Springer Finance

480 11 Introduction to Jump Processes

where of course as II Jill -+ 0 the number of points in li must approach infinity.
In general, X, X can be random (i.e., can depend on the path of X).
However, in the case of Brownian motion, we know that W, W = T does
not depend on the path. In the case of an Ito integral I(T) = J 0
T
r(s)dW(s)
with respect to Brownian motion, [J, I] (T) = J: T^2 (s)ds can depend on the
path because r( s) can depend on the path.
We will also need the concept of cross variation. Let XI(t) and X 2 (t) be
jump processes. We define

n-I

Cn(XI, X 2 ) = L (XI(tj+l) - XI(ti))(X 2 (ti+1) - X 2 (ti))

j=O

and

Theorem 11.4.7. Let XI(t) = XI(O) + h(t) + RI(t) + JI(t) be a jump pro­

cess, where II(t) = f� n(s) dW(s), RI(t) = f� BI(s) ds, and JI(t) is a right­

continuous pure jump process. Then Xl(t) = XI (0) + h (t) + RI (t) and

Jl> JI = Jl> JI + Jl> JI = 1

T

rf(s) ds + L (LlJI(s))

2

.

(^0) 0<s:$T
(11.4.11)
Let X 2 (t) = X 2 (0) + l 2 (t) + R2(t) + J 2 (t) be another jump process, where
h(t) = f� T2(s) dW(s), R2(t) = f�^82 (s) ds, and J 2 (t) is a right-continuous
pure jump process. Then Xi (t) = X 2 (0) + l 2 (t) + R2(t), and
JI,J 2 = [Xf,Xil(T) + JI,J 2
= 1
T
n(s)r 2 (s) ds + L LlJI(s)Llh(s). (11.4.12)
(^0) O<s:$T
PROOF: We only need to prove (11.4.12) since (11.4.11) is the special case of
(11.4.12) in which x 2 =XI. We have
n-I

Cn(XI, X 2 ) = L (XI(tjH)-XI(ti)) (X 2 (ti+1) - X 2 (ti))

j=O

n-I

= L (Xf(tj+l)-xc(tj) + JI(tj+I)-JI(tj))

j=O

X (XHtj+l)-Xi{tj) + h(tj+l)-h(tj))