Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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460 8 Poisson Processes and Ruin Theory
Definition 8.2.1.1 Let F be a given filtration and λ a positive constant.
The process N is an F-Poisson process with intensity λ if N is an F-
adapted process, such that for all positive numbers (t, s), the r.v. N
t+s
−N
t
is
independent of F
t
and follows the Poisson law with parameter λs.
The random measure μ associated with a Poisson process is such that
μ(Λ) is almost surely finite for any bounded set Λ (the number of jumps in
any finite interval of time is almost surely finite), and E(μ(Λ)) = λ|Λ| where
|Λ| is the Lebesgue measure of the set Λ.
We now recall some properties of Poisson processes.
• The time T
n
when the n
th
-jump of N occurs is the sum of n independent
exponential r.v’s, hence it has a Gamma law with parameters (n, λ):
P(T
n
∈ dt)=
(λt)
n−1
(n − 1)!
λe
−λt
1
{t>0}
dt,
and its Laplace transform, for μ>−λ, is given by
E(e
−μT
n
)=
λ
λ + μ
n
.
• From the properties of the Poisson distribution, it follows that for every
t>0,
E(N
t
)=λt, Var (N
t
)=λt
and for every x>0,t≥ 0,u,α∈ R
E(x
N
t
)=e
λt(x−1)
; E(e
iuN
t
)=e
λt(e
iu
−1)
; E(e
αN
t
)=e
λt(e
α
−1)
. (8.2.1)
• Conditionally on (N
t
= n), the law of (T
1
,T
2
,...,T
n
) is a multinomial
distribution on [0,t].
• Let, for t fixed and i ≥ 1, T
(t)
i
:= T
N
t
+i
− t where T
N
t
+i
is the time of
the i-th jump which occurs after t. The sequence of times (T
(t)
i
,i ≥ 1) has
thesamelawas(T
i
,i≥ 1). This property is called the lack of memory of the
Poisson process.
Exercise 8.2.1.2 Let N be a Poisson process. Prove that N
t
t
−1
→ λ a.s.
when t goes to infinity.
Exercise 8.2.1.3 Let N be a Poisson process and T
n
its n-th jump time.
Prove that
P(T
n
≥ s|F
t
)=1
s≤T
n
≤t
+ 1
t<T
n
∞
s−t
λ(λu)
n−1−N
t
(n − 1 − N
t
)!
e
−λu
1
{u≥0}
du .
8.2 Standard Poisson Process 461
8.2.2 Martingale Properties
From the independence of the increments of the Poisson process, we derive
the following martingale properties:
Proposition 8.2.2.1 Let N be an F-Poisson process. For each α ∈ R,for
each bounded Borel function h, the following processes are F-martingales:
(i) M
t
:=N
t
− λt,
(ii) M
2
t
− λt =(N
t
− λt)
2
− λt, (8.2.2)
(iii) exp(αN
t
− λt(e
α
− 1)), (8.2.3)
(iv) exp
t
0
h(s)dN
s
− λ
t
0
(e
h(s)
− 1)ds
,
(v)
t
0
h(s)dM
s
,
(vi)
t
0
h(s)dM
s
2
− λ
t
0
h
2
(s)ds .
Proof: Let s<t. From the independence of the increments of the Poisson
process, we obtain:
(i) E(M
t
−M
s
|F
s
)=E(N
t
−N
s
) −λ(t −s) = 0, hence M is a martingale.
(ii) The martingale property of M and the independence of the increments
of the Poisson process imply
E(M
2
t
− M
2
s
|F
s
)=E[(M
t
− M
s
)
2
|F
s
]=E[(N
t
− N
s
− λ(t − s))
2
|F
s
]
= E[(N
t
− N
s
)
2
] − λ
2
(t − s)
2
= E[N
2
t−s
] − λ
2
(t − s)
2
=VarN
t−s
,
hence,
E(M
2
t
− M
2
s
|F
s
)=λ(t − s) ,
and the process (M
2
t
− λt, t ≥ 0) is a martingale.
(iii) From the form of the Laplace transform of N
t
given in (8.2.1)andthe
independence of the increments, E[exp[α(N
t
−N
s
)−λ(t −s)(e
α
−1)] |F
s
]=1,
hence the martingale property of the process in (iii).
Assertions (iv-v-vi) can be proved first for elementary functions h of the
form h =
i
a
i
1
]t
i
,t
i+1
]
and by then passing to the limit for general bounded
Borel functions h.
Exercise 8.2.2.2 Prove that, for any β>−1, any bounded Borel function
h, and any bounded Borel function ϕ valued in ] − 1, ∞[, the processes
462 8 Poisson Processes and Ruin Theory
exp[ln(1 + β)N
t
− λβt]=(1+β)
N
t
e
−λβt
,
exp
t
0
h(s)dN
s
+ λ
t
0
(1 − e
h(s)
)ds
=exp
t
0
h(s)dM
s
+ λ
t
0
(1 + h(s) − e
h(s)
)ds
,
exp
t
0
ln(1 + ϕ(s))dN
s
− λ
t
0
ϕ(s)ds
=exp
t
0
ln(1 + ϕ(s))dM
s
+ λ
t
0
(ln(1 + ϕ(s)) − ϕ(s))ds
,
are martingales.
Hint: These formulae are “avatars” of those of Proposition 8.2.2.1.
Exercise 8.2.2.3 Prove (without using the following Proposition!) that the
process (
t
0
N
s−
dM
s
,t ≥ 0) is a martingale, and that the process
t
0
N
s
dM
s
is not a martingale.
Definition 8.2.2.4 The martingale (M
t
= N
t
− λt, t ≥ 0) is called the
compensated process of N ,andλ the intensity of the process N.
Remarks 8.2.2.5 (a) Note that the process M is a discontinuous martingale
with bounded variation.
(b) We give an example of a martingale which is not square integrable.
Let X
t
=
t
0
1
√
s
dM
s
. The process X is a martingale, however, it is not square
integrable.
The previous Proposition 8.2.2.1 can be generalized to predictable integrands:
Proposition 8.2.2.6 Let N be an F-Poisson process and let H be an F-
predictable bounded process. Then the following processes are martingales:
(i) (HM)
t
=
t
0
H
s
dM
s
=
t
0
H
s
dN
s
− λ
t
0
H
s
ds
(ii) (HM)
2
t
− λ
t
0
H
2
s
ds
(iii) exp
t
0
H
s
dN
s
+ λ
t
0
(1 − e
H
s
)ds
=: E(HM)
t
=1+
t
0
E(HM)
s−
H
s
dM
s
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(8.2.4)
Proof: One establishes (8.2.4) for predictable processes (H
t
,t ≥ 0) of the
form H
t
= K
S
1
]S,T ]
(t)whereS and T are two stopping times and K
S
is
F
S
-measurable. In that case,
t
0
H
s
dM
s
= K
S
(M
T ∧t
− M
S∧t
)
8.2 Standard Poisson Process 463
and the martingale property follows. Then, one passes to the limit. The same
procedure can be applied to prove that the two processes (ii) and (iii) of (8.2.4)
are martingales.
We have used in (iii) the notation E(HM)
t
for the Dol´eans-Dade
exponential of the martingale
H
s
dM
s
.
Comments 8.2.2.7 (a) If H satisfies E(
t
0
|H
s
|ds) < ∞, the process in (i) is
still a martingale.
(b) The results of Exercise 8.2.2.3 are now quite clear: in general, the
martingale property (8.2.4) does not extend from predictable to adapted
processes H. Indeed, from the definition of the stochastic integral w.r.t. N,
and the fact that for every fixed s, N
s
− N
s−
=0, Pa.s.,
t
0
(N
s
− N
s−
)dM
s
=
t
0
(N
s
− N
s−
)dN
s
− λ
t
0
(N
s
− N
s−
)ds
= N
t
− λ
t
0
(N
s
− N
s−
)ds = N
t
.
Hence, the left-hand side, where one integrates the adapted (unpredictable)
process N
s
− N
s−
with respect to the martingale M, is not a martingale.
Equivalently, the process
t
0
N
s
dM
s
=
t
0
N
s−
dM
s
+ N
t
,
is not a martingale.
(c) Property (i) of Proposition 8.2.2.6 enables us to prove that the jump
times (T
i
,i ≥ 1) are not predictable. Indeed, if T
1
were a predictable
stopping time, then the process (1
{t<T
1
}
,t≥ 0) would be predictable, however
t
0
1
{s<T
1
}
dM
s
= −λ(t ∧T
1
) is not a martingale. More generally, assume that
T
i
is predictable. Then, (
t
0
1
[T
i
]
(s)dM
s
,t≥ 0) would be a martingale and
E
t
0
1
[T
i
]
(s)dN
s
= E(1
T
i
≤t
(N
T
i
− N
T
i
−
)) = P(T
i
≤ t)
would be equal to E
t
0
1
[T
i
]
(s)λds
= 0, which is absurd.
Remark 8.2.2.8 Note that (i) and (ii) of Proposition 8.2.2.1 imply that
the process (M
2
t
− N
t
; t ≥ 0) is a martingale. Hence, there exist (at least)
two increasing processes A such that (M
2
t
− A
t
,t ≥ 0) is a martingale. The
increasing process (λt, t ≥ 0) is the predictable quadratic variation of M
(denoted M), whereas the increasing process (N
t
,t ≥ 0) is the optional
quadratic variation of M (denoted [M]). For any μ ∈ [0, 1], the process
(μN
t
+(1− μ)λt; t ≥ 0) is increasing and the process
464 8 Poisson Processes and Ruin Theory
M
2
t
− (μN
t
+(1− μ)λt)=M
2
t
− λt − μ(N
t
− λt)
is a martingale. (See Section 9.2 for the definition of quadratic variation if
needed.)
8.2.3 Infinitesimal Generator
Proposition 8.2.3.1 The Poisson process is a process with independent
and stationary increments, and hence is a Markov process; its infinitesimal
generator L is given by
L(f)(x)=λ[f (x +1)− f (x)] ,
where f is a bounded Borel function.
Proof: The Markov property follows from
E(f(N
t
)|F
N
s
)=E(f(N
t
− N
s
+ N
s
)|F
N
s
)=F (t − s, N
s
)
where F (u, x)=E(f (x + N
u
)) and t ≥ s. We recall the definition of the
infinitesimal generator:
L(f)(x) = lim
t→0
1
t
(E(f(x + N
t
)) − f (x)) .
Hence, from E(f(x + N
t
)) =
∞
n=0
f(x + n)P(N
t
= n), we obtain
1
t
(E(f(x + N
t
)) − f (x)) = e
−λt
∞
n=0
f(n + x) − f (x)
t
(λt)
n
n!
.
From
e
−λt
n≥2
(λt)
n
n!
≤
λ
2
t
2
2
the limit of
1
t
(E(f(x + N
t
)) − f (x)) when t goes to 0 is equal to the limit of
e
−λt
λt
t
(f(x +1)− f (x)), that is to λ(f(x +1)− f(x)).
Therefore, for any bounded Borel function f, the process
C
f
t
= f(N
t
) − f (0) −
t
0
L(f)(N
s
)ds
is a martingale (see Proposition 1.1.14.2). Using that
f(N
t
) − f (0) =
t
0
(f(N
s−
+1)− f(N
s−
)) dN
s
, (8.2.5)
the martingale (C
f
t
,t≥ 0) can be written as a stochastic integral with respect
to the compensated martingale (M
t
= N
t
− λt, t ≥ 0) as
C
f
t
=
t
0
[f(N
s−
+1)− f(N
s−
)]dM
s
.
8.2 Standard Poisson Process 465
Comment 8.2.3.2 Processes with independent and stationary increments
are called L´evy processes, the reader may refer to Chapter 11 for a more
extended study.
Exercise 8.2.3.3 Extend formula (8.2.5) to functions f defined on R
+
× N
that are C
1
with respect to the first variable, and prove that if β is a constant
with β>−1andL
t
= exp(log(1 + β)N
t
− λβt), then dL
t
= L
t
−
βdM
t
.
More generally, let L
t
=(1+a)
N
t
e
−λat
for a ∈ R.ProvethatL satisfies
dL
t
= L
t
−
adM
t
, i.e.,
L
t
=1+
t
0
L
s
−
adM
s
=1+a
t
0
L
s
−
dN
s
− λa
t
0
L
s
−
ds .
Note that, for a<−1, L
t
takes values in R. The process L is the Dol´eans-Dade
exponential of the martingale aM.
Exercise 8.2.3.4 Let T>0befixedandletϕ :[0,T] → R be a bounded
Borel function and N a Poisson process. Prove that there exist a predictable
process h and a constant c such that
exp
T
0
ϕ(s)dN
s
= c +
T
0
h
s
dN
s
.
Hint: Set Z
t
=
t
0
ϕ(s)dN
s
. Then,
de
Z
t
=
e
Z
t
−
+ϕ(t)
− e
Z
t
−
dN
t
.
The reader may be interested to compare this simple result with the
predictable representation theorem in Subsection 8.3.5.
8.2.4 Change of Probability Measure: An Example
If N is a Poisson process with constant intensity λ, then, from Exercises 8.2.2.2
and 8.2.3.3,forβ>−1, the process L defined by
L
t
=(1+β)
N
t
e
−λβt
is a strictly positive martingale with expectation equal to 1. Let Q be the
probability defined via Q|
F
t
= L
t
P|
F
t
.From
E
Q
(x
N
t
)=E
P
(L
t
x
N
t
)=e
−λβt
E
P
([(1 + β)x]
N
t
) = exp((1 + β)λt(x − 1))
we deduce that the r.v. N
t
follows the Poisson law with parameter (1 + β)λt
under Q.Lett
1
< ···<t
i
<t
i+1
< ···<t
n
and let (x
i
,i≤ n) be a sequence
of positive real numbers. The equalities
466 8 Poisson Processes and Ruin Theory
E
Q
n
i=1
x
N
t
i+1
−N
t
i
i
= E
P
e
−λβt
n
i=1
((1 + β)x
i
)
N
t
i+1
−N
t
i
= e
−λβt
n
i=1
e
−λ(t
i+1
−t
i
)
e
λ(t
i+1
−t
i
)(1+β)x
i
=
n
i=1
e
(1+β)λ(t
i+1
−t
i
)(x
i
−1)
establish that, under Q, N
t
i+1
− N
t
i
law
= N
t
i+1
−t
i
is a Poisson r.v. with
parameter (1 + β)λ(t
i+1
− t
i
)andthatN has independent increments.
Therefore, the process N is a Q-Poisson process with intensity equal to
(1 + β)λ. Let us state this result as a proposition:
Proposition 8.2.4.1 Let Π
λ
be the probability on the canonical space which
makes the coordinate process a Poisson process with intensity λ. Then, the
following absolute continuity relationship holds:
Π
(1+β)λ
|
F
t
=
(1 + β)
N
t
e
−λβt
Π
λ
|
F
t
.
Comment 8.2.4.2 One should note the analogy between the change of
intensity of Poisson processes and the change of drift of a BM under a change
of probability. However, let us point out a major difference. If Q is equivalent
to P, we know that if B is a P-BM and
B is the martingale part of B
under Q,thenB
2
t
− t is a P-martingale and
B
2
t
− t is a Q-martingale (in
other words the brackets are the same, i.e., B =
B). If Q is equivalent
to P,andM
t
= N
t
− λt the compensated martingale associated with a
Poisson process, the process M
2
t
−λt is a P-martingale and the P-(predictable)
bracket of M is λt. We have proved above that the Q-(predictable) bracket of
M
t
= N
t
−(1+β)λt is (1+β)λt. Hence, the predictable bracket is no longer the
same under a change of probability. See Section 9.4 for a general Girsanov
theorem and Subsection 11.3.1 for the case of L´evy processes.
8.2.5 Hitting Times
Let x>0andT
x
=inf{t, N
t
≥ x}. Then, for n −1 <x≤ n, the hitting time
T
x
=inf{t, N
t
≥ n} =inf{t, N
t
= n} is equal to the time of the n
th
-jump of
N, and hence has a Gamma (n, λ)law.
Exercise 8.2.5.1 Let X
t
= N
t
+ ct. Compute P(inf
s≤t
X
s
≤ a). One should
distinguish the cases c>0andc<0.
8.3 Inhomogeneous Poisson Processes 467
8.3 Inhomogeneous Poisson Processes
8.3.1 Definition
Instead of considering a constant intensity λ as before, now (λ(t),t≥ 0) is an
R
+
-valued Borel function satisfying
t
0
λ(u)du < ∞, ∀t and
∞
0
λ(u)du = ∞.
An inhomogeneous Poisson process N with intensity λ is a counting
process with independent increments which satisfies, for t>s,
P(N
t
− N
s
= n)=e
−Λ(s,t)
(Λ(s, t))
n
n!
(8.3.1)
where Λ(s, t)=Λ(t) − Λ(s)=
t
s
λ(u)du,andΛ(t)=
t
0
λ(u)du.
If (T
n
,n≥ 1) is the sequence of successive jump times associated with N,
the law of T
n
is:
P(T
n
≤ t)=
1
(n − 1)!
t
0
exp(−Λ(s)) (Λ(s))
n−1
dΛ(s) .
It can easily be shown that an inhomogeneous Poisson process with determin-
istic intensity is an inhomogeneous Markov process. Moreover, since N
t
has a
Poisson law with parameter Λ(t), one has E(N
t
)=Λ(t), Var(N
t
)=Λ(t). For
any real numbers u and α, for any t ≥ 0,
E(e
iuN
t
) = exp((e
iu
− 1)Λ(t)),
E(e
αN
t
) = exp((e
α
− 1)Λ(t)) .
An inhomogeneous Poisson process can be constructed as a deterministic
time changed Poisson process, i.e., if
N is a Poisson process with constant
intensity equal to 1, then N
t
=
N
Λ(t)
is an inhomogeneous Poisson process
with intensity Λ.
We emphasize that we shall use the term Poisson process only when dealing
with the standard Poisson process, i.e., when Λ(t)=λt.
8.3.2 Martingale Properties
The martingale properties of a standard Poisson process can be extended to
an inhomogeneous Poisson process:
Proposition 8.3.2.1 Let N be an inhomogeneous Poisson process with
deterministic intensity λ and F
N
its natural filtration. The process
M
t
= N
t
−
t
0
λ(s)ds, t ≥ 0
468 8 Poisson Processes and Ruin Theory
is an F
N
-martingale. The increasing function Λ(t):=
t
0
λ(s)ds is called the
(deterministic) compensator of N.
Let φ be an F
N
-predictable process such that E(
t
0
|φ
s
|λ(s)ds) < ∞ for ev-
ery t. Then, the process (
t
0
φ
s
dM
s
,t≥ 0) is an F
N
-martingale. In particular,
E
t
0
φ
s
dN
s
= E
t
0
φ
s
λ(s)ds
. (8.3.2)
As in the constant intensity case, for any bounded F
N
-predictable process H,
the following processes are martingales:
(i)(HM)
t
=
t
0
H
s
dM
s
=
t
0
H
s
dN
s
−
t
0
λ(s)H
s
ds ,
(ii)(HM)
2
t
−
t
0
λ(s)H
2
s
ds ,
(iii)exp
t
0
H
s
dN
s
−
t
0
λ(s)(e
H
s
− 1)ds
.
8.3.3 Watanabe’s Characterization of Inhomogeneous
Poisson Processes
The study of inhomogeneous Poisson processes can be generalized to the case
where the intensity is not absolutely continuous with respect to the Lebesgue
measure. In this case, Λ is an increasing, right-continuous, deterministic
function with value zero at time zero, and it satisfies Λ(∞)=∞.IfN is a
counting process with independent increments and if (8.3.1) holds, the process
(N
t
−Λ(t),t≥ 0) is a martingale and for any bounded predictable process φ,
the equality E(
t
0
φ
s
dN
s
)=E(
t
0
φ
s
dΛ(s)) is satisfied for any t. This result
admits a converse.
Proposition 8.3.3.1 (Watanabe’s Characterization.) Let N be a count-
ing process and Λ an increasing, continuous function with value zero at time
zero. Let us assume that the process (M
t
:=N
t
−Λ(t),t≥ 0) is a martingale.
Then N is an inhomogeneous Poisson process with compensator Λ.Itisa
Poisson process if Λ(t)=λt.
Proof: Let s<tand θ>0.
e
θN
t
− e
θN
s
=
s<u≤t
e
θN
u
− e
θN
u−
=
s<u≤t
e
θN
u−
(e
θ
− 1)ΔN
u
=(e
θ
− 1)
]s,t]
e
θN
u−
dN
u
=(e
θ
− 1)
]s,t]
e
θN
u−
dM
u
+
]s,t]
e
θN
u
dΛ(u)
.
8.3 Inhomogeneous Poisson Processes 469
By relying on the fact that the first integral is a martingale,
E(e
θN
t
− e
θN
s
|F
s
)=(e
θ
− 1)E
]s,t]
e
θN
u
dΛ(u)|F
s
=(e
θ
− 1)
]s,t]
E (e
θN
u
|F
s
)dΛ(u) .
Let s be fixed and define φ(t)=E(e
θN
t
|F
s
). Then, for t>s,
φ(t)=φ(s)+(e
θ
− 1)
t
s
φ(u)dΛ(u) .
Solving this equation leads to
φ(t)=e
θN
s
exp
(e
θ
− 1)
t
s
dΛ(u)
.
This shows that the process N has independent increments and that, for s<t,
the r.v. N
t
− N
s
has a Poisson law with parameter Λ(t) − Λ(s).
8.3.4 Stochastic Calculus
In this section, M is the compensated martingale of an inhomogeneous Poisson
process N with deterministic intensity (λ(s),s≥ 0). From now on, we restrict
our attention to integrals of predictable processes, even if the stochastic
integral is defined in a more general setting.
Integration by Parts Formula
Let x and y be two predictable processes and define two processes X and Y
as
X
t
= x +
t
0
x
s
dN
s
,Y
t
= y +
t
0
y
s
dN
s
.
The jumps of X (resp. of Y ) occur at the same times as the jumps of N and
ΔX
s
= x
s
ΔN
s
,ΔY
s
= y
s
ΔN
s
. The processes X and Y are of finite variation
and are constant between two jumps. Then, it is easy to check that
X
t
Y
t
= xy +
s≤t
Δ(XY )
s
= xy +
s≤t
X
s−
ΔY
s
+
s≤t
Y
s−
ΔX
s
+
s≤t
ΔX
s
ΔY
s
We shall write this equality as
X
t
Y
t
= xy +
t
0
Y
s−
dX
s
+
t
0
X
s−
dY
s
+[X, Y ]
t