Let E be a normed R -vector space, ( X t ) t ≥ 0 be an E-valued càdlàg Lévy process on a filtered probability space ( Ω , A , ( F t ) t ≥ 0 , P ), B ∈ B ( E ) with 0 ∉ B ¯ and N t ( ω ) := | { s ∈ ( 0 , t ] : Δ X s ( ω ) ∈ B } | = ∑ s ∈ [ 0 , t ] Δ X s ( ω ) 1 B ( Δ X s ( ω ) )for ω ∈ Ω and t ≥ 0.How do we see that t ↦ N t ( ω ) is càdlàg?

Question

Let   E be a normed       R  -vector space,   (      X    t        )          t      ≥      0       be an   E-valued càdlàg Lévy process on a filtered probability space   (  Ω  ,      A    ,  (            F        t        )          t      ≥      0        ,  P  ),   B  ∈      B    (  E  ) with   0  ∉      B    ¯   and      N    t    (  ω  )  :=      |          {      s      ∈      (      0      ,      t      ]      :      Δ              X        s            (      ω      )      ∈      B      }        |    =      ∑                                                      s              ∈              [              0              ,                            t              ]                                                          Δ                              X                s                            (              ω              )                                                1    B    (  Δ      X    s    (  ω  )  )for   ω  ∈  Ω and   t  ≥  0.How do we see that   t  ↦      N    t    (  ω  ) is càdlàg?

Alexzander Bowman · Accepted Answer

In order for   N to be cadlag, it should be assumed that   X has finite jump activity in   B, i.e.   ν  (  B  )  &amp;lt;  ∞, where   ν is the Lévy measure of   X (otherwise, it blows up to infinity immediately). And in such case, this is rather straightforward: with probability 1, the number of jumps in   B is locally finite, and       N    t    =      ∑          n      ≥      1                  1              [              τ        n            ,      ∞      )        (  t  ), where       τ    n   is the time of   nth jump in   B; this is clearly cadlag.

Let E be a normed <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R

Answered question

Answer & Explanation

New Questions in Pre-Algebra