We will develop this point of view which

We will develop this point of view (which unfortunately can’t be made rigorous, but provides the proper
intuition). Hence, reviewing from above, with Brownian motion we would have:
dz ( t ) = z ( t + dt ) − z ( t ) ∼ N (0 ,dt ) . (1.17)
Note that since Brownian motion has continuous sample paths, z ( t − ) = z ( t ). However, for a Poisson process,
we should think of differentials as
dπ ( t ; λ ) = π ( t + dt ; λ ) − π ( t − ; λ ) ∼ Poisson ( λdt ) (1.18)
in order to make sure that we capture jumps.
Don’t forget that we also have the binary model approximations of Figures 1.3 and 1.4. Those binary
models provide the proper intuition, and in both cases, sums of them will limit as Brownian motion or a
Poisson process. For the differential, we simply replace ∆ t by dt in (1.8) and (1.10), giving
dz ≈
braceleftbigg √
dt w.p. 1 / 2
−
√
dt w.p. 1 / 2
(1.19)
and
dπ ( λ ) ≈
braceleftbigg
1 w.p. λdt
0 w.p. 1 − λdt
. (1.20)
1.2.3 Compound Poisson Process
When Poisson processes jump, they jump up by 1. We can generalize this and allow them to jump randomly.
Let π ( t ; λ ) be a Poisson process with jump times t
1
,t
2
,... . Construct a new process π
Y
( t ; λ ), by assigning
jump Y
1
at time t
1
, Y
2
at time t
2
, etc. where Y
1
, Y
2
, ... are iid random variables. This process can be
written as
π
Y
( t ; λ ) =
π ( t ; λ )
summationdisplay
i =0
Y
i
. (1.21)
That is, at time t it is the sum of π ( t ; λ ) iid copies of Y , where π ( t ; λ ) is a standard Poisson process. Processes
of this form can also conveniently be written as integrals,
π
Y
( t,λ ) =
π ( t ; λ ) summationdisplay
i =0
Y
i
=
integraldisplay
t
0
Y
s
dπ ( s ; λ ) . (1.22)
For this reason, we represent the differential form of a compound Poisson process by Ydπ ( t,λ ). That is, we
may write
dπ
Y
= Ydπ. (1.23)
Following along the lines of the binary approximation to a Poisson process as in Figure 1.4, an infinitesimal
model of a compound Poisson process can be thought of as
Ydπ ( λ ) ≈
braceleftbigg
Y
i
w.p. λdt
0 w.p. 1 − λdt
(1.24)
and a heuristic infinitesimal picture of this is given in Figure 1.6.
1.2.4 Ito Stochastic differential equations
Stochastic integrals can be defined in different ways. The most useful for us is the Ito stochastic integral. At
this point, I will not delve into the depths of the stochastic integral (because often people are never able to
return!), but merely provide the intuition that you should take away when considering stochastic differential
equations.

8 CHAPTER 1. BASIC BUILDING BLOCKS AND STOCHASTIC DIFFERENTIAL EQUATION MODELS
Figure 1.6: Infinitesimal model of a compound Poisson process.
A stochastic differential equation will be written as:
dx ( t ) = a ( x ( t ) ,t ) dt + b ( x ( t ) ,t ) dz ( t ) (1.25)
where in this case, it is being driven by Brownian motion z ( t ). (At this stage, I will ignore the technical
conditions that must be placed on a and b in order to make such an equation well defined.) We will interpret
this equation as follows:
x ( t + dt ) − x ( t ) = a ( x ( t ) ,t ) dt + b ( x ( t ) ,t )( z ( t + dt ) − z ( t )) . (1.26)
Since z ( t ) has independent increments, and a ( x ( t ) ,t ) and b ( x ( t ) ,t ) are evaluated at time t , they are inde-
pendent of dz ( t ) = z ( t + dt ) − z ( t ). This is important! It allows us to do the following simple calculations