11.4 Jump Processes and Their Integrals 477trend), then the gain we make from investing in the asset should also be a
martingale. The stochastic integral is this gain.
In the context of processes that can jump, we still want the stochastic
integral with respect to a martingale to be a martingale. However, we see
in Example 11.4.4 that this is not always the case. The integrator M(t) in
that example is a martingale (see Theorem 11.2.4), but the integral N(t) in
(11.4.5) is not because it goes up but cannot go down.
An agent who invests in the compensated Poisson process M ( t) by choos
ing his position according to the formula 4>(s) = <1N(s) has created an arbi
trage. To do this, he is holding a zero position at all times except the jump
times of N(s), which are also the jump times of M(s), at which times he holds
a position one. Because the jumps in M(s) are always up and our investor
holds a long position at exactly the jump times, he will reap the up side gain
from all these jumps and have no possibility of lo ss.
In reality, the portfolio process 4>(s) = <1N(s) cannot be implemented
because investors must take positions before jumps occur. No one without
insider information can arrange consistently to take a position exactly at the
jump times. However, 4>( s) depends only on the path of the underlying process
M up to and including at time s and does not depend on the future of the
path. That is the definition of adapted we used when constructing stochastic
integrals with resp ect to Brownian motion. Here we see that it is not enough
to require the integrand to be adapted. A mathematically convenient way
of formulating the extra condition is to insist that our integrands be left
continuous. That rules out 4'( s) = <1N ( s). In the time interval between jumps,
this process is zero, and a left-continuous process that is zero between jumps
must also be zero at the jump times.
We give the following theorem without proof.
Theorem 11.4.5. Assume that the jump process X(s) of {1 1.4.1}-{11.4.3} is
a martingale, the integrand 4>(s) is left-continuous and adapted, and
lE 1 t F^2 (s)4>^2 (s) ds < 00 for all t 2:: 0.Then the stochastic integral J� 4>(s) dX(s) is also a martingale.
The mathematical literature on integration with respect to jump processes
gives a slightly more general version of Theorem 11.4.5 in which the integrand
is required only to be predictable. Roughly speaking, such processes are those
that can be gotten as the limit of left-continuous processes. We shall not need
this more general concept.
Note that although we require the integrand 4>( s) to be left-continuous in
Theorem 11.4.5, the integrator X(t) is always taken to be right-continuous,
and so the integral J� 4>( s) dX ( s) will be right-continuous in the upper limit of
integration t. The integral jumps whenever X jumps and 4> is simultaneously
not zero. The value of the integral at time t includes the jump at time t if
there is a jump; see (ll.4.4).