Springer Finance

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11.4 Jump Processes and Their Integrals 481

= E (Xf(tj+l)- Xf(tj)) (Xi(tj+l)- Xi(tj))

j=O

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+ E (Xf(tj+I)- Xf(tj))(J2(tj+l)- J2 (tj))

j=O

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+ E (J1 (tj+1) - J 1(tj))(Xi(tj+l)-Xi(tj))

j=O

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+ E (J1 (ti+d - J 1 (tj)) (J2(tj+1)- h(tj)). (11.4.13)

j=O

We know from the theory of continuous processes that

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lim L (Xf(ti+d - Xf(tj)) (Xi{tj+l)- Xi(tj)) = [XL Xi] (T)

IIIIII-+ (^0) j=O
= 1
T
n(s)H(s)ds.
We shall show that the second and third terms appearing on the right­
hand side of (11.4.13) have limit zero as llllll -+ 0, and the fourth term has
limit
J1, J2 = L L1J (^1) (s)L1J 2 (s).
O<s�T
We consider the second term on the right-hand side of (11.4.13):
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E (Xf(tj +l)-Xf(tj )) (J 2(tj+l )- J2 (tj))
j=O
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::; �ax IXf(tj+l)- Xf(tj)l· L ih(tj+l)- J2(tj)l
O�J�n-1.
J= 0
::; �ax O<J<n-1 IXf(tj+l)- Xf(tj)l· L IL1J 2 (s)l.

• 
  
  
  
  • O<s�T
  
  
  
  

As llllll -+ 0, the factor maxo�j�n-1 IXf(tj+l) -Xf(tj)l has limit zero,
whereas Eo<s<T IL1J2(s)l is a finite number not depending on ll. Hence, the
second term on the right-hand side of (11.4.13) has limit zero as llllll -+ 0.
Similarly, the third term on the right-hand side of (11.4.13) has limit zero.
Let us fix an arbitrary w E fl, which fixes the paths of these processes, and
choose the time points in ll so close together that there is at most one jump of
J 1 in each interval (ti, ti+l], at most one jump of J 2 in each interval (tj, ti+1],
and if J 1 and J 2 have a jump in the same interval, then these jumps are

simultaneous. Let A 1 denote the set of indices j for which (tj, t (^3) +1] contains a