or reindexed by $k = j + 1$ when required), while $G 1_{(0, ρ_{k}]}$ has finite energy $E \int_{0}^{t} G_{s}^{2} 1_{(0, ρ_{k}]} d s \leq E A_{t \land ρ_{k}} \leq k$ .

3.1F3step 1.1step 1.2step 2.1

Steps 1.1, 1.2 and 2.1 verify the hypotheses of [F3] for the process $N = (H \cdot B)^{τ}$ and the localizing sequence $(ρ_{k})$ : $N$ is adapted and continuous with $N_{0} = 0$ , and $N^{ρ_{k}} = (H \cdot B)^{τ \land τ_{k}}$ is indistinguishable from the finite-energy integral of $G 1_{(0, ρ_{k}]}$ for every $k$ . Therefore $N$ is indistinguishable from the localized integral $G \cdot B$ , which is exactly the identity $(H \cdot B)_{t \land τ} = \int_{0}^{t} 1_{[0, τ]} H d B$ up to indistinguishability.

4.1F2F5step 3.1given∎

The special cases are consistent: for $τ \equiv \infty$ one has $1_{[0, τ]} \equiv 1$ and the identity is the definition of the localized integral; for finite-energy $H$ it is the finite-energy stopping identity [F2] used in the proof; and for a deterministic $τ \equiv t_{0}$ it recovers the convention $\int_{0}^{t} H 1_{[0, t_{0}]} d B = (H \cdot B)_{t \land t_{0}}$ . AC enters only through the declared ambient interfaces [F5], and the localizing sequence $(τ_{n})$ is canonical.

Source notes

Van der Vaart, Lemma 5.28, proves the finite-energy stopping identity, and Theorem 5.36 plus Lemma 5.33 extends it to the localized integral. The proof here packages the extension as an application of the characterization clause of the localized integral, with the canonical localizing sequence $ρ_{k} = τ_{k}$ of $H$ : the stopping identity for each $τ_{k}$ is step 1.2, and the a.e. integrand identity of step 2.1 expresses $N^{τ_{k}}$ as the integral of $G 1_{(0, τ_{k}]}$ , so clause 3 of Localized Ito integral applies with $ρ_{k} ↑ \infty$ .

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Sources

Aad van der Vaart, Martingales, Diffusions and Financial Mathematics, Lemma 5.28 and Theorem 5.36 (standard reference, not scraped)