Springer Finance

476 11 Introduction to Jump Processes

where

tP(t)dX(t) = tl>(t) dl(t) + tl>(t) dR(t) + tl>(t) dJ(t)

= tf>(t) dXc(t) + tf>(t) dJ(t),

tl>(t) dl(t) = tl>(t)F(t) dW(t), tl>(t) dR (t) = tl>(t)8(t) dt,

tl>(t) dXc(t) = tl>(t)F(t) dW(t) + tl>(t)8(t) dt.

Example 11.4.4. Let X(t) = M(t) = N(t)->..t, where N(t) is a Poisson process

with intensity>.. so that M(t) is the compensated Poisson process of Theorem

11.2.4. In the terminology of Definition 11.4.2, I(t) = 0 , xc(t) = R(t) = ->..t,

and J(t) = N(t). Let tl>(s) = LlN(s) (i.e., tl>(s) is 1 if N has a jump at time

s, and tf>( s) is zero otherwise). For s E (0, t] , tf>( s) is zero except for finitely

many values of s, and thus

However,

Therefore,

1 t tl>(s) dN(s) = L (.6.N(s))

2

= N(t).

(^0) O<s�
t
lot tl>(s) dM(s) =->.lot tl>(s) ds +lot tl>(s) dN(s) = N(t).
For Brownian motion W(t), we defined the stochastic integral
I(t) =lot F(s) dW(s)
(11.4.5)
0
in a way that caused I(t) to be a martingale. To define the stochastic integral,
we approximated the integrand F(s) by simple integrands Fn(s), wrote down
a formula for
ln(t) =lot Fn(s) dW(s),
and verified that, for each n, In(t) is a martingale. We defined I(t) as the limit
of In(t) as n -+ oo and, because it is the limit of martingales, I(t) also is a
martingale. The only conditions we needed on F(s) for this construction were
that it be adapted and that it satisfy the technical condition lE J; F^2 ( s) ds <
oo for every t > 0.
This construction makes sense for finance because we ultimately replace
F(s) by a position in an asset and replace W(s) by the price of that asset.
If the asset price is a martingale (i.e., it is pure volatility with no underlying