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Stochastic Processes in Credit Risk Modelling

Discussion Paper 05/05

Roberto Casarin CEREMADE, Dept. of Mathematics

University Paris IX (Dauphine) and

Dipartimento di Scienze Economiche Università di Brescia

Stochastic Processes in Credit Risk Modelling

Roberto Casarin‡

CEREMADE, Dept. of MathematicsUniversity Paris IX (Dauphine)

andDept. of EconomicsUniversity of Brescia

March 10, 2005

Abstract

In credit risk modelling, jump processes are widely used to de-scribe both default and rating migration events. This work ismainly a review of some basic definitions and properties of thejump processes intended for a preliminary step before more ad-vanced lectures on credit risk modelling. We focus on the Poissonprocess and some generalisations, like the compounded and thedouble stochastic Poisson processes, which are widely used fordescribing the time-inhomogeneous dynamic either of the defaultprocess or of the credit rating transition. As such, much of thematerial is not new, but focused and organized from a credit riskperspective. Moreover it contains detailed proofs of some funda-mental results. Other original contributions come from examplesand simulated studies, which help the reader to better understandthe features of the described processes.

Keywords: Markov process, Jump process, Poisson process, Coxprocess, Doubly stochastic Poisson process.

MSC: 60G55, 60J75, 91B28.

‡Address: CEREMADE, University Paris Dauphine, 16 place du Marechal de Lattrede Tassigny, 75775 Paris Cedex 16. E-mail: casarin@ceremade.dauphine.fr. This workcomes from a revised version of my M.Sc. thesis in Applied Mathematics (DEA MASE)at University Paris Dauphine and ENSAE. I would like to thank Christian Gourieroux,Francesco Menoncin and Diana Barro for all the useful suggestions. Only the authorremains responsible for any omission and mistake.

1

1 Introduction

In this work we survey definitions and basic properties of some stochasticprocesses involved in credit risk modelling. We mainly focus on point pro-cesses which are widely used in risk theory and which play now an importantrole also in the actuarial and reduced form framework to credit risk mod-elling. These processes are used in many disciplines, like physics, biology,image analysis, operations research (queuing theory), hydrology (see Cal-enda, Napolitano (1999)), volcanology (see Jaquet, Carniel (2001), Jaquet etal. (2000)), etc., in order to describe the random occurrence of discrete valueevents over both time and space. In credit risk models point processes aresuitable to capture the discontinuity induced by a default or credit migrationevent and play an important role in both defaultable claim pricing and creditportfolio loss evaluation.

Point stochastic processes have already been applied to many other finan-cial problems like: actuarial or insurance mathematics (see Grandell (1991)),Extreme Value Theory (see Embrechts, Kluepperlberg and Mikosch (1997)),stochastic volatility and term structure models. Finally note that anotherfundamental process involved in credit risk models is the traditional diffu-sion process which is not treated in detail in this work. We refer the interestedreader to the Karatzas and Shreve (1988), Revuz and Yor (1999).

In more detail point processes have been widely used in the recent creditrisk literature on both structural and reduced form approach, but in thelatter they play a crucial role. For an application within the structural ap-proach, see Zhou (1997). For their use within the reduced form approachsee Lando (1998) and Jarrow, Lando and Turnbull (1997). In the credit risktheory the occurrence of the default event is of interest because it is a sourceof risk that affects the current value of the defaultable securities. In the re-duced form model the default of a firm is described through a point process.Furthermore the intensity of the process can be made dependent on a set ofexogenous state variables. This particular reduced form approach to creditrisk evaluation, is called intensity based.

In early papers on the intensity based approach (see Jarrow, Turnbull(1995)) the default event is modelled as the first jump of a Poisson process,which is assumed to be independent of the other financial variables in themodel. In more recent contributions, some generalisations of the Poisson pro-cess have been introduced. In particular Lando (1998) proposes to use a Coxprocess(or doubly stochastic Poisson process, see Grandell (1976), (1991)),which allows for a time-inhomogeneous dynamic of the stochastic intensitydriving the default. Moreover the intensity is made dependent on a set ofvariables, like macroeconomic factors or financial market variables.

2

A possible criticism to this framework, when we consider a single creditposition, is that in the underlying probability model more than one defaultmay occur. In practice, within the reduced form models, only the first jumpof the default process is of interest. Another way to overcome this limit,in discrete time, is to apply duration models (see Gourieroux and Monfort(2002)). When we consider a portfolio of homogeneous credit positions thejump process represents the multiple defaults event occurring in the portfolioand represent a powerful modelling tool.

The work is structured as follows. In Section 2, following Bouleau (2000),we introduce some basic definitions and properties of a Markov process. InSection 3 some Markov point processes, like homogeneous and compoundedPoisson processes and their associated martingales, are defined. In particularthe Poisson counting model is defined also through a differential argument,which is particularly useful in order to understand the relation between thecounting process, the inter-arrival distribution and the waiting time distri-bution. In Section 4 we treat in more detail the Poisson stochastic process.It represents a simple, but meaningful probability model, that can also beeasily extended towards more ”realistic” stochastic models. Section 5 givesa more general definition of point process and Section 6 concludes.

2 Markov jump process

In this section we show that Markov jump processes are Markov processeswhere the time length of the permanence in each state is described by arandom variable. In credit risk modelling, the p.d.f. of this r.v. is strictlyrelated to the probability of default. Let us introduce a probability space(Ω,F , P) and a state space (E, E). If E is assumed to be countable then acontinuous time Markov process defines as follows.

Definition 2.1. (Markov Process) - A stochastic process (Xt)t≥0 with valuesin (E, E), is a Markov process if ∀ t, u ≥ 0 and ∀A ∈ E, P Xt+u ∈ A | Xs, s ≤ t,= P Xt+u ∈ A | Xt, P-a.s..

If E is a finite countable set then Ps,t is a stochastic matrix defined as follows1:

Ps,t =

P (Xt = y | Xs = x) if P (Xs = x) 6= 0Ix=y if P (Xs = x) = 0

(1)

1We denote with Ix=y the indicator function, which is defined as: Ix=y = 1 if x =y , 0 otherwise

3

which satisfies the Chapman-Kolmogorov equation:

P Xu = z | Xs = x = (2)

=∑

y∈E

P Xt = y | Xs = xP Xu = z | Xt = y (3)

=∑

y∈E

Ps,t (x, y) Pt,u (y, z) ⇔ Ps,u (x, z) = Ps,tPt,u (4)

Now, let us assume the state space E is not countable and let Eb be the setof all bounded E-measurable functions with values in R, then the definitionof Markov process modifies as follow.

Definition 2.2. (Markov Process) - Given a state space (E, E) with its Borelσ-algebra, and a family of transition probabilities (Ps,t)0<s<t on (E, E) suchthat ∀s < t < u, Ps,u = Ps,tPt,u, and a stochastic process (Xt)t≥0

defined on

(E, E) with values in (Ω,F , P), then(

Ω,F , P, (Xt)t≥0

)

is a Markov process,with transition probability Ps,t if ∀ f ∈ Eb, ∀ s < t:

E (f (Xt) | Xu, u < s) = Ps,tf (Xs)

=

∫

x∈E

f (x) Ps,t (Xs, dx) = E (f (Xt) | Xs) P − a.s.. (5)

If the process has an initial distribution π, then Pπ and Eπ(·) denote thetransition probability and the expected value, respectively. If the initialdistribution is a Dirac mass δx, the previous notation becomes Px andEx(·). The Markov process is said to be homogeneous if:

Ps,t = Ps+h,t+h, ∀s < t,∀h > 0 (6)

and we will write: P0,t = Pt and Ps,t = Pt−s. Note that the transitionprobability defines a semi-group operator:

PtPs = Ps+t, ∀s, t > 0 (7)

Definition 2.3. (Jump process) - Given a transition semi-group operator Pt

defined on the space (E, E), a Markov process with transition probability Pt

is a Jump process if ∀π, initial probability, almost every trajectory is rightcontinuous and constant, except in a countable set of jumps.

Let Ft = σ (Xs, s ≤ t) a sub σ-algebra of F . Then from previous definitions,the following properties hold:

1. Ex (f (Xt)) = Ptf (x), ∀f ∈ Eb and Px (Xt ∈ A) = Pt (x,A), ∀A ∈ E

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2. Eπ (f (Xt) | Fs) = Eπ (f (Xt) | Xs) = Pt−sf (Xs) = EXs(f (Xt−s)) Pπ-

a.s.

3. Let f0, ..., fn be a sequence of functions in Eb then ∀ t0 < t1 < . . . < tn:

Eπ (f0 (Xt0) . . . fn (Xtn)) =

∫

x0∈E

π (dx0) f (x0)

∫

x1∈E

Pt1 (x0, dx1) f1 (x1)

∫

x2∈E

Pt2−t1 (x1, dx2) f2 (x2) . . .

∫

xn∈E

Ptn−tn−1 (xn−1, dxn) fn (xn) (8)

Theorem 2.1. (Simple Markov property) - Let (Xt)t≥0 be a Markov jump

process and A a measurable set of trajectories, that is A ⊂ Ω = ER+ andA ∈ F = E⊗R+. The trajectories X·+t (which stays for s → Xs+t) satisfy thefollowing property: Pπ (X·+t ∈ A | Ft) = PXt

(X· ∈ A), Pπ-a.s..

Before introducing the strong Markov property we need to define a stoppingtime and some of its properties. Stopping times widely apply to credit riskmodelling. An example is the definition of the default or rating transitionevents as the first time an underlying stochastic process (the firm’s value)crosses an exogenously fixed barrier.

Definition 2.4. (Stopping time) - A random variable τ : Ω → R+ = R+ ∪+∞ is called stopping time if τ ≤ t ∈ Ft, ∀t.

The first entrance time, defined as D = inf t ≥ 0 : Xt ∈ F, where F ⊂ E,is an example of stopping time. Let F = (Ft)t≥0 be a filtration, thena stopping time τ is F-predictable if there exists an increasing sequenceof stopping time τnn≥0 such that τn < τ and lim τn = τ . A stoppingtime τ is called F-inaccessible if for any F-predictable stopping time S,P ω ∈ Ω|τ (ω) = S (ω) = 0.

Definition 2.5. (Stopping time σ-algebra) - Given a stopping time τ the as-sociated σ-algebra is defined as follows: Fτ =

A ∈ ER+ |A ∩ τ ≤ t ∈ Ft, ∀t

.

Theorem 2.2. Let τ be a stopping time, then Fτ is a σ-algebra.

Theorem 2.3. Let τ be a stopping time and (Xt)t a Markov process, thenXτ is Fτ -measurable.

From previous definitions and theorems, the following property holds for aMarkov jump process.

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Theorem 2.4. (Strong Markov property) - A jump process (Xt)t≥0 is strongly

Markovian. This means that ∀τ , Fτ -stopping time and ∀A ∈ ER+:

P (X·+τ ∈ A ∩ τ < ∞ | Fτ ) = Iτ<∞PXT(X· ∈ A) . (9)

The following result shows that in a jump process with initial value x, thewaiting time for the next jump is an exponentially distributed random time.Furthermore the new value of the process at the instant of the jump (Xτ ) isindependent to the random time of the jump.

Theorem 2.5. Let τ = τxc be the first enter time in the complementary setof x, with x ∈ E, then: (a) Px (τ > s) = e−λ(x)s, where λ (x) is a positiveor null function defined on E; (b) under the probability Px, if λ (x) 6= 0 thenXτ and τ are independent.

The function λ (x) is called intensity of the process. In the following weassume: ‖ λ (x) ‖∞= sup

x∈E|λ (x) | < ∞. Under the previous assumption and

from Theorem 2.5 the expected value of the stopping time is: Ex

(

Txc

)

=

(λ (x))−1.

Theorem 2.6. - Let (Xt)t be a jump process with bounded intensity λ (x).Then it exists a stochastic matrix P such that the transition semi-group of theprocess can be written as: Pt = exp (tIλ (P − I)), where Iλ is the by intensityfunction product operator. Furthermore the matrix P is defined as follows:

P (x, y)

Px

(

XTxc = y)

if λ (x) 6= 0

Ix=y if λ (x) = 0(10)

and in a neighbour of t = 0: Px (Xt = x) = 1 − λ (x) t + ε1 (x, y) t andPx (Xt ∈ A) = λ (x) t+P (x,A)+ tε2 (x, t, A), ∀A ⊂ xc, where ε1 (x, t) andε2 (x, t, A) converge towards zero, uniformly in x and in A, when t → 0.

In the following of the section we show a general method for building aMarkov jump process. This method allows us to completely specify the jumpprocess by means of a transition probability P and of an intensity functionλ (x). We require this function to be bounded, but it is possible to define thejump process also with an unbounded intensity (see Bouleau (2000) for moredetails). Theorem 2.7 will be useful also in the definition of Poisson process(see next section).

Theorem 2.7. (Markov jump process) - Let (Ω,F , P) be a probability space,assume (Zn)n≥0 and (τn (x))n≥1, with x ∈ E, are two sequences of randomvariables such that:

6

1. Z0, Z1, ..., Zn, ... is a Markov chain with transition matrix P and initialprobability π;

2. ∀x ∈ E, τn (x) is a variable such that P (τn (x) > t) = λ (x) e−λ(x)t,∀ t ≥ 0;

3. τn (x) is a measurable function of (ω, x);

4. the processes (Zn)n≥0, (τ1 (x))x∈E, (τ2 (x))x∈E, . . . are independent;

then the following series:

Xt = Z0It<τ1(Z0) + Z1Iτ1(Z0)≤t<τ1(Z0)+τ2(Z1) + . . . +

+ZnIτ1(Z0)+...+τn(Zn−1)≤t<τ1(Z0)+...+τn+1(Zn) + . . . (11)

defines a jump process with initial probability measure π and transition semi-group Pt = etIλ(P−I).

By removing the hypothesis of exponentially distributed random time, it ispossible to extend the definition of Markov jump process toward the defini-tion of a semi-Markov jump process (see Dacunha-Castelle, Duflo (1993) forfurther details).

3 Poisson Process

It is possible to define a Poisson counting process (or simply Poisson process)as a particular case of Markov counting process (or Markov jump process).Thus all the properties defined in previous paragraphs apply.

Definition 3.1. (Poisson counting process) - A Poisson process is a Markovjump process with values in N, constant intensity λ and transition matrixP (n,m) = I(m−n)=1.

This specification of the transition probability makes the process to be ”non-decreasing” with transition semi-group: Pt = etλ(P−I). In order to proposesome generalizations, it is useful to give another definition of Poisson process,which is equivalent to that one in Definition 3.1.

Definition 3.2. (Homogeneous Poisson process) - Let (Ω,F , P) be a probabil-ity space and (Nt)t≥0 a stochastic process defined on it. Let Ft = σ Ns, s ≤ tthe σ-field generated by the process, then (Nt)t≥0 is called homogeneous Pois-son process with intensity λ, if the following two conditions are satisfied:

1. ∀ 0 ≤ s < t the increment (Nt − Ns) is independent of the σ-field Fs;

7

2. ∀ 0 ≤ s < t the increment (Nt − Ns) has the following Poisson distri-bution:

P (Nt − Ns) = k | Fs = P (Nt − Ns) = k =(λ (t − s))k

k!e−λ(t−s)

with k = 0, 1, 2, . . ..

The process is called homogeneous because the increment distribution is in-variant with respect to time shifts. In the following we propose some gener-alisations of the Poisson model and show two alternative derivations of thePoisson stochastic model: through a sum of memoryless process and througha differential argument. These two methods highlight the role of the pa-rameters and random time distributions involved in the probabilistic model,underlying this kind of process.In the first method the Poisson process is built through the use of a sequenceof random times, which describe the arrival phenomena. The key idea followsfrom a simple observation. Let τk = inf

t > τk−1 : (Nt − Nτk−1) = 1

be astopping time, then from Definition 3.2, the random variable ξk = τk −τk−1, which represents the inter-arrival time, is exponentially distributed withparameter λ. This suggests that a Poisson process can be constructed alsoby specifying the inter-arrival time distribution.

Theorem 3.1. - Let ξ1, ξ2, . . . be a sequence of random variables defined onthe (Ω,F , P) with common exponential distribution with parameter λ > 0.The process (Nt)t≥0, defined by setting Nt = 0 if t < ξ1 and:

Nt =∞∑

k=1

kIξ1+ξ2+...+ξk≤t<ξ1+ξ2+...+ξk+1 (12)

otherwise, is a Poisson process with intensity λ.

Proof. - It is sufficient to prove that the Laplace transform of the process(Nt)t≥0 is equal to eλt(u−1) (see Theorem 3.4), which is the Laplace trans-form of a Poisson process with constant intensity λ. The process counts thenumber of the events in the interval ]0, t]. Define τi = ξ1 + ... + ξi, then thenumber of events in the interval is:

Nt =∞∑

k=1

kIξ1+ξ2+...+ξk≤t<ξ1+ξ2+...+ξk+1

=∞∑

k=1

kIτk≤t<τk+1 =∞∑

k=1

I]0,t] (τk) (13)

8

Therefore the Laplace transform of the process is:

E(

uNt)

= E

(

u

∞∑

k=1I]0,t](τk)

)

= limm→∞

m∏

k=1

E(

uI]0,t](τk))

(14)

= limm→∞

∞∫

0

g (ξ1) g (ξ1 + ξ2) . . . g (ξ1 + . . . + ξm) λe−λξ1dξ1 . . . λe−λξmdξm

where g (τi) = I]0,t] (τi). Consider now, the following change of variable:vi = ξ1 + . . . + ξi, with i = 1, . . . ,m, which is a bijective transformation onthe partition v1 < v2 < ... < vm of R

m+ . The integral becomes:

E(

uNt)

= limm→∞

1

(m − 1)!

∫

v1<v2<...<vm

g (v1) . . . g (vm) λme−λvmdv1 . . . dvm

= limm→∞

1

(m − 1)!

∫

0<v1<vm0<v2<vm

...0<vm−1<vm

g (v1) . . . g (vm) λme−λvmdv1 . . . dvm (15)

the number of equal integrals is given by the coefficient (m − 1)!, whichrepresents the total number of permutations of first m − 1 variables. TheJacobian of the transformation is equal to 1. It follows:

E(

uNt)

= limm→∞

1

(m − 1)!

∞∫

0

α∫

0

g (v) dv

m−1

g (α) λme−λαdα

= limm→∞

∞∫

0

1

m!

λ

α∫

0

g (v) dv

m

λe−λαdα

= limm→∞

∞∫

0

1

m!

λ

α∫

0

uI]0,t](v)dv

m

λe−λαdα

= limm→∞

∞∫

0

1

m!(λ (α + t (u − 1)))m λe−λαdα

= limm→∞

(

1 + . . . +1

m!(λt (u − 1))m

)

= eλt(u−1).

9

From Fig. 2-5 in Appendix A, we note that an higher level of λ producesmore frequent jumps. Furthermore note that a λ < 1 means a longer timeto the next jump. The process, (Nt)t≥0, defined in Theorem 3.1 counts thenumber of events in the interval ]0, t] and from Theorem 2.7 it is a Markovjump process. The construction of the process through the specification of theinter-arrival distribution is susceptible of generalisations towards the notionof renewal processes (see Grandell (1991)).From Theorem 3.1 we note that the time at which the events (arrivals) occur,is a random variable. In particular the inter-arrival time, T, follows an ex-ponential distribution. For this distribution the memoryless property holds:P (T > s + t | T > t) = P (T > s), or equivalently, by Bayes theorem:

P (T > s + t | T > t) =

=P (T > s + t ∩ T > t)

P (T > t)=

P (T > s + t)

P (T > t)

⇔ P (T > s + t) = P (T > s) P (T > t) (16)

Another distribution strictly related to the Poisson probability model is thewaiting time distribution.Denote the waiting time for the i-th event, with:

τi = ξ1 + . . . + ξi. (17)

Its cumulative density function is:

Fτn= P (τn ≤ t) = 1 − P (τn > t)

= 1 − P (Nt ≤ n − 1) = 1 −n−1∑

k=0

(λt)k e−λt

k!. (18)

By differentiating the c.d.f. with respect to t we obtain:

fτn=

d

dt

[

1 −n−1∑

k=0

(λt)k e−λt

k!

]

= λe−λt −n−1∑

k=1

kλ (λt)k−1 e−λt

k!+

n−1∑

k=1

λ (λt)k e−λt

k!

= λe−λt

[

1 −n−1∑

k=1

(λt)k−1

(k − 1)!+

n−1∑

k=1

(λt)k

k!

]

= λe−λt

[

1 −n−1∑

k=1

(λt)k−1

(k − 1)!+

n−2∑

k=0

(λt)k

k!

]

= λe−λt

[

1 − 1 +(λt)n−1

(n − 1)!

]

=e−λtλntn−1

(n − 1)!(19)

10

which is the gamma distribution: Ga (n, λ), also known in risk theory asErlang distribution.The following useful properties hold for a Poisson process.

Theorem 3.2. - Let Ft = σ Ns, s ≤ t the σ-field generated by the Poissonprocess (Nt)t, then the processes:

(a) Nt = Nt − λt (compensated Poisson process)(b) Mt = (Nt − λt)2 − λt = N2

t − λt(c) At = eaNt−λt(ea−1) (exponential martingale process) with a ∈ R;

are Ft-martingales.

Proof.

(a) Note that the intensity of the process plays the role of compensatorof the Poisson process and this is a general result in point process

theory. We need to prove that E

(

Nt − Ns | Fs

)

= 0, with t ≥ s, or

equivalently: E (Nt − Ns | Fs)− λ (t − s) = 0. From the Definition 3.2of Poisson process:

P (Nt − Ns | Fs) = P (Nt − Ns) = Poi (λ (t − s)) ,

thus EP (Nt − Ns | Fs) = EP (Nt − Ns) = λ (t − s).

(b) We need to prove that:

E (Mt − Ms | Fs) = E(

(Nt − λt)2 − (Ns − λs)2 − λ (t − s) | Fs

)

= 0.

From the Definition 3.2 of Poisson process it follows that:

P (Nt − Ns | Fs) = P (Nt − Ns) = Poi (λ (t − s)) ,

thus:

EP

(

(Nt − Ns − λ (t − s))2 | Fs

)

= EP

(

(Nt − Ns − λ (t − s))2) =

= Var (Nt − Ns) = λ (t − s) .

Now it is sufficient to observe that:

EP

(

(Nt − Ns − λ (t − s))2 | Fs

)

=

= EP

(

(Nt − λt)2 + (Ns − λs)2 − 2 (Nt − λt) (Ns − λs) | Fs

)

= EP

(

(Nt − λt)2 + (Ns − λs)2 | Fs

)

− 2EP ((Nt − λt) (Ns − λs) | Fs)

and because (Nt − λt) is a Ft-martingale and (Ns − λs) is Fs -measurable,

EP

(

(Nt − λt)2 + (Ns − λs)2 | Fs

)

− 2EP ((Ns − λs) (Ns − λs) | Fs) =

= EP

(

(Nt − λt)2 − (Ns − λs)2 | Fs

)

= EP

(

Nt − Ns | Fs

)

11

(c) The proof is similar to previous points. Here it is sufficient to note that

EP

(

eaNt−aNs | Fs

)

= eλ(t−s)(ea−1).

Just to give a graphical representation of these process, in the Appendix Bwe simulate a path from each of the previous martingales, associated to aPoisson process with intensity λ = 0.5. For an introduction to the simulationmethod for the Poisson process we refer the interested reader to Reply (1987).Note however that the following property is particularly useful, because itshows an easy way to simulate a path from a Poisson process.

Theorem 3.3. - Let 0 = t0 < t1 < ... < tk = t be a partition of the interval[0, t] and (Nt)t≥0 a Poisson process, then conditionally to Nt = n the randomvariables Nt1−Nt0 , Nt2−Nt1 , . . . , Ntk−Ntk−1

, Nt1 are independent, uniformlydistributed in the interval [0, t].

The following differential derivation of the Poisson model (see Ross (1997)),could help to understand other simulation methods, which have been pro-posed in the literature.The process Nt describes the number of events in a time interval [0, t]. As-sume the following hypotheses hold:

(a) Nt=0 = 0;

(b) ∀ t1 ≤ t2 < t3 ≤ t4 the number of events that occur in disjoint intervals[t1, t2] and [t3, t4] are independent;

(c) the probability distribution of the number of events Nt, that occur inthe interval [t, s], depends only on the length of the interval, (s − t),and not on the time location;

(d) assume a time interval of length dt, then P (Nt,t+dt = 1) = λdt + o (dt)and P (Nt,t+dt ≥ 2) = o (dt).

Consider a partition of the time interval [0, t] in n in disjoint subintervals oflength t/n. There are two possible ways of obtaining k events in the interval:

(a) k intervals contain exactly one event and the other n − k intervals donot contain events;

(b) there exists at least one interval that contains two or more events.

12

In case (b) the probability is:

P (k events in [0, t] ∩ at least 1 interval contain 2 or more events)

≤ P (at least 1 interval contains 2 or more events)

= P

(

n⋃

j=1

j-th interval contains 2 or more events

)

≤n∑

j=1

P (j-th interval contains 2 or more events) =n∑

j=1

o (t/n)

= nt

n

o (t/n)

t/n= t

o (t/n)

t/n

n→∞−→ 0

because when n → ∞, t/n → ∞ and o(t/n)t/n

→ 0.

In case (a) the probability of k intervals with one events and n− k intervalswith zero events is given by a binomial probability model. By assumption theP (1 event) = λt/n + o (t/n) while P (0 events) = 1 − P (1 event or more) =1 − λt/n − o (t/n) − o (t/n). Thus the probability is:

P (k intervals with 1 event, n − k intervals with 0 events) =

=

(

nk

)

(λt/n + o (t/n))k (1 − λt/n − o (t/n))n−k (20)

When n → ∞, the limiting probability is:

P (Nt = k) = limn→∞

n!

k! (n − k)!

1

nk

(

λt +o (t/n)

t/nt

)k

· (21)

·

(

1 −t

n

(

λ −o (t/n)

t/n

))n(

1 −t

n

(

λ −o (t/n)

t/n

))−k

=1

k!(λt)k e−λt

because:

limn→∞

n!

k! (n − k)!

1

nk= lim

n→∞

1

k!

(n

n

)

(

n − 1

n

)

. . .

(

n − k

n

)

=1

k!(22)

limn→∞

(

λt +o (t/n)

t/nt

)k

= (λt)k (23)

limn→∞

(

1 −t

n

(

λ +o (t/n)

t/n

))n

= limx→∞

(

1 +1

x+

o (x)

x

)−λtx

= e−λt (24)

limn→∞

(

1 −t

n

(

λ −o (t/n)

t/n

))−k

= 1. (25)

13

We conclude this section with the definition of Laplace transform of a Poissonprocess. It is useful to characterize the Poisson process through its Laplacetransform, in order to analyse some powerful generalisations, like the Cox andthe compound Poisson processes (see Section 4), which are used in credit riskmodelling.

Theorem 3.4. (Laplace transform) - Let (Nt)t≥0 be a Poisson process withintensity λ, then its Laplace transform is:

ΨN = E(

e−Nt)

=∞∑

k=0

e−k (λt)k e−λt

k!= e−λt

∞∑

k=0

(e−1λt)k

k!= e−λt(1−e−1).

(26)

4 Generalizations of the Poisson Process

Before proposing some generalisations of the homogeneous Poisson processbased on a redefinition of the underlying probability model, we introducethe operations of superposition and composition of Poisson processes andcharacterize the result of such operations. The operation of superposition isuseful in order to decompose the intensity function in more simple functionsbefore using simulation techniques. The operation of composition, throughrandom sums, is a fundamental tool used in risk theory and also in theactuarial approach to credit risk modelling (see Credit Suisse (1997) creditportfolio approach), because it allows to aggregate through a random sum aset of competing risk sources.

Theorem 4.1. (Superposition) - Let N1t, N2t, ..., Nnt be a sequence of n in-dependent Poisson processes, with intensities λ1, λ2, ..., λn respectively. Thesuperposition of these processes:

Mt = N1t + N2t + ... + Nnt (27)

gives a Poisson process with intensity: λ = λ1 + ... + λn.

Proof. Follows immediately by calculating the Laplace transform of the pro-cess Mt.

Definition 4.1. (Compound Poisson process) - Let ξ1, ξ2, ξ3, . . . be a sequenceof independent and identically distributed random variables with c.d.f. Fξ andLaplace transform E

(

euξ)

= g (u). Let Nt be a Poisson process with intensityλ independent of the sequence ξ1, ξ2, ξ3, . . .. The process:

Mt =Nt∑

k=1

ξk (28)

14

is called compound Poisson process.

Theorem 4.2. - The compound Poisson process Mt has Laplace transforme−λt(1−g(u)).

Proof.

E(

eMtu)

=∞∑

j=0

(λt)j

j!e−λt

E

(

exp

(

uNt∑

k=1

ξk

)

| Nt = j

)

=∞∑

j=0

(λt)j

j!e−λt

E

(

exp

(

u

j∑

k=1

ξk

))

=∞∑

j=0

(λt)j

j!e−λt

j∏

k=1

E(

euξk)

=∞∑

j=0

(λt)j

j!e−λt (g (u))j

=∞∑

j=0

(λtg (u))j

j!e−λt = e−λt(1−g(u)) (29)

From Definition 3.2 it is possible to build other definitions of Poisson processthrough a modification of the hypothesis on the intensity parameter. Afirst simple extension follows by assuming the intensity of the arrivals is adeterministic function of the time.

Definition 4.2. (Deterministic intensity) - Let (Ω,F , P) be a probabilityspace and (Nt)t≥0 a stochastic process. Let Gt = σ Ns, s ≤ t the σ-fieldgenerated by the process, then (Nt)t≥0 is called in-homogeneous Poisson pro-cess with intensity function (or hazard rate function) λ (t) : R+ → R+ if thefollowing two conditions are satisfied:

1. ∀ 0 ≤ s < t the increment (Nt − Ns) is independent of the σ-field Gs;

2. ∀ 0 ≤ s < t the increment (Nt − Ns) has the following Poisson distri-bution:

P (Nt − Ns) = k | Fs = P (Nt − Ns) = k =(Λ (t) − Λ (s))k

k!e−(Λ(t)−Λ(s))

with k = 0, 1, 2, ....

15

The function: Λ (t) =t∫

0

λ (u) du is called intensity measure (or hazard func-

tion, or cumulative intensity) (see for example Cox and Oakes (1984), Zacks(1992), Ibrahim, Chen and Sinha (2001), for an introduction to survivalmodels in probability and statistics). Note that the increments of an in-homogeneous Poisson process are no longer stationary unless the intensityΛ(t) is a linear function of t. In the following, we give two simple examplesof time-dependent deterministic intensity function.

Example 1. (Piecewise constant intensity) - Let 0 = t0 < t1 < . . . < tn =T be a partition of the time interval [0, T ], it is possible to construct anin-homogeneous Poisson process simply by assuming that the intensity is apiecewise constant function:

λ (t) =n−1∑

i=0

λiIti<t≤ti+1.

In Figure 9, Appendix C, we show a simulated path from a process with apiecewise constant intensity. The time interval is divided in two sub-intervals,notice from the simulated path, that the intensity of the jump is higher in thesecond sub-interval because an higher intensity parameter is assumed.

Example 2. (Weibull intensity) - A widely used intensity function is thepower function with two parameters: λ (t) = αβtβ−1.

This kind of intensity characterizes a renewal process with a Weibull inter-arrival time distribution (see Section 1). In Fig. 11 and 12, we simulatesome paths from this process through a change of time technique and underdifferent assumptions on the parameters. In particular notice that if β < 1then the probability of the jump events decreases with time, while if β >1 then the probability increases with time. The intermediate case β = 1corresponds to a constant intensity function. The same considerations followfrom Fig.1, which shows the value of the cumulated intensity function fordifferent values of the parameters.The change of time is a general method for building an in-homogeneousPoisson process. This method is also very useful in order to simulate froman in-homogeneous Poisson process. Let (Nt)t≥0 be a standard Poisson pro-cess, with constant intensity, λ = 1, defined on the filtered probability space

(Ω,F , G, P), where G = (Gt)t≥0 and Gt = σ

Ns, s ≤ t

. Introduce an op-

erational time scale function, given by the cumulated intensity, Λ (t), andassume it is a continuous function (generally obtained by cumulating overtime the intensity rate function as in Definition 4.2). The process defined

16

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

1

2

3

α=0.5, β=0.5α=0.5, β=1.0α=0.5, β=1.5

Figure 1: Power change of time for three Weibull intensity processes, with:(α = 0.5, β = 0.5), (α = 0.5, β = 1) and (α = 0.5, β = 1.5).

by the following time-change operation: Nt = Nt Λ (t)def= NΛ(t) is an in-

homogeneous Poisson process with intensity Λ (t). The proof follows fromDefinition 3.2 of Poisson process, where the increments are defined as:

P(NΛ(t) − NΛ(s)) = k =(Λ (t) − Λ (s))k

k!e−(Λ(t)−Λ(s))

which is equivalent to the definition of in-homogeneous Poisson process withintensity Λ (t). The homogeneous Poisson process is stationary because itdoes not allow for fluctuations in the mean number of events, but in orderto produce a more realistic stochastic model it is convenient to modify thishypothesis. The introduction of a deterministic time-dependent intensity isa first generalization, but it also possible to introduce random fluctuationsin the intensity function. Assume the hazard function Λ (ω) is a randomvariable with p.d.f. fΛ, then through the usual time-change operation weobtain the following process.

Definition 4.3. (Mixed Poisson process) - Let (Nt)t≥0 be a standard ho-mogeneous Poisson process with intensity λ = 1 and Λ (ω) a random vari-able with P (Λ > 0) = 1, independent of (Nt)t≥0. Then the process Nt =

Nt (Λ (ω) t)def= NΛ(ω)t is called a mixed Poisson process. The random vari-

able Λ (ω) is called structure variable.

17

Notice that a mixed Poisson process has stationary increments, however theindependent increments condition does not hold, unless the distribution of Λiis concentrated at one point: fΛ = δΛ∗. Assume 0 ≤ t1 < t2 < t3 < t4 ≤ tthen the covariance between the increments is:

E ((Nt2 − Nt1) (Nt4 − Nt3)) − E (Nt2 − Nt1) E (Nt4 − Nt3) =

= E [E ((Nt2 − Nt1) (Nt4 − Nt3) | Λ)] +

−E [E (Nt2 − Nt1 | Λ)] E [E (Nt4 − Nt3 | Λ)]

= E (Λ (t2 − t1) Λ (t4 − t3)) − E (Λ (t2 − t1)) E (Λ (t4 − t3))

= Var (Λ) (t2 − t1) (t4 − t3) . (30)

A common choice for the distribution of the structural variable Λ (ω) is thegamma distribution Γ (α, β) whose density function is: fΛ (y) = βα

Γ(α)yα−1e−βy

with y ≥ 0. Thus the resulting mixed Poisson process is a Polya process.

Theorem 4.3. (Polya process) - Let (Nt)t≥0 be a mixed Poisson process andlet the structural variable Λ (ω) follow a gamma distribution Γ (α, β), thenthe process (Nt)t≥0 is a Polya process (or negative binomial process).

Proof. In order to show that the resulting process is a Polya process it issufficient to show that the distribution of the process is a negative binomial:

P (N (t) = n) =

∞∫

0

P (N (t) = n | Λ = λ) fΛ (λ) dλ

=

∞∫

0

P

(

N (tΛ) = n | Λ = λ)

fΛ (λ) dλ =

=

∞∫

0

(λt)n e−λt

n!

βα

Γ (α)e−βλλα−1dλ

=tn

n!

βα

Γ (α)

Γ (α + n)

(β + t)α+n

∞∫

0

(β + t)α+n

Γ (α + n)e−λ(β+t)λn+α−1dλ

=(α + n − 1)!

(α − 1)!n!

(

t

β + t

)n(β

β + t

)α

= N eg (α, p) with p =β

β + t. (31)

18

Figure 13, in Appendix C, exhibits some paths simulated from a Polya pro-cess. We take a Gamma distributed intensity, with parameter α = 3/2 andβ = 2/3. Note that the intensity’s mean value is αβ = 1, but the process be-haves differently with respect an homogeneous Poisson with intensity λ = 1,because the random nature of the Polya process intensity allows for a highheterogeneity of the paths. This effect can be noted by a direct comparisonbetween Figure 13 and Figure 14 where some samples have been simulatedfrom a homogeneous Poisson with intensity λ = 1.The doubly stochastic Poisson process is another useful generalisation of thePoisson process. The double stochastic Poisson is also known in the liter-ature as conditional Poisson process (see Bremaud (1981)). See Cox andIsham (1980), Kingman (1993) for further details on the doubly stochasticPoisson as point process and Grandell (1976) and (1991) for the applicationsof this kind of processes in risk theory. A particular case of doubly stochas-tic Poisson process is the Cox process (see Grandell (1991)), which is usedin the reduced form approach to credit risk modelling (see Lando (1999)).The conditional Poisson process, introduced in probability theory by Cox in1955, allows to model more realistically the random occurrence of events likethe default event in credit risk model, or the number of contingent claims inthe actuarial models. In this kind of process the intensity is not constant,but varies stochastically over time. In particular the intensity is assumed tobe a stochastic process. It can be specified also through a dynamic model,where some state variables (business cycle variables, macro-market variables,financial indexes, etc. ), determine its evolution either over time or space.The stochastic nature of the intensity causes the variance of the process to begreater than the variance of a homogeneous Poisson process with the sameexpected intensity measure. This feature of the Cox processes is referred, inthe literature on point processes and survival models, as overdispersion. Letus build the conditional Poisson process through the following definitions.

Definition 4.4. (Hazard process) - Let (Ω,G, G, P) be a filtered probabilityspace and let F be a sub-filtration of G. We called hazard process an Ft-adapted, right-continuous, increasing stochastic process (Λt)t≥0 with P-a.s.Λ0 = 0 and Λ∞ = ∞.

The hazard process can be specified also in the following way:

Λt =

t∫

0

λsds, with t ≥ 0

where (λt)t≥0 is a Ft-progressively measurable process with locally Riemannintegrable realizations. The process (λt)t≥0 is called intensity process (or

19

hazard rate process). Although it is possible to define a conditional Poissonprocess through the usual random time-change operation, we prefer to defineit by specifying the increment distribution.

Definition 4.5. (Doubly stochastic Poisson process) - Let (Ω,G, P, G) be afiltered probability space and (Nt)t≥0 a stochastic process defined on it. Let F

be a sub-filtration of G, then (Nt)t≥0 is called F-conditional Poisson processwith hazard process (Λt)t≥0 if the following two conditions are satisfied:

1. ∀ 0 ≤ s < t, conditionally F∞ on the increment (Nt−Ns) is independentof the σ-field Gs;

2. ∀ 0 ≤ s < t the increment (Nt − Ns) has the following Poisson dis-tribution: P (Nt − Ns) = k | Gs ∨ F∞ = P (Nt − Ns) = k | F∞ =(Λt−Λs)

k

k!e−(Λt−Λs) with k = 0, 1, 2, ... where F∞ = σ Nu : u ∈ R+.

We adapt the definition of overdispersion, given in Kingman (1993), to ourdefinition of Poisson process, we obtain an equality which shows that thevariance of a Cox process is greater than the variance of a Poisson processwith the same expected intensity measure. Let 0 = t0 < t1 < t2 < .... < tn =T be a partition of the time interval [0, T ], and introduce, as in Definition4.5, a random measure:

Λ (]0, t]) = Λt =

t∫

0

λ (u) du.

We need to prove that Var (Nt) ≥ E (Nt). Observe that E (Nt) = EΛ (EP (Nt | Λt)) =EΛ (Λt) and compute:

E(

(Nt)2) = EΛ

(

EP

(

(Nt)2 | Λt

))

= EΛ

(

Λt + Λ2t

)

= EΛ (Λt) + VarΛ (Λt) + (EΛ (Λt))2 ⇔

⇔ E(

(Nt)2)− (EΛ (Λt))

2 = EΛ (Λt) + VarΛ (Λt) ⇔

⇔ E(

(Nt)2)− (E (Nt))

2 = E (Nt) + VarΛ (Λt) ⇔

⇔ Var (Nt) = E (Nt) + VarΛ (Λt) . (32)

The desired result follows immediately.The hazard process plays an important role in the martingale approach tocredit risk because it is the compensator of the associated doubly stochas-tic Poisson process (see for example Bielecki and Rutkowski (2002)). Thefollowing theorem states the result.

20

Theorem 4.4. - Let (Nt)t≥0 be a doubly stochastic Poisson process with

intensity (Λt)t≥0, then the compensated process Nt = Nt − Λ (t) is a G-martingale.

Proof. We need to prove that EP(Nt−Ns | Fs) = 0 with t ≥ s, or equivalentlyE (Nt − Ns − Λt | Gs) = 0. From the Definition 4.5 of doubly stochasticPoisson process: P (Nt − Ns | Gs ∨ F∞) = P (Nt − Ns | F∞). Thus

EP (Nt − Ns | Gs) − EP (Λt − Λs | Gs) =

= EP (EP (Nt − Ns | Gs ∨ F∞) | Gs) − EP (Λt − Λs | Gs)

= EP (EP (Nt − Ns | F∞) | Gs) − EP (Λt − Λs | Gs)

= EP (Λt − Λs | Gs) − EP (Λt − Λs | Gs) = 0.

In some applications like the credit risk models it is useful to consider aparticular kind of conditional Poisson process which is called Cox process.In this kind of process the intensity process, λt = h (t,Xt), is a continuousfunction of the time and of a vector of state variables Xt which are R

d-valued stochastic processes independent of the Poisson process. A simpleway of constructing the filtrations related to the Cox process is the following:Ht = σ Ns : s ≤ t, Ft = σ Xs : s ≤ t and finally Gt = Ht∨Ft. We presentsome examples of intensity process and refer to Bielecki and Rutkowski (2002)for a more complete analysis of the hazard function (or process) modelling.

Example 3. (Intensity diffusion process) - In the continuous time model ofSchonbucher (2000) the intensity process λt is the solution of the followingSDE: dλt = λtµdt + λtσdWt.

Example 4. (Proportional hazard rate process) - As in the regression modeldue to Cox (1972), the hazard rate process depends generally on both timeand state variables: λt = h (t,Xt). When the process can be factorised asfollows: λt = h (t,Xt) = h0 (t) eg(Xt) the model is called proportional hazard.The function h0 (t) is called baseline hazard function and the function g (x)is generally assumed to be linear: g (x) = β ′x, with β vector of coefficients.

Given a Cox process it is possible to determine the density function of theassociated random time.

Theorem 4.5. - Let (Ω,G, G, P) be a filtered probability space with Gt =Ht ∨ Ft and (Xt)t≥0 a R

d-valued continuous stochastic process. Let λ (Xt) :

Rd → R+ be the intensity process and E an exponential random variable,

21

independent of X, with p.d.f.: P (E ≥ t) = e−t. The random time definedas:

τ = inf

t ≥ 0 :

t∫

0

λ (Xs) ds ≥ E

(33)

has the following conditional distribution function given the σ-field Ft:

P (τ > s | Ft) = exp

−

s∫

0

λ (Xu) du

(34)

The unconditional distribution is given by:

P (τ > s) = E

exp

−

s∫

0

λ (Xu) du

. (35)

5 Point processes

The Poisson process can intuitively described as a sequence of unitary jumps,which occur at random on a time interval. Thus the process can be viewedas a random partition of points on the set R+. In this paragraph we gen-eralize the Poisson process by modifying the assumption on the randomlypartitioned set and on the probability function associated to the subsets.The resulting processes are called point processes. Let (E, E) be a space withthe borel σ-algebra, z1, z2, z3, ..., zi, ... a sequence of point and B a subset ofE. Then the point measure of the set B is given by:

m (B) =∞∑

i=1

Izi∈B

Definition 5.1. (Point process) - Let Mp = m (B) : E → N be the set of allthe finite point measures defined on (E, E) and let Mp = σ m : m → m (A) , A ∈ E.A point process is a random variable defined as:

N (ω) :

[

(Ω,F , P) → (Mp,Mp)ω 7−→ N (ω)

(36)

Note that the probability PN , defined on (Mp,Mp), is the image of the prob-ability P, through the process N . The probability, PN , is determined by thejoint probability of the random vector (N (ω,B1) , N (ω,B2) , . . . , N (ω,Bk)),

22

where B1, B2, . . . , Bk ∈ E . If the N is a point process, the concentration mea-sure (or intensity) µ is defined as: µ (A) = E (N (ω,A)), ∀A ∈ E . It is alsopossible to construct a Poisson point process by specifying the concentrationmeasure of the process. The following proposition states the result.

Theorem 5.1. (Point Poisson process) - Let µ be a σ-finite positive measuredefined on (E, E), then it exists a point process N with values in Mp (E) suchthat:

1. ∀A ∈ E the random variable N (ω,A) follows a Poisson distributionwith intensity µ (A)

2. A1, A2, ..., Ak ∈ E such that Ai ∩ Aj = ∅ ∀i 6= j the random variablesN (ω,A1) , N (ω,A2) , ..., N (ω,Ak) are independent.

We report the proof due to Bouleau (2000) of previous theorem because itshows the importance of the Laplace transform in the study of the pointprocesses.

Proof. Assume the measure µ is finite: µ (E) = θ < ∞, then by normalizingthe measure we obtain a probability: µ0 = µ/θ. Assume a point processdefined on the following space: (Ω,F , P) = (N,P (N) , P (θ)) × (E, E , µ0)

N,where P (θ) represents the Poisson distribution with parameter θ. The pointprocess is defined as follows:

N (ω) =

Y (ω)∑

i=1

δXi(ω)

with Y,X1, X2, ..., Xi, ... ∈ (N,P (N) , P (θ)) × (E, E , µ0)N

To show that N is a Poisson process it is sufficient to calculate its Laplacetransform:

E(

uN(ω,A))

= E

u

Y (ω)∑

i=1IA(Xi)

= E

Y (ω)∏

i=1

uIA(Xi)

= E

(

∞∑

n=0

P (Y = n)n∏

i=1

uIA(Xi)

)

=∞∑

n=0

e−θ θn

n!

n∏

i=1

E(

uIA(Xi))

23

=∞∑

n=0

e−θ θn

n!(1 · P (Xi /∈ A) + u · P (Xi ∈ A))n

=∞∑

n=0

e−θ θn

n!(1 · (1 − µ0 (A)) + u · µ0 (A))n

=∞∑

n=0

e−θ (θ (1 + (u − 1) µ0 (A)))n

n!= e−θ+θ(1+(u−1)µ0(A))

= eθ(u−1)µ(A)

θ = e(u−1)µ(A)

which is the Laplace transform of a Poisson process with intensity µ (A).Furthermore if the A1, A2, ..., Ak ∈ E are independent then:

E

(

uN(ω,A1)1 u

N(ω,A2)2 ...u

N(ω,Ak)k

)

= e(u1−1)µ(A1)e(u2−1)µ(A2)...e(uk−1)µ(Ak) (37)

for the properties of the Laplace transform also the random variables N (ω,A1),N (ω,A2), . . ., N (ω,Ak) are independent.

We conclude this section by giving the Laplace transform of a point Poissonprocess. It is a generalization of the result stated in Theorem 3.4.

Theorem 5.2. (Laplace transform) - The Laplace transform of a point Pois-son process with concentration measure µ is given by:

ΨN = exp

−

∫

(

1 − e−f(x))

dµ (x)

. (38)

6 Conclusion

Many processes are now involved in credit risk modelling. Much attentionin the literature has been devoted to diffusion processes, while other kindof processes, like jump processes might be less known. Jump processes playan important role in both defaultable claim pricing and credit portfolio lossevaluation, because they are suitable to capture the discontinuity induced bya default or credit migration event. In this work we focus on the homoge-neous and in-homogeneous Poisson processes, highlighting main features anddifferences. Some analytical and simulated simple examples help the readerto understand the behaviour of the processes analysed.

24

Appendix A - The Intensity Parameter

In this appendix the present some paths simulated from a Poisson process,with the following intensities: λ = 0.05, 0.5, 1, 1.5. Observe how higher isthe value of the intensity parameter, more frequent are the jumps.

0 10 20 30 40 50Time

0

1

2

3

4

5

6

-

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-

Poisson process, Lambda=0.50 Time interval (0,50)

Figure 2: Simulated path from a Poisson process with λ = 0.05.

0 10 20 30 40 50Time

0

6

12

18

24

30

N(t)

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_

Poisson Process, Lambda=0.5 Time interval (0,50)

Figure 3: Simulated path from a Poisson process with λ = 0.50.

25

0 10 20 30 40 50Time

0

10

20

30

40

N(t)

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Poisson Process, Lambda=1 Time interval (0,50)

Figure 4: Simulated path from a Poisson process with λ = 1.00.

0 10 20 30 40 50Time

0

20

40

60

N(t)

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Poisson Process, Lambda=1.5 Time interval (0,50)

Figure 5: Simulated path from a Poisson process with λ = 1.50.

26

Appendix B -Martingales associated to a Pois-

son process

In the following we exhibits some paths simulated from the martingale pro-cesses defined in Theorem 3.2:a) Nt = Nt − λt (compensated Poisson process);b) Mt = N2

t − λt;c) At = eaNt−λt(ea−1) (exponential martingale process).The simulated martingale processes are associated to the Poisson processgiven in Fig. 4 (with λ = 0.5). The exponential martingale is represented(Fig. 8) in logarithmic scale.

0 10 20 30 40 50Time

-2

3

8

13

18

23

λ

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Martingale associated to a Poisson process Lambda=0.50, Time interval (0,50)

Figure 6: A path simulated from the compensated Poisson process, Nt =Nt − λt.

27

0 10 20 30 40 50Time

-30

-20

-10

0

λ

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Martingale associated to a Poisson process Lambda=0.50, Time interval (0,50)

Figure 7: A path simulated from the martingale Mt = N2t − λt.

0 10 20 30 40 50Time

0.0

0.5

1.0

1.5

2.0

λ

! "#

-

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Martingale associated to a Poisson process Lambda=0.50, Time interval (0,50)

Figure 8: A path simulated from the exponential martingale process, At =eaNt−λt(ea−1).

28

Appendix C - In-homogeneous Poisson Pro-

cess

In the following we show some simulated examples of the in-homogeneousPoisson process. We carried out the simulation by means of a change of timetechnique. In the first example (Fig. 9) we assume that the Poisson processhas the following piecewise constant intensity function:

λ(t) =

λ1 if 0 ≤ t ≤ 100λ2 if 100 < t ≤ 150

with λ1 = 0.05, λ2 = 1.5 (39)

Observe that the probability of a jump of the process is higher in the secondtime interval (see Figure 9). In the second example (Figures 10 and 11) westill assume a time dependent deterministic intensity function. In particularthe intensity is the same of a renewal process with a Weibull inter-arrivaltime (duration time) distribution:

λ (t) = αβtβ−1 (40)

If you compare the simulated path of a standard Poisson process (Fig. 10)with the path of the in-homogeneous processes (Fig. 11 and 12), you can seethe effect of the parameter beta on the random times stream. When betais greater then 1, the probability of a jump increases with time (Fig. 12),when beta is less then 1 the probability decreases in t (Fig. 11). In the lastexample (Fig. 13) we simulate a Polya process.

29

5 30 55 80 105 130 155Time

0

20

40

60

80

$% &'

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In-homogeneous Poisson processλ1=0.05 (t ≤ 100), λ2=1.50 (t>100)

Figure 9: A Poisson process with piecewise constant intensity function.

0 10 20 30 40 50Time

0

40

80

120

() *+

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-

Homogeneous Poisson processLambda=1, Time interval (0,50)

Figure 10: A Poisson process with Weibull intensity: standard Poisson sam-ple path.

30

0 10 20 30 40 50Time

0

40

80

120

,- ./

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In-homogeneous Poisson process α=0.40, β=0.99

Figure 11: A Poisson process with Weibull intensity. Change of time withα = 0.4 and β < 1.

0 10 20 30 40 50Time

0

40

80

120

01 23

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In-homogeneous Poisson process α=0.40, β=1.50

Figure 12: A Poisson process with Weibull intensity. Change of time withα = 0.4 and β > 1.

31

0 10 20 30 40 50

0

20

40

60

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Samples simulated from a Poisson process, Lambda=1 Time interval (0,50)

Figure 13: Samples from a Poisson process, with intensity λ = 1.

0 10 20 30 40 50

0

10

20

30

40

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Samples simulated from a Polya process time interval (0,50)

Figure 14: Mixed Poisson Process: Polya process with intensity Λ ∼Ga(

32, 2

3

)

.

32

Appendix D - Cox Process

We simulate a particular kind of Cox process, called Markov modulated Pois-son process, which is used in some credit risk models. It is defined as:

EP (Nt|Ft) = Λt with Λt =

t∫

0

α (Xu) du (41)

with intensity function:

α (Xu) :

[

1, ...,m → α1, ..., αmXu = i 7→ α (Xu = i) = αi

(42)

The process Xu is a Markov chain with values in 1, . . . ,m and with transitionmatrix (Qi,j)i,j=1,...,m. We simulate the process trough the following steps:

1. simulate a sequence of random time τ1, . . . , τn, where n = sup i|τi ≤ T;

2. simulate a path of the Markov chain X1, . . . , Xn with initial value X0;

3. make a time-change τ ∗i = τi/α (Xi).

In this numerical example we pose:

Q =

(

0.8 0.20.2 0.8

)

, α1 = 1.7, α2 = 0.2. (43)

The values of the intensity process are exhibited in Fig. 15 while the simu-lated values of the Cox process are drown in Fig. 16.

33

0 50 100 150 200Time

0.0

0.5

1.0

1.5

45 67

Intensity process modelled as a 2-states Markov chain Transition probabilities: p11=0.80, p21=0.20

Figure 15: Two State Markov Chain.

0 50 100 150 200Time

0

20

40

60

8 9:; <=

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Cox processλ1=1.70 (X(t)=1), λ2=0.20 (X(t)=2)

Figure 16: Cox Process.

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