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I have the following definition of exercise time. Let $T\in\mathbb{N}$ with $T>0$, let $(\Omega,\mathcal{F})$ be a probability space with the $\sigma$-algebra $\mathcal{F}=2^{\Omega}$ and let $(\mathcal{F}_t)_{t=0,...,T}$ be a filtration such that $\mathcal{F}_0=\left\{\emptyset,\Omega\right\}$ and $\mathcal{F}_T=\mathcal{F}$.

Definition. An exercise time $\tau$ is any random variable

$$ \tau:\Omega\rightarrow\left\{0,1,2,..,T\right\} $$

such that

$$ \left\{\tau=t\right\}\in\mathcal{F}_t,~~\textit{ for all }t=0,...,T.~~~~~~~~ (1) $$

I get lost in the interpretation of the measurability condition (1). Any example/clarification would be appreciated.

AlmostSureUser
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    You might find this useful: http://math.stackexchange.com/questions/331410/why-is-stopping-time-defined-as-a-random-variable –  Nov 20 '15 at 15:36
  • Another possible example that came to my mind is the following. The condition (1), if I am correct, should be equivalent to say that the indicator function $1_{\left{\tau(\omega)=t\right}}$ must be $\mathcal{F}t$-adapted and hence if $\mathcal{F}_t$ is, for example, the filtration generated by a stochastic process $S_t$ we get that, for all $t$, there exists a deterministic measurable function $h_t$ such that $$ 1{\left{\tau(\omega)=t\right}} = h_t\left(S_0,S_1,...,S_t\right), $$ whence the idea that the occurrence of the event $\tau=t$ depends on the path of the process $S_t$. – AlmostSureUser Nov 21 '15 at 10:39

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