Supplement to " Testing Whether Jumps Have Finite or Infinite Activity
نویسندگان
چکیده
When ≥ 2 these processes are finite-valued, but not necessarily so when 2. In this case we have to be careful: if = inf( : () =∞) and 0 = inf( : 0() = ∞) then both processes () and 0() are null at 0 and finite-valued on [0 ) and [0 0) respectively, the first continuous and the second one càdlàg with jumps not bigger than 1, and they are infinite on (∞) and (0∞). However, if 0 ∞ or 0 0 ∞ we may have () or 0()0 finite or infinite, and the left limits satisfy ()− = () and 0()0− ≤ 0()0 ≤ 0()0− + 1. Note that 0(0) is the number of jumps over [0 ], whereas (0) = R 0 (R) . Lemma 3. For any random time and any ≥ 0, the two sets {() =∞} and { 0() =∞} are a.s. equal. Proof. Set = inf( : () ≥ ) and 0 = inf( : 0() ≥ ), and consider the sets = {() ∞} and 0 = { 0() ∞}. Observe that () ≤ and ()0 ≤ + 1, which entail = ∪≥1{ ≤ } and 0 = ∪≥1{ ≤ 0 }. Moreover () is the predictable compensator of 0(), so for any stopping time we have E(() ) = E( 0() ). We deduce that 0() ∞ and ()0 ∞ a.s., and the almost sure equality = 0 is then obvious.
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