For a filtration F 0 ⊂ ⋯ ⊂ F t ⊂ ⋯ ⊂ F ∞ . We know that if X is F 0 -measurable, then X is also F t -measurable for t ≥ 0.I have a question about the martingale X t (which F t measurable) is that given a stopping time T,why 1 T = ∞ X ∞ is F T -measurable?It is clear that 1 T = ∞ is F T -measurable. But because X ∞ is F ∞ -measurable. How can we say X ∞ is also F T -measurable?

Question

For a filtration             F        0    ⊂  ⋯  ⊂            F        t    ⊂  ⋯  ⊂            F              ∞      . We know that if   X is             F        0  -measurable, then   X is also             F        t  -measurable for   t  ≥  0.I have a question about the martingale       X    t   (which             F        t   measurable) is that given a stopping time   T,why       1          T      =      ∞            X          ∞       is             F              T      -measurable?It is clear that       1          T      =      ∞       is             F              T      -measurable. But because       X          ∞       is             F              ∞      -measurable. How can we say       X          ∞       is also             F              T      -measurable?

szabadotttt3dg · Accepted Answer

X    ∞    does not need to be             F        T   measurable, but       1          T      =      ∞            X    ∞   is. We can show this directly from the definition:We want to show   {      1          T      =      ∞            X    ∞    ∈  B  }  ∈            F        T   for any Borel set B. By the definition of             F        T   this means we must show   {      1          T      =      ∞            X    ∞    ∈  B  }  ∩  {  T  ≤  t  }  ∈            F        t   for all   t  &amp;gt;  0. On the event   {  T  ≤  t  },       1          T      =      ∞            X    ∞    =  0, so  {      1          T      =      ∞            X    ∞    ∈  B  }  ∩  {  T  ≤  t  }  =  {  T  ≤  t  }if   0  ∈  B and  {      1          T      =      ∞            X    ∞    ∈  B  }  ∩  {  T  ≤  t  }  =  ∅if   0  ∉  B. In either case,   {      1          T      =      ∞            X    ∞    ∈  B  }  ∩  {  T  ≤  t  }  ∈            F        t  , so       1          T      =      ∞            X    ∞   is             F        T   measurable.

For a filtration <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mat

Answered question

Answer & Explanation

New Questions in Pre-Algebra