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Fractional analysics
Fractional analysics
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Let $\tau$ be a random variable, which is defined on the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F})_{t\in T}, P)$ with values in $T$. In most cases, $T=[0,\infty]$. Then $\tau$ is called a stopping time (with respect to the filtration $(\mathcal{F})_{t\in T}$), if the following condition holds: $\{\tau\leq t\}\in (\mathcal{F})_t $ for all $t\in T$. Can someone explain a little bit what is the difference $\{\tau\leq t\}\in (\mathcal{F})_t $ and $\{\tau<t\}\in (\mathcal{F})_t $? Can we replace $\{\tau\leq t\}\in (\mathcal{F})_t $ using $\{\tau<t\}\in (\mathcal{F})_t $ in the definition of stopping time?

Let $\tau$ be a random variable, which is defined on the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F})_{t\in T}, P)$ with values in $T$. In most cases, $T=[0,\infty]$. Then $\tau$ is called a stopping time (with respect to the filtration $(\mathcal{F})_{t\in T}$), if the following condition holds: $\{\tau\leq t\}\in (\mathcal{F})_t $ for all $t\in T$. Can someone explain a little bit what is ? Can we replace $\{\tau\leq t\}\in (\mathcal{F})_t $ using $\{\tau<t\}\in (\mathcal{F})_t $ in the definition of stopping time?

Let $\tau$ be a random variable, which is defined on the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F})_{t\in T}, P)$ with values in $T$. In most cases, $T=[0,\infty]$. Then $\tau$ is called a stopping time (with respect to the filtration $(\mathcal{F})_{t\in T}$), if the following condition holds: $\{\tau\leq t\}\in (\mathcal{F})_t $ for all $t\in T$. Can someone explain a little bit what is the difference $\{\tau\leq t\}\in (\mathcal{F})_t $ and $\{\tau<t\}\in (\mathcal{F})_t $? Can we replace $\{\tau\leq t\}\in (\mathcal{F})_t $ using $\{\tau<t\}\in (\mathcal{F})_t $ in the definition of stopping time?

Source Link
Fractional analysics
Fractional analysics
  • 461
  • 2
  • 9

What is the difference $\{\tau\leq t\}\in (\mathcal{F})_t $ and $\{\tau<t\}\in (\mathcal{F})_t $ in the definition of stopping time?

Let $\tau$ be a random variable, which is defined on the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F})_{t\in T}, P)$ with values in $T$. In most cases, $T=[0,\infty]$. Then $\tau$ is called a stopping time (with respect to the filtration $(\mathcal{F})_{t\in T}$), if the following condition holds: $\{\tau\leq t\}\in (\mathcal{F})_t $ for all $t\in T$. Can someone explain a little bit what is ? Can we replace $\{\tau\leq t\}\in (\mathcal{F})_t $ using $\{\tau<t\}\in (\mathcal{F})_t $ in the definition of stopping time?