In Steele's Stochastic Calculus and Financial Applications the Itô integral is defined at a first stage for integrands belonging to the set $\mathcal{H}^2$ comprised of all $f\colon\Omega\times[0,T]\to\mathbb{R}$ that satisfy the following conditions:
- $f$ is jointly measurable, that is, $f$ is $\mathcal{F}_T\times\mathcal{B}$ measurable (where $\{\mathcal{F}_t\colon\,t\in[0,T]\}$ is the standard Brownian filtration and $\mathcal{B}$ is the Borel $\sigma$-field on $[0,T]$)
- $f$ is adapted, that is $\omega\mapsto f(\omega,t)$ is $\mathcal{F}_t$ measurable for each $t\in[0,T]$
- $\int_\Omega \int_0^T f(\omega,t)^2\,\mathrm{d}t\,\mathrm{d}P(\omega) <\infty$
He then goes on to extend the domain of the Itô integral to include all functions belonging to the set $\mathcal{L}^2_{\mathrm{LOC}}$ comprised of jointly measurable, adapted $f$ for which $$P\bigg(\omega:\,\int_0^T f(\omega,t)^2\,\mathrm{d}t < \infty \bigg) = 1.$$
Proposition 7.1 then states that for $f\in \mathcal{L}^2_{\mathrm{LOC}}$ the sequence of r.v's $(\tau_n)$ defined by $$\tau_n(\omega) = \inf\bigg\{s:\,\int_0^s f(\omega,t)^2\,\mathrm{d}t \geq n \mbox{ or } s\geq T\bigg\}$$ is localizing for $f$, that is to say, the $\tau_n$ are stopping times such that
- $\tau_n \leq \tau_{n+1}$ almost surely
- $P\big(\bigcup_{n=1}^\infty[\tau_n = T]\big) = 1$
- The functions $f_n$ defined by $f_n(\omega,t) = f(\omega,t)\,\mathbb{I}_{[\tau_n\geq t]}(\omega)$, belong to $\mathcal{H}^2$ for all $n$.
Next, local Martingales are introduced as follows: an $\{\mathcal{F}_t\}$-adapted process $\{M_t:\,0\leq t<\infty\}$ is said to be a local Martingale provided there is a nondecreasing sequence $(\tau_n)$ of stopping times with the property that $\lim \tau_n = \infty$ almost surely and such that for each $n$ the process $$\{M_{t\wedge\tau_n} - M_0:\, 0\leq t<\infty\}$$ is a Martingale with respect to $\{\mathcal{F}_t:\, 0\leq t<\infty\}$.
The above definition is given so that it keeps up with tradition (it is the definition of a local Martingale given in most textbooks), but it appears to break the line of exposition where the stochastic processes being considered had time parameter varying over $[0, T]$. In particular, Proposition 7.7 states that, for any $f\in\mathcal{L}^2_{\mathrm{LOC}}$, there is a continuous local martingale $(X_t)$ such that $$P\bigg(\omega:\,X_t(\omega) = \int_0^t f(\omega,s) B_{\mathrm{d}s}(\omega)\bigg)=1,$$ where the localizing sequence can be taken as in Proposition 7.1
Now, clearly the $\tau_n$ defined in Prop. 7.1 are such that $\tau_n(\omega) \leq T$, and so the condition $\tau_n\uparrow \infty$ cannot hold. My question is: should I use a different definition of local Martingale when considering a process with time parameter varying on $[0,T]$ (for instance, requiring only that $\tau_n\uparrow T$ almost surely)? Otherwise, how can I adapt the statement of Proposition 7.7 so that is is correct? (Perhaps taking $\tau_{n+1} = \tau_n + 1$ after the sequence reaches $T$?)