We can show the following: If E is a normed R -vector space, x : [ 0 , ∞ ) → E is càdlàg, B ⊆ E ∖ { 0 }, τ 0 := 0 and τ n := inf { t &gt; τ n − 1 : Δ x ( t ) ∈ B } ⏟ =: I n for n ∈ N , then1. τ 1 ∈ ( 0 , ∞ ];2. If n ∈ N , then either I n = ∅ and hence τ n = ∞ or τ n ∈ I n .Now if X is any E-valued càdlàg Lévy process on a filtered probability space ( Ω , A , ( F t ) t ≥ 0 , P ), we can similarly define I n and τ n with x replaced by X. ( Δ X t ) t ≥ 0 is clearly F -adapted and ( X t 1 + s − X t 1 ) s ≥ 0 is a Lévy process with respect to the filtration ( F t 1 + s ) s ≥ 0 with the same distribution as X for all t 1 &gt; 0.Are we able to show that τ 1 is A -measurable? Or are we even able to show that τ 1 is measurable with respect to the right-continuous filtration F t + := ⋂ ε &gt; 0 F t + ε ? The latter is equivalent to showing that { τ 1 &lt; t } ∈ F t for all t &gt; 0. Can we show this?

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We can show the following: If   E is a normed       R  -vector space,   x  :  [  0  ,  ∞  )  →  E is càdlàg,   B  ⊆  E  ∖  {  0  },       τ    0    :=  0 and      τ    n    :=  inf                              {          t          &amp;gt;                      τ                          n              −              1                                :          Δ          x          (          t          )          ∈          B          }                ⏟                    =:                    I        n            for   n  ∈      N  , then1.       τ    1    ∈  (  0  ,  ∞  ];2. If   n  ∈      N  , then either       I    n    =  ∅ and hence       τ    n    =  ∞ or       τ    n    ∈      I    n  .Now if   X is any   E-valued càdlàg Lévy process on a filtered probability space   (  Ω  ,      A    ,  (            F        t        )          t      ≥      0        ,  P  ), we can similarly define       I    n   and       τ    n   with   x replaced by   X.  (  Δ      X    t        )          t      ≥      0       is clearly       F  -adapted and   (      X                  t        1            +      s        −      X                  t        1                  )          s      ≥      0       is a Lévy process with respect to the filtration   (            F                      t        1            +      s            )          s      ≥      0       with the same distribution as   X for all       t    1    &amp;gt;  0.Are we able to show that       τ    1   is       A  -measurable? Or are we even able to show that       τ    1   is measurable with respect to the right-continuous filtration             F              t      +        :=      ⋂          ε      &amp;gt;      0                  F              t      +      ε      ? The latter is equivalent to showing that   {      τ    1    &amp;lt;  t  }  ∈            F        t   for all   t  &amp;gt;  0. Can we show this?

nelppeazy9v3ie · Accepted Answer

The process of jumps   Δ  X is progressively measurable since it is the difference of two progressively measurable processes. Then, for a measurable subset   B  ⊆  E the hitting time       τ    1    =  inf  {  t  &amp;gt;  0  :  Δ  X  (  t  )  ∈  B  } is a stopping time according to the Début theorem. Note: The theorem assumes that the probability space is complete.

Ashley Fritz · Accepted Answer

Thank you for your answer. I was hoping that there would be a more elementary answer which doesn&#039;t rely on the Début theorem and hence doesn&#039;t require to assume completness (and right-continuity, which I guess you&#039;ve forgotten to mention) of the filtration.

We can show the following: If E is a normed <mrow class="MJX-TeXAtom-ORD"> <mi mathva

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