(H \cdot M)_{t} = \sum_{k = 0}^{n - 1} Z_{t_{k}} (M_{t \land t_{k + 1}} - M_{t \land t_{k}}) .

Common refinement of the deterministic partitions makes this definition independent of the representation. Each summand is a martingale increment multiplied by information already available at its left endpoint, so the resulting process is a continuous square-integrable martingale starting at zero. Its terminal value is the same sum with

t

replaced by

t_{n}

Write

Δ_{k} M = M_{t_{k + 1}} - M_{t_{k}}

. If

j < k

, then

Z_{t_{j}} Δ_{j} M Z_{t_{k}}

F_{t_{k}}

-measurable. By the conditional expectation identity for increments of a martingale,

E [Z_{t_{j}} Δ_{j} M Z_{t_{k}} Δ_{k} M] = 0.

(2)

The conditional bracket isometry for stopped martingale increments gives

E [(Δ_{k} M)^{2} ∣ F_{t_{k}}] = E ([M]_{t_{k + 1}} - [M]_{t_{k}} ∣ F_{t_{k}}) .

(3)

For completeness, this identity follows by applying the optional stopping theorem to

M

and

M^{2} - [M]

and expanding the square. The latter process is a true uniformly integrable martingale here: the Doob L2 maximal inequality bounds

E sup_{t} ∣ M_{t} ∣^{2}

, stopped square identities and Fatou's lemma give

E [M]_{\infty} < \infty

, and its absolute supremum is bounded by

sup_{t} ∣ M_{t} ∣^{2} + [M]_{\infty}

. Consequently all the stopped identities pass to the limit.

Multiplying the conditional identity by

Z_{t_{k}}^{2}

, taking expectations, and using the vanished cross terms proves the Itô isometry:

E [(H \cdot M)_{\infty}^{2}] = k \sum E [Z_{t_{k}}^{2} ([M]_{t_{k + 1}} - [M]_{t_{k}})] = E \int_{0}^{\infty} H_{s}^{2} d [M]_{s} .

(4)

To extend the stochastic integral, use the finite quadratic-variation measure

ν_{M} (C) = E \int 1_{C} d [M]

on the predictable sigma-algebra. The density of simple predictable processes for finite measures gives approximations to every

H \in L^{2} (ν_{M})

. The Itô isometry makes the corresponding terminal integrals Cauchy in

L^{2}

, and the Doob L2 maximal inequality makes the whole processes Cauchy in expected squared supremum. Their limit defines a continuous martingale, and the isometry persists. For a locally bounded predictable process, choose increasing stopping times tending to infinity on which it is bounded, also restrict the time horizon, and perform this construction after stopping. The isometry shows that the stopped definitions agree on overlaps, so they patch into a uniquely defined continuous local martingale