Springer Finance

478 11 Introduction to Jump Processes

Example 11.4.6. Let N(t) be a Poisson process with intensity A, let M(t) =

N(t)- At be the compensated Poisson process, and let

be 1 up to and including the time of the first jump and zero thereafter. Note

that cJ> is left-continuous. We have

t cJ>(s) dM(s) = {-At,^0 � t < St ,

}0 1 - ASt, t ;::::St

= ll[s 1 ,oo)(t) - A (t 1\ St). (11.4.6)

The notation t 1\ S1 in (11.4.6) denotes the minimum of t and S1. See Figure

11.4.1.

1

t

Fig. 11.4.1. Hrs1,ooJ (t)- A(t A SI).

We verify the martingale property for the process llrs.,oo)(t)-A(t 1\ St) by

direct computation. For 0 � s < t, we have

If S1 � s, then at time s we know the value of S1 and the conditional ex­
pectations above give us the random variables being estimated. In particular,
the right-hand side of (11.4.7) is 1 -ASt = llrs.,oo)(s)-A(s 1\ St), and the
martingale property is satisfied. On the other hand, if S 1 > s, then

IP'{ St � tiF(s)} = 1 -IP'{ St > tiSt > s} = 1 - e->.(t-s), (11.4.8)

where we have used the fact that S 1 is exponentially distributed and used the

memorylessness (11.2.3) of exponential random variables. In fact, the memo­

rylessness says that, conditioned on S 1 > s, the density of S 1 is

a a

• 8
  u
  IP'{St > uiSt > s} = -
  au

e->.(u-s) = Ae->.<u-s), u > s.

It follows that, when S1 > s,