Let S , T be two stopping times. Then S ∨ T and S ∧ T are salso topping times with respect to F S ∧ T = F S ∩ F T . Futhermore, { S ≤ T } ∈ F S ∧ T and { S = T } ∈ F S ∧ T . For a stopping time τ ∈ F t and A ∈ F τ . How to show that A ∩ { τ ≤ t } ∈ F τ ∧ t ?

Question

Let   S  ,  T be two stopping times. Then   S  ∨  T and   S  ∧  T are salso topping times with respect to             F              S      ∧      T        =            F              S        ∩            F              T      . Futhermore,   {  S  ≤  T  }  ∈            F              S      ∧      T       and   {  S  =  T  }  ∈            F              S      ∧      T      . For a stopping time   τ  ∈            F        t   and   A  ∈            F              τ      . How to show that   A  ∩  {  τ  ≤  t  }  ∈            F              τ      ∧      t      ?

Leland Ochoa · Accepted Answer

For any   t  ≥  0,   τ  ∧  t is also a sopping time. If   A  ∈            F        τ  , then by definition of             F        τ  , for any   s  ≥  0,   A  ∩  {  τ  ≤  s  }  ∈            F        s  .To show that   A  ∩  {  τ  ≤  t  } belongs to             F              τ      ∧      t       notice that for any   s  ≥  0,                                          (                          A        ∩        {        τ        ≤        t        }                              )                          ∩        {        τ        ∧        t        ≤        s        }                            =        A        ∩                              (                          {        τ        ≤        t        }        ∩        {        τ        ∧        t        ≤        s        }                              )                                                            =        A        ∩        {        τ        ≤        s        ∧        t        }        ∈                              F                                t            ∧            s                          ⊂                              F                    s                    This means that   A  ∩  {  τ  ≤  t  }  ∈            F              τ      ∧      t      .

Let S , T be two stopping times. Then S ∨ T and S &#

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