1
$\begingroup$

  1. I want to prove that for stopping times $S,T$ : $$ \{ \min(S,t) < \min(T,t) \} \in \mathcal{F}_t$$

  2. I want to prove implication for both directions ( I took it from the lecture notes of the course I study):

A random time $T$ is an $\mathbb{F}^+$ -stopping time iff for all $t>0$ one has $\{ T

I have deleted my useless old ideas. I would like to apologise for mess in comments and confusion I did to saz and Ansel B.

Below, I am trying to write down rigorously without missing steps what Ansel B said about the problem number 1.

Firstly, to prove that $ \{T \leq t \} \in \mathcal{F}_T $ I want to see that: $$ \{T \leq t \} \in \{F \in \mathcal{F}_\infty | F \cap \{T \leq t\} \in \mathcal{F}_t, \forall t > 0 \}$$

Is it trivial, or I should say something on that? My explanation to this step is: as $\mathcal{F}_\infty = \sigma(\mathcal{F}_t, t \geq 0 )$ we indeed have $F$ inside of even $\{T \leq t\}$ as $ \mathcal{F}_\infty$ is created by $\mathcal{F}_t$-s.

By analogy, we say that $ \{ \min(T,t) \leq t \} \in \mathcal{F}_{\min(T,t)} $.

Moreover $\mathcal{F}_{\min(T,t)} \subset \mathcal{F}_{t} $. ( Is it trivial?)

Similarly with $ \{ \min(S,t) \leq t \} \in \mathcal{F}_{\min(S,t)} \subset \mathcal{F}_{t}$.

Finally 2 subsets of $\mathcal{F}_{t}$: $ \{ \min(S,t) \leq t \} $ and $ \{ \min(T,t) \leq t \} $ together lead to 4 cases:

$$\{S \leq T\}, \{t \leq T\}, \{S \leq t\}, \{t \leq t\}$$ where first case is by definition correct, as we choose from the beginning $ S \leq T$. Second is impossible, Third is definion of stopping time $S$, hence is always in $\mathcal{F}_{t}$. Fourth is trivial case and also in $\mathcal{F}_{t}$.

Therefore we can conclude that $ \{ \min(S,t) < \min(T,t) \} \in \mathcal{F}_t$.

  • 0
    Are you sure about statement 2? I doubt that it is correct as stated now. (Certainly, we have $\{T \leq t\} \in \mathcal{F}_{t+}$, but, as far as I can say, $\{T \leq t\} \in \mathcal{F}_t$ does, in general, not hold true.) – 2017-02-16
  • 0
    @saz, random time $T$ is an $F^+$ - stopping time iff for all $t>0$ one has $\{T – 2017-02-16
  • 0
    Saz is correct your second statement is always true when the filtration is right continuous. – 2017-02-16
  • 0
    @StochasticIntegrationStudent I'm not asking for the definition of a random time; I claim that the second assertion is, in general, not correct. – 2017-02-16
  • 0
    Dear @saz, I hope I made the question clear now. Sorry for confusion I did. – 2017-03-05

1 Answers 1

1

First note that if $S$ and $T$ are stopping times then so is $\min(S,T)$ as $\{\min(S,T)>t\}=\{S>t\}\cap\{T>t\}\in\mathcal{F}_t$. Also it is very easily seen that any stopping time $T$ is $\mathcal{F}_T$ measurable. And for a fixed $t$, $t$ is trivally a stopping time, thus $\min(T,t)$ is $\mathcal{F}_{\min(T,t)}\subset\mathcal{F}_t$-measurable, similarly with $\min(S,t)$, so $\{\min(S,t)<\min(T,t)\}\in\mathcal{F}_t$.

  • 0
    Dear @Ansel B, I took a pause to digest material. First, thank you for reply. Secondly, I found answer a bit difficult to understand and decided to extend it in my question. Could you please look up at my question again and say if that is precisely what you meant by answering the question? If you may, could you also reply on my questions I added in between the proof. – 2017-03-05