Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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500 8 Poisson Processes and Ruin Theory
V
t
= e
rt
E(σW)
t
(x + c
t
0
e
−rs
[E(σW)
s
]
−1
ds). The optional stopping theorem
applied to the bounded martingale
M
t
= E(1
m≥0
|F
t
)
and the strong Markov property lead to
Ψ(x)=P(m ≥ 0) = M
0
= E(M
T
)=E(Ψ(S
T
)) .
Hence,
Ψ(x)=E(Ψ(S
τ
)1
τ<τ
∗
)+E(Ψ(S
τ
∗
)1
τ
∗
<τ
)
= E(Ψ (V
τ
− Y
1
)1
τ<τ
∗
)+E(Ψ(V
τ
∗
(1 + Y
∗
1
))1
τ
∗
<τ
)
=
∞
0
dtλe
−λt
E(Ψ(V
t
− Y
1
)) P(t<τ
∗
)
+
∞
0
dtλ
∗
e
−λ
∗
t
E(Ψ(V
t
(1 + Y
∗
1
))) P(t<τ)
=
∞
0
e
−(λ+λ
∗
)t
(λE[Ψ(V
t
− Y
1
)] + λ
∗
E[Ψ(V
t
(1 + Y
∗
1
))]) dt .
Employing the change of variable t =4σ
−2
s,
Ψ(x)=
4
σ
2
∞
0
e
−4σ
−2
(λ+λ
∗
)s
(λΥ (s)+λ
∗
Υ
∗
(s)) ds ,
where
Υ (s)=E[Ψ (X
s
− Y
1
)],Υ
∗
(s)=E[Ψ(Z
s
(1 + Y
∗
1
))]
and
X
s
= e
2(as+B
s
)
x +
4c
σ
2
s
0
e
−2(at+B
t
)
dt
,B
s
=
σ
2
W
4σ
−2
s
,
where a =
2r
σ
2
− 1. Hence, using the symmetry of BM,
Υ (s)=E
Ψ
e
−2(B
s
−as)
x +
4c
σ
2
s
0
e
2(B
t
−at)
dt
− Y
1
.
Therefore
4
σ
2
∞
0
e
−4σ
−2
(λ+λ
∗
)s
λΥ (s)ds
=
λ
λ + λ
∗
E
+
Ψ
e
−2(B
Θ
−aΘ)
x +
4c
σ
2
Θ
0
e
2(B
t
−at)
dt
− Y
1
,
where Θ is an exponential random variable, independent of B, with parameter
4(λ + λ
∗
)σ
−2
. The law of the pair
8.8 Marked Point Processes 501
e
−2(B
Θ
−aΘ)
,
Θ
0
e
2(B
t
−at)
dt
was presented in Corollary 6.6.2.2. It follows that
4
σ
2
∞
0
e
−4σ
−2
(λ+λ
∗
)s
λΥ (s)ds =
∞
0
∞
0
γ
2y
2+α+a
p
α
u
(1,y)D(y,u)dydu .
The study of the second term can be carried out by the same method.
Comment 8.7.3.2 See the main papers of Kl¨uppelberg [526], Paulsen [700],
Paulsen and Gjessing [701], Yuen et al [871], the books of Asmussen [23],
Embrechts et al. [322], Mikosch [650] and Mel’nikov [639] and the thesis of
Loisel [602]. Many applications to ruin theory can be found in Gerber and his
co-authors, e.g., in [387].
8.8 Marked Point Processes
We now generalize compound Poisson processes, introducing briefly a class of
processes which are no longer L´evy processes: we introduce a spatial dimension
for the size of jumps which are no longer i.i.d. random variables; moreover,
the time intervals between two consecutive jumps are no longer independent.
Let (E,E) be a measurable space and (Ω,F, P) a probability space.
8.8.1 Random Measure
Definition 8.8.1.1 A random measure ϑ on the space R
+
×E is a family
of positive measures (ϑ(ω; dt, dx); ω ∈ Ω) defined on R
+
× E such that, for
[0,t] ×A ∈B⊗E, the map ω → ϑ(ω;[0,t],A) is F-measurable, and satisfying
ϑ(ω; {0}×E)=0.
8.8.2 Definition
Let (Z
n
) be a sequence of random variables taking values in the measurable
space (E, E), and (T
n
) an increasing sequence of positive random variables,
with - to avoid explosion - lim
n
T
n
=+∞. We define the marked point
process N = {(T
n
,Z
n
)} by:foreachBorelsetA ⊂ E,
N
t
(A)=
n
1
{T
n
≤t}
1
{Z
n
∈A}
.
We associate with N a random measure μ by
μ(·;[0,t],A)=N
t
(A) .
502 8 Poisson Processes and Ruin Theory
The natural filtration of N is
F
N
t
= σ(N
s
(A),s≤ t, A ∈E) .
Let H be a map
(t, ω, z) ∈ (R
+
,Ω,E) → H(t, ω, z) ∈ R .
The map H is predictable if it is P⊗Emeasurable. The random counting
measure μ(ω; ds, dz) acts on the set of predictable processes H as
(Hμ)
t
=
]0,t]
E
H(s, z)μ(ds, dz)=
n
H(T
n
,Z
n
)1
{T
n
≤t}
=
N
t
(E)
n=1
H(T
n
,Z
n
) ,
wherewehavedroppedω in the notation.
Definition 8.8.2.1 The compensator of μ is the (up to a null set) unique
random measure ν such that, for every predictable process H,
(i) the process Hν is predictable,
(ii) if moreover, the process |H|μ is increasing and locally integrable, the
process (Hμ − Hν) is a local martingale.
The existence of a compensator is established in Br´emaud and Jacod [125],
Jacod and Shiryaev [471] and Kallenberg [505].
We now assume that E = R
d
. The compensator admits an explicit repre-
sentation: let G
n
(dt, dz) be a regular version of the conditional distribution
of (T
n+1
,Z
n+1
) with respect to F
N
T
n
= σ{((T
1
,Z
1
),...(T
n
,Z
n
)}. Then,
ν(dt, dz)=
n
1
{T
n
<t≤T
n+1
}
G
n
(dt, dz)
G
n
([t, ∞[×R
d
)
. (8.8.1)
A proof can be found in Prigent [725], Chapter 1 Proposition 1.1.30.
Comment 8.8.2.2 See Br´emaud and Jacod [125], Br´emaud [124], Prigent
[725], Jacod [467] and Jacod and Shiryaev [471] for more details on marked
point processes.
Warning 8.8.2.3 The notation in various papers in the literature can be
very different from the above: authors may use N or N for various quantities.
8.8 Marked Point Processes 503
8.8.3 An Integration Formula
Let dX
t
= β
t
dt +
E
γ(t, z)μ(dt, dz), where β and γ are predictable and let F
be a C
1,1
function. Then
dF (t, X
t
)=∂
t
Fdt+ β
t
∂
x
Fdt
+
E
(F (t, X
t
−
+ γ(t, z)) − F (t, X
t
−
)) μ(dt, dz)
or,inanintegratedform
F (t, X
t
)=F (0,X
0
)+
t
0
∂
t
F (s, X
s
)ds +
t
0
β
s
∂
x
F (s, X
s
)ds
+
N
t
(E)
n=1
[F (T
n
,X
T
n
) − F (T
n
,X
T
−
n
)] .
8.8.4 Marked Point Processes with Intensity and Associated
Martingales
In what follows, we assume that, for every A ∈E, the process N
t
(A)admits
the F-predictable intensity λ
t
(A), i.e., there exists a predictable process
(λ
t
(A),t≥ 0) such that
N
t
(A) −
t
0
λ
s
(A)ds
is a martingale. (The most common form of intensity is λ
t
(A)=α
t
m
t
(A)
where α
t
is a positive predictable process and m
t
a deterministic probability
measure on (E,E). In that case, ν(dt, dz)=α
t
m
t
(A)dt. We shall say that the
marked point process admits (α
t
,m
t
(dz)) as P-local characteristics.)
If X
t
:=
N
t
(E)
n=1
H(T
n
,Z
n
)whereH is an F-predictable process that
satisfies
E
]0,t]
E
|H(s, z)|λ
s
(dz)ds
< ∞
the process
X
t
−
t
0
E
H(s, z)λ
s
(dz)ds =
]0,t]
E
H(s, z)[μ(ds, dz) − λ
s
(dz)ds]
is a martingale and in particular
E
]0,t]
E
H(s, z)μ(ds, dz)
= E
]0,t]
E
H(s, z)λ
s
(dz)ds
.
504 8 Poisson Processes and Ruin Theory
8.8.5 Girsanov’s Theorem
Let μ be the random measure of a marked point process with intensity of
the form λ
t
(A)=α
t
m
t
(A)wherem
t
is, as above, a deterministic probability
measure on (E,E). Let (ψ
t
,h(t, z)) be two predictable positive processes such
that
t
0
ψ
s
α
s
ds < ∞,
E
h(t, z)m
t
(dz)=1.
Let L be the local martingale solution of
dL
t
= L
t
−
E
(ψ
t
h
t
(z) − 1)(μ(dt, dz) − α
t
m
t
(dz)dt) .
If E(L
t
) = 1, setting Q|
F
t
= L
t
P|
F
t
, the marked point process has the Q-local
characteristics (ψ
t
α
t
,h(t, z)m
t
(dz)).
Exercise 8.8.5.1 Prove Proposition 8.6.6.1 using the above result.
8.8.6 Predictable Representation Theorem
Let (Ω,F, F, P) be a probability space where F is the filtration generated by
the marked point process N. Then, any (P, F)-square integrable martingale
M admits the representation
M
t
= M
0
+
t
0
E
H(s, x)(μ(ds, dx) − λ
s
(dx)ds)
where H is a predictable process such that
E
t
0
E
|H(s, x)|
2
λ
s
(dx)ds
< ∞.
See Br´emaud [124] for a proof. More generally
Proposition 8.8.6.1 Let W be a Brownian motion M, N a marked point
process and F
t
= σ(W
s
, F
N
s
; s ≤ t) completed.
Let μ(ds, dz)=μ(ds, dx) − λ
s
(dx)ds. Then, any (P, F)-local martingale
has the representation
M
t
= M
0
+
t
0
ϕ
s
dW
s
+
t
0
E
H(s, z)μ(ds, dz) (8.8.2)
where ϕ is a predictable process such that
t
0
ϕ
2
s
ds < ∞ and H is a predictable
process such that
t
0
E
|H(s, x)|λ
s
(dx)ds < ∞.IfM is a square integrable
martingale, each term on the right-hand side of the representation (8.8.2) is
square integrable, and
8.9 Poisson Point Processes 505
E
t
0
ϕ
s
dW
s
2
= E
t
0
ϕ
2
s
ds
E
t
0
E
H(s, z)μ(ds, dz)
2
= E
t
0
E
H
2
(s, z)λ
s
(dz)ds
Proof: We refer to Kunita and Watanabe [550], Kunita [549], and to Chapter
III in the book of Jacod and Shiryaev [471].
Comment 8.8.6.2 Bj¨ork et al. [103] and Prigent [725, 724] gave the first
applications to finance of Marked point processes, which are now studied by
many authors, especially in a BSDE framework.
Exercise 8.8.6.3 Check that the process
S
t
=exp
t
0
β
s
−
1
2
σ
2
s
ds +
t
0
σ
s
dW
s
n,T
n
≤t
(1 + γ(T
n
,Z
n
))
is a solution of
dS
t
= S
t
−
β
t
dt + σ
t
dW
t
+
E
γ(t, y)μ(dt, dy)
,
where μ is the random measure associated with the marked point process
N = {(T
n
,Z
n
)}.
8.9 Poisson Point Processes
We end this chapter with a brief section on Poisson point processes, which
are of major importance in the study of Brownian excursions.
8.9.1 Poisson Measures
Let (E,E) be a measurable space. A random measure μ on (E,E) is a Poisson
measure with intensity ν,whereν is a σ-finite measure on (E,E), if
(i) for every set B ∈Ewith ν(B) < ∞, μ(B) follows a Poisson distribution
with parameter ν(B), and
(ii) for disjoint sets B
i
,i≤ n, the variables μ(B
i
),i≤ n are independent.
Example 8.9.1.1 Let π be a probability measure, (Y
k
,k ∈ N) i.i.d. random
variables with law π and N a Poisson variable with mean m, independent of
the Y
k
’s. The random measure
N
k=1
δ
Y
k
is a Poisson measure with intensity
ν = mπ. Here, δ
y
is the Dirac measure at point y.
506 8 Poisson Processes and Ruin Theory
8.9.2 Point Processes
Let (E, E) be a measurable space and δ an additional point. We introduce
E
δ
= E ∪δ, E
δ
= σ(E, {δ}).
Definition 8.9.2.1 Let e be a stochastic process defined on a probability space
(Ω,F, P), taking values in (E
δ
, E
δ
). The process e is a point process if:
(i) the map (t, ω) → e
t
(ω) is B(]0, ∞[) ⊗F-measurable,
(ii) the set D
ω
= {t : e
t
(ω) = δ} is a.s. countable.
For every measurable set B of ]0, ∞[×E,weset
N
B
(ω):=
s≥0
1
B
(s, e
s
(ω)) .
In particular, if B =]0,t] × Γ ,wewrite
N
Γ
t
= N
B
= Card{s ≤ t : e(s) ∈ Γ }.
Let the space (Ω, P) be endowed with a filtration F. A point process is
F-adapted if, for any Γ ∈E, the process N
Γ
is F-adapted. For any Γ ∈E
δ
,
we define a point process e
Γ
by
e
Γ
t
(ω)=e
t
(ω)ife
t
(ω) ∈ Γ
e
Γ
t
(ω)=δ otherwise
Definition 8.9.2.2 A point process e is discrete if N
E
t
< ∞ a.s. for every t.
It is said to be σ-discrete if there is a sequence E
n
of sets with E = ∪E
n
such
that each e
E
n
is discrete.
8.9.3 Poisson Point Processes
Definition 8.9.3.1 An F-Poisson point process e is a σ-discrete point
process such that:
(i) the process e is F-adapted,
(ii) for any s and t and any Γ ∈E, N
Γ
s+t
− N
Γ
t
is independent from
F
t
and distributed as N
Γ
s
.
In particular, for any disjoint family (Γ
i
,i =1,...,d), the d-dimensional
process (N
Γ
i
t
,i =1, ··· ,d) is a Poisson process. Moreover, if N
Γ
is finite
almost surely, then E(N
Γ
t
) < ∞ and the quantity
1
t
E(N
Γ
t
) does not depend
on t.
Definition 8.9.3.2 The σ-finite measure on E defined by
n(Γ )=
1
t
E(N
Γ
t
)
is called the characteristic measure of e.
8.9 Poisson Point Processes 507
If n(Γ ) < ∞, the process N
Γ
t
− tn(Γ )isanF-martingale.
Proposition 8.9.3.3 (Compensation Formula.) Let H be a predictable
positive process (i.e., measurable with respect to P⊗E
δ
) vanishing at δ.Then
E
+
s≥0
H(s, ω, e
s
(ω))
,
= E
∞
0
ds
E
H(s, ω, u)n(du)
.
If, for any t, E
t
0
ds
E
H(s, ω, u)n(du)
< ∞, the compensated process
s≤t
H(s, ω, e
s
(ω)) −
t
0
ds
E
H(s, ω, u)n(du)
is a martingale.
Proof: By the Monotone Class Theorem, it is enough to prove this formula
for H(s, ω, u)=K(s, ω)1
Γ
(u). In that case, N
Γ
t
− tn(Γ )isanF-martingale.
Proposition 8.9.3.4 (Exponential Formula.) If f is a B⊗E-measurable
function such that
t
0
ds
E
|f(s, u)|n(du) < ∞ for every t, then,
E
⎡
⎣
exp
⎛
⎝
i
0<s≤t
f(s, e
s
)
⎞
⎠
⎤
⎦
=exp
t
0
ds
E
(e
if(s,u)
− 1)n(du)
.
Moreover, if f ≥ 0,
E
⎡
⎣
exp
⎛
⎝
−
0<s≤t
f(s, e
s
)
⎞
⎠
⎤
⎦
=exp
−
t
0
ds
E
(1 − e
−f(s,u)
)n(du)
.
8.9.4 The Itˆo Measure of Brownian Excursions
Let (B
t
,t ≥ 0) be a Brownian motion and (τ
s
) be the inverse of the local
time (L
t
) at level 0. The set ∪
s≥0
]τ
s
−
(ω),τ
s
(ω)[ is (almost surely) equal to
the complement of the zeros set {u : B
u
(ω)=0}. The excursion process
(e
s
,s≥ 0) is defined by
e
s
(ω)(t)=1
{t ≤ τ
s
(ω) − τ
s
−
(ω)}
B
(τ
s
−
(ω)
+ t)(ω)
,t≥ 0.
This is a path-valued process e : R
+
× Ω → Ω
∗
,where
Ω
∗
= {ε : R
+
→ R : ∃V (ε) < ∞, with ε(V (ε)+t)=0, ∀t ≥ 0
ε(u) =0, ∀0 <u<V(ε),ε(0) = 0,εis continuous }.
Hence, V (ε) is the lifetime of ε.
508 8 Poisson Processes and Ruin Theory
The starting point of Itˆo’s excursion theory is that the excursion process is
a Poisson Point Process; its characteristic measure n, called Itˆo’s measure,
evaluated on the set Γ , i.e., n(Γ ), is the intensity of the Poisson process
N
Γ
t
:=
s≤t
1
e
s
∈Γ
.
The quantity n(Γ )isthepositiverealγ such that N
Γ
t
− tγ is an (F
τ
t
)-
martingale.
Here, are some very useful descriptions of n:
• Itˆo: Conditionally on V = v, the process
(|
u
|,u≤ v)
is a BES
3
bridge of length v. The law of the lifetime V under n is
n
V
(dv)=
dv
√
2πv
3
.
Thanks to the symmetry of Brownian motion, a full description of n is
n(d )=
∞
0
n
V
(dv)
1
2
(Π
v
+
+ Π
v
−
)(d )
where Π
v
+
(resp. Π
v
−
) is the law of the standard Bessel Bridge (resp. the
law of its negative) with dimension 3 and length v.
• Williams:LetM ( )=sup
u≤v
|
u
|. Then, conditionally on M = m,the
two processes (
u
,u ≤ T
m
)and(
V −u
,u ≤ V − T
m
) are two independent
BES
3
processes considered up to their first hitting time of m,and
n
M
(dm)=n(M( ) ∈ dm)=
dm
m
2
.
We leave to the reader the task of writing a disintegration formula for n
with respect to n
M
.
Comment 8.9.4.1 See Jeanblanc et al. [483] for applications to decomposi-
tion of Brownian paths and Feynman-Kac formula. At the moment, there are
very few applications to finance of excursion theory. One can cite Gauthier
[376] for a study of Parisian options, and Chesney et al. [173] for Asian-
Parisian options.
9
General Processes: Mathematical Facts
In this chapter, we consider studies involving c`adl`ag processes. We pay par-
ticular attention to semi-martingales with respect to a given filtration; these
processes will always be taken with c`adl`ag paths. We present the definition
of stochastic integrals with respect to a square integrable martingale, and we
extend the definition to stochastic integrals with respect to a local martingale.
Then, we introduce semi-martingales, quadratic covariation processes for
semi-martingales and some general versions of Itˆo’s formula and Girsanov’s
theorem. We give necessary and sufficient conditions for the existence of an
equivalent martingale measure. We end the chapter with a brief survey of
valuation in an incomplete market.
The reader may refer to the lectures of Itˆo[464], Meyer [647]andthe
books of Kallenberg [505] and Protter [727]. The general theory of stochastic
processes is presented in Dellacherie [240], Dellacherie and Meyer [242, 244],
Dellacherie, Maisonneuve and Meyer [241] and He et al. [427]. More advanced
results are given in Jacod and Shiryaev [471]andBichteler[88]. The survey
papers of Kunita [549] and Runggaldier [750] are excellent introductions to
this subject with finance in view. Applications to finance can be found in
Prigent [725] and Shiryaev [791].
We assume that a filtered probability space (Ω, F, F, P) is given, with
some filtration F, satisfying the usual conditions. We recall that the notation
X
mart
= Y means that X − Y is a local martingale.
9.1 Some Basic Facts about c`adl`ag Processes
9.1.1 An Illustrative Lemma
To justify why c`adl`ag processes are considered throughout the development
of semi-martingales, we note the very elementary lemma:
M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial
Markets, Springer Finance, DOI 10.1007/978-1-84628-737-4
9,
c
Springer-Verlag London Limited 2009
509