In a paper, I find the following situation:
Let $(\Omega,\mathcal{G},\mathbb{Q})$ be a probability space. $\mathbb{Q}$ is supposed to be a risk neutral measure. Suppose that $\left(\mathcal{F}_t\right)_{t\geq 0}$ is a filtration with the full market information at time $t$.
Also, it is supposed that the random variable $\tau$ represents the default time of an obligor.
The paper state that $\tau$ is 'adapted' to the filtration $\left(\mathcal{F}_t\right)_{t\geq 0}$. My question: How I am supposed to undertand $\tau$?
I see $\tau$ defined as a stochastic proccess $\tau_t$ where $\tau_t$ is $\mathcal{F}_t$-medible and $\tau_t\in\{0,t\}$, and $\mathbb{Q}(\tau_t=0|\tau_{t'}>0)=1$ with $t>t'$.
Is my vision right?
$\tau$ here strikes me as a Stopping Time. In that case, it is not a process but a random variable a) defined on the entire space of paths, b) measurable with respect to the filtration $\mathcal{F}_t$ in the sense that: $$\{\tau \leq t\}$$ is $\mathcal{F}_t$ measurable for all $t$.
It is strange for me to think of it as a standard adapted process. Say for a path $\omega$, the value of $\tau$ is $s$. I presume one could then define $\tau_t$ to be $\infty$ for $t < s$ and $s$ for $t \geq s$. Then $\{\tau_t \leq t\}$ is indeed $\mathcal{F}_t$ measurable.