Springer Finance

11.4 Jump Processes and Their Integrals 4 79

AlE[tAS1IF(s)] = AlE[tAS1IS 1 > s]

= A^21

00

(t A u)e- >. (u-s) du

= A^21

t

ue->.(u-s)du + A^21

oo

te->.(u-s)du

= -Aue->.<u-s) l

u=t

+A 1

t

e->.(u-s>du- Ate->.<u-s) lu

=oo

u=s 8 u=t

= As -Ate->.(t-s) -e->.(u-s) 1::: + Ate->.(t-s)

= As-e->.(t-s) + 1. (11.4.9)

Subtracting (11.4.9) from (11.4.8), we obtain in the case 81 > s that

This completes the verification of the martingale property for the stochastic
integral in (11.4.6).
Note that if we had taken the integrand in (11.4.6) to be H[o,st)(t), which is
right-continuous rather than left-continuous at sl' then we would have gotten

(11.4.10)

According to (11.4.9) with s = 0,

lE[-A(t A SI)] = e->.t-1,

which is strictly decreasing in t. Consequently, the integral (11.4.10) obtained
from the right-continuous integrand n[o,st) (t) is not a martingale. 0

11.4.2 Quadratic Variation

In order to write down the It6-Doeblin formula for processes with jumps, we
need to discuss quadratic variation. Let X(t) be a jump process. To compute
the quadratic variation of X on [0, T], we choose 0 = to < h < t 2 < · · · <
tn = T, denote the set of these times by II = {to, h, ... , tn}, denote the
length of the longest subinterval by IIIIII = maxj(tj+l -tj), and define
n- 1
Qrr(X) = L (X(tj+I)- X(tj)t
j=O

The quadratic variation of X on [0, T] is defined to be

[X, X](T) = lim Qrr(X),

IIITII--+0