Closed-form formulas for the distribution of the jumps of doubly-stochastic Poisson processes

0.19 MB·

by  Arturo Valdivia

Closed-form formulas for the distribution of the jumps of
doubly-stochastic Poisson processes
7 Arturo Valdivia∗
1
0
2 January 4, 2017

n
a
J

3

Abstract
]
R
P Westudytheobtainmentofclosed-formformulasforthedistributionofthejumps
h. of a doubly-stochastic Poisson process. The problem is approached in two ways. On
t the one hand, we translate the problem to the computation of multiple derivatives of
a
m the Hazardprocess cumulant generating function; this leads to a closed-formformula
[ written in terms of Bell polynomials. On the other hand, for Hazardprocesses driven

1 by Lévy processes, we use Malliavin calculus in order to express the aforementioned
v
distributions in an appealing recursive manner. We outline the potential application
7
1 of these results in credit risk.
7
0
Keywords: doubly-stochastic Poisson process; Bell polynomials; Malliavin calcu-
0
. lus; Credit risk; Hazard process; Integrated non-Gaussian OU process.
1
0 AMS MSC 2010: 60G22,60G51, 60H07, 91G40.
7
1
:
v
1 Introduction
i
X
r
a
Consider an ordered series of random times τ ... τ accounting for the sequenced
1 m
≤ ≤
occurence of certain events. In the context of credit risk, these random times can be seen
as credit events such as the firm’s value sudden deterioration, credit rate downgrade, the
firm’s default, etcetera. The valuation of defaultable claims (see [5, 18]) is closely related
to computation of the quantities
P(τ > T ), t 0, n = 1,...,m,
n t
|F ≥
where the reference filtration F = ( ) accounts for the information generated by all
t t≥0
F
state variables.
∗Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, E-08007 Barcelona, Spain. E-mail:
[email protected]
1
An interesting possibility to model these random times consists in considering τ ,...,τ
1 m
as the succesive jumps of a doubly-stochastic Poisson process (DSPP). That is, a time-
changedPoissonprocess(P ) ,wherethetimechange(Λ ) isanon-decreasingcàdlàg
Λt t≥0 t t≥0
F-adaptedprocessstartingatzero; andthePoissonprocess(P ) hasintensity rateequal
t t≥0
to 1, and it is independent of F. We refer to (Λ ) as the Hazard process.
t t≥0
The purpose of this note is to study the obtainment of closed-form formulas for the distri-
butions of the n-th jump of a doubly-stochastic Poisson process. We address the problem
from two different approaches. First, we relate this problem to the computation of the
first n derivatives of the Hazard process cumulant generating function. As shown below,
the result is written in closed-form in terms of Bell polynomials on the aforementioned
derivatives —see [6, 12, 16] for details on these polynomials.
Theorem 1.1. For 0 t < T, denote the cumulant generating function of Λ by
T
≤
Ψ(u) := logE[exp iuΛ ].
T t
{ }|F
If Λ has a finite conditional n-th moment (i.e., E[Λn ] < ), then the following
T T|Ft ∞
equation holds true
n−1eΨ(i) ∂Ψ ∂kΨ
P(τ > T ) = 1 B (i),..., (i) , (1.1)
n |Ft {τn>t} k!ik k ∂u ∂uk
k=0 (cid:18) (cid:19)
X
where B is the k-th Bell polynomial.
k
In light of this result, two considerations are in order. On the one hand, it is desirable
to consider a model for (Λ ) having a cumulant generating function Ψ being analytic
t t≥0
(aroundi),sothatarbitraryjumpsofthedoubly-stochasticPoissonprocesscanbehandled.
Ontheotherhand, itisstraightforward tocompute(1.1)inclosed-formgivenatratactable
expression for the cumulant generating function Ψ. See examples in Section 2.
As a second approach, we compute the aforementioned distributions direcly, by means of
the Malliavian calculus. For this approach we consider a strictly positive pure-jump Lévy
process (L ) with Lévy measure ν, and having moments of all orders —see [1, 17] and
t t≥0
[7] for a general exposition about Lévy processes and Malliavin calculus. We then assume
that the Hazard process is of the form
t
Λ = σ(s,z)N(ds,dz), t 0, (1.2)
t
ˆ0 ˆR0 ≥
e
where N is the compensated Poisson random measure associated (L ) , and σ is a de-
t t≥0
terministic function, integrable with respect to N. Assume further that F is given by the
e
natural filtration generated by the driving Lévy process (L ) . In this setting, we have
t t≥0
e
the following result.
2
Theorem 1.2. The conditional distribution of the n-th jump of doubly-stochastic Poisson
process with Hazard process satisfying (1.2) is given by
n−1 k j
Λ
P(τ > T )= 1 eΛt t m (t) ,
n |Ft {τn>t}  j!(k j)! k−j 
k=0j=0 −
XX
 
where the quantities m ,m ,...,m are given recursively according to
0 1 n
T
m (t) := exp e−σ(s,z) 1+σ(s,z) dsν(dz) , (1.3)
0 (cid:26)ˆt ˆR0h − i (cid:27)
and for r 1
≥
T
m (t) = m (t) (e−σ(s,z) 1)σ(s,z)dsν(dz) (1.4)
r+1 r ˆt ˆR0 −
r r T
+ m (t) e−σ(s,z)σk+1(s,z)dsν(dz).
k=1(cid:18)k(cid:19) r−k ˆt ˆR0
X
The rest of the paper is organized as follows. In Section 2 we present relevant examples
appearing the literature. Finally in Section 3 we provide the proofs of our results.
Let us remark that eventhough our study is motivated by the valuation of defaultable
claims, our results can potentially be also used in other areas; see for instance [2, 14, 21]
and references therein.
2 Examples
In many traditional models (e.g., [8]) the Hazard processes (Λ ) is assumed to be abso-
t t≥0
lutely continuous with respect to the Lebesgue measure, that is,
t
Λ := λ ds, t 0, (2.1)
t ˆ s ≥
0
where the process (λ ) is usually refer to as the hazard rate, and it is seen as the instan-
t t≥0
taneous rate of default in the credit risk context. The following two examples show how to
use Theorem using two prominent particular cases for the hazard rate —and consequently
for the Hazard process.
Example 2.1. The integrated square-root process (ΛintSR) (see [9]) defined by means
t t≥0
of (2.1) where the hazard rate is given by the solution of
dλSR = ϑ(κ λSR)dt+σ λSRdW ,
t − t t t
q
3
where (W ) is a Brownian motion, and we assume σ > 0 and ϑκ σ2 in order to
t t≥0
≥
ensure that (λSR) remains positive. Take now F as the natural filtration generated by
t t≥0
(W ) . Itis well-known that the correspondent Hazard process has an analytic cumulant
t t≥0
generating function given by
ΨintSR(u) := A(u,T t)+λSRB(u,T t), T t 0,
t
− − ≥ ≥
where the functions A and B are given by
2ϑκ 2γe12(γ+ϑ)(T−t) 2γ(e−γ(T−t) 1)
A(u,T t) = log , and B(u,T t)= −
− σ2 (γ +ϑ)e−γ(T−t) 2γ! − (γ +ϑ)e−γ(T−t) 2γ
− −
with γ := γ(u) := √ϑ2 2iuσ2. The simplicity of ΨintSR allows to compute its par-
−
tial derivatives involved in (1.1). And finally we can use the n-th Bell polynomial B
n
characterization given by
n−1 x n−1 x n−1 x n−1 x n−1 x
0 1 1 2 2 3 ··· n−2 n−1 n−2 n
1 n−2 x n−2 x n−2 x n−2 x
(cid:0) −(cid:1) (cid:0) 1 (cid:1) 1 (cid:0) 1 (cid:1) 2 ··· (cid:0)n−3(cid:1) n−2 n(cid:0)−2 (cid:1)n−1
0 1 n−3 x n−3 x n−3 x
Bn(x1,...,xn) := det .. (cid:0) −..(cid:1) (cid:0) 1..(cid:1) 1 ··· (cid:0)n−4(cid:1).. n−3 (cid:0)n−3(cid:1).. n−2,
 . . (cid:0) .(cid:1) (cid:0) (cid:1). (cid:0) (cid:1). 
 
 0 0 0 1 x 1 x 
 ··· 0 1 1 2 
 0 0 0 1 0 x 
 ··· (cid:0)−(cid:1) (cid:0)0(cid:1) 1 
 
(cid:0) (cid:1)
where in each column the remaining entries below the 1 are equal to zero. For instance,
−
one can easily see that the first three Bell polynomials are B (x ) = x , B (x ,x ) =
1 1 1 2 1 2
x2+x and B (x ,x ,x ) = x3+3x x +x .
1 2 3 1 2 3 1 1 2 3
Example2.2. Theintegrated non-GaussianOrnstein-Uhlenbeck processes (see[3])defined
by means of
1 1 t
ΛintOU := (1 e−ϑt)λ + (1 e−ϑ(t−s))dL , t 0, (2.2)
t ϑ − 0 ϑ ˆ − ϑs ≥
0
where ϑ,λ > 0 are free parameters, λ being random, and (L ) a non-decreasing pure-
0 0 t t≥0
jump positive Lévy process. Equivalently, we can consider again the model in (2.1) where
this time (λ ) is given by the solution of
t t≥0
dλ = ϑλ dt+dL , λ > 0.
t t ϑt 0
−
An interesting property of this Hazard rate process is that it has continuous sample paths.
It can be shown that
iuλ T u
ΨintOU(u) := 0(1 e−ϑT)+ϑ k (1 e−ϑ(T−s)) ds, T 0, (2.3)
L
ϑ − ˆ ϑ − ≥
0
(cid:16) (cid:17)
4
where we take = σ(λ ), that is, the σ-algebra generated by λ . Particular cases
0 0 0
F
of interest are the following. On the one hand, we have the so-called Gamma(a,b)-OU
process which is obtained by taking (L ) as a Compound Poisson process
t t≥0
Zt
L = x , t 0,
t n
≥
n=1
X
where (Z ) is a Poisson process with intensity aϑ, and (x ) is a sequence of inde-
t t≥0 n n≥1
pendent identically distributed Exp(b) variables. In this case, the correspondent Hazard
process in (2.2) has a finite number of jumps in every compact time interval. Moreover,
the equation (2.3) becomes
iuλ ϑa b
ΨintOU (u) := 0(1 e−ϑT)+ blog iuT .
Gamma ϑ − iu ϑb b iu(1 e−ϑT)!− !!
− − ϑ −
On the other hand, we have the so-called Inverse-Gausssian(a,b)-OU process (see [13]
and Tompkins and Hubalek (2000)) which is obtained by taking (L ) as the sum of two
t t≥0
independentprocesses,(L = L(1)+L(2)) ,where(L(1)) isanInverse-Gausssian(1a,b)
t t 2 t≥0 t t≥0 2
(2)
process, and (L ) is a Compound Poisson process
t t≥0
Zt
L(2) = b−1 x2, t 0,
t n ≥
n=1
X
where (Z ) is a Poisson process with intensity 1ab, and (x ) is a sequence of in-
t t≥0 2 n n≥1
dependent identically distributed Normal(0,1) variables. In this case, the correspondent
Hazard process in (2.2) jumps infinitely often in every interval. Moreover, the equation
(2.3) becomes
iuλ 2aiu
ΨintOU(u) := 0(1 e−ϑT)+ A(u,T),
IG ϑ − bϑ
where, using c:= 2b−2iuϑ−1, the function A is defined by
−
1 1+c(1 e−ϑT)
A(u,T) := − −
c
p
1 1 1+c(1 e−ϑT) 1
+ arctanh − − arctanh .
√1+c " p c !− (cid:18)√1+c(cid:19)#
In both of the cases above, we can see that the simplicity of Ψ allows to compute (1.1) in
a straightforward way.
This traditional approach reduces the analytical tractability of the model, along with its
parameters calibration. Indeed, suffices to say the Laplace transform of a Hazard process
as in (2.1) is known in closed-form only for a reduced number of Hazard rates models.
5
That is one the reasons why in more recent contributions the modelling focus is set on
the Hazard process itself, without requiring to make a reference to the Hazard rate —see
for intance [4, 13]. In this line, consider a Hazard process (Λ ) as given in (1.2). The
t t≥0
following example provides an explicit computation the quantities involved in Theorem
1.2.
Example 2.3. (CMY Hazard process) In the financial literature, the CMY process —or
one-sided CGMY process [15]—withparameters C,M > 0andY < 1referstothepositive
pure-jump Lévy process (LCMY) having Lévy measure ν given by
t t≥0 CMY
Ce−Mz
ν (z) := 1 .
CMY z1+Y {z>0}
The Gamma process and the Inverse Gaussian process can be seen as particular cases by
taking Y = 0 and Y = 1, respectively, see [18].
2
Consider now a Hazard process of the form
t
ΛCMY := σ(s)dLCMY, t 0.
t ˆ s ≥
0
This is equivalent to take, in (1.2), a function σ is of the form σ(s,z) = zσ(s). Then the
quantities in (1.3) and (1.4) are given by
exp C T Γ(1 Y)MY−1σ(s)+Γ( Y) (M +σ(s))Y MY ds , Y = 0
mC0MY(t) = expnC´tT σ(s)− log 1+ σ(s) ds −, (cid:2) − (cid:3) o Y 6= 0
 t M − M
´
n (cid:16) (cid:17) o

and
T
mCMY(t)= mCMY(t)CΓ(1 Y) σ(s) (M +σ(s))Y−1 MY−1 ds
n+1 n − ˆ −
(cid:20) t (cid:21)
n n (cid:0) T (cid:1)
+ mCMY(t)CΓ(k+1 Y) σk+1(s)(M +σ(s))Y−(k+1)ds.
k n−k − ˆ
k=1(cid:18) (cid:19) t
X
for n 1.
≥
Finally, letusremarkthatwewhenconsideringamodellike(1.2),thequantities appearing
in Theorem 1.1 and Theorem 1.2 can be related according to the following.
Example 2.4. Let the Hazard process (Λ ) be given as in (1.2). It can be seen that in
t t≥0
this case (Lemma 3.2 below) the cumulant generating function is given by
T
Ψ(u) = iuΛ + eiuσ(s,z) 1 iuσ(s,z) dsν(dz).
t
ˆt ˆR0h − − i
6
Consequently, if the function σ has finite moments
T
σk(s,z)dsν(dz) < , k = 1,...,n, (2.4)
ˆ0 ˆR0 ∞
then the n-th derivative of Ψ is given by
1∂Ψ T
= Λ + eiuσ(s,z) 1 σ(s,z)dsν(dz),
t
i ∂u ˆt ˆR0h − i
and
1 ∂kΨ T
= eiuσ(s,z)σk(s,z)dsν(dz), k = 2,...,n.
ik ∂uk ˆt ˆR0
Indeed, theseequations canbeobtained by succesivedifferentiation under theintegral sign
due to the assumption (2.4).
3 Proofs
Let us start by the construction of the doubly-stochastic Poisson process that we shall
consider in what follows.
Let F = ( ) denote our reference filtration; we shall assume that it satisfies the usual
t t≥0
F
conditionsofP-completenessandright-continuity. Letthei.i.d. randomvariablesη ,...,η
1 m
be exponentially distributed with parameter 1, all being independent of . Then the
∞
F
n-th jump of the doubly-stochastic Poisson process with Hazard rate (Λ ) can be char-
t t≥0
acterized as
τ = inf t > 0 : Λ η +...+η . (3.1)
n t 1 n
{ ≥ }
This construction leads to the following expression for the conditional distribution of the
DSPP n-th jump
n−1
1
P(τ > T )= e−ΛT Λj , T 0. (3.2)
n |FT j! T ≥
j=0
X
Indeed, by construction,
n n−1 j
Λ
P(τ > t )= P η > Λ = e−Λt t,
n ∞ j t ∞
| F (cid:12) F ! j!
Xj=1 (cid:12) Xj=0
(cid:12)
(cid:12)
since conditioned to the random variable η +.(cid:12)..+η has an Gamma distribution. The
∞ 1 n
F
result then follows by preconditioning to —recall that (Λ ) is F-adapted.
t t t≥0
F
7
Notice first that by conditioning (3.2) to we get
t
F
n−1
1
P(τ > T ) = E Λj e−ΛT , T t 0. (3.3)
n |Ft j! T Ft ≥ ≥
Xj=0 h (cid:12) i
(cid:12)
(cid:12)
Then the purpose of Theorem 1.1 and Theorem 1.2 is to provide a way to compute the
conditional expectations in the equation above.
3.1 Proof of Theorem 1.1
Let µ stand for the conditional (to ) law of Λ , so that the assumption on the n-th
t t T
F
conditional moment reads
xnµ (dx) < .
t
ˆR ∞
As in the unconditional case (cf. [11, Theorem 13.2]), the condition above ensures that
the conditional characteristic function
ϕ(u;t,T) := E[exp iuΛ ]
T t
{ }|F
has continuous partial derivatives up to order n, and furthermore the following equation
holds true
1 ∂kϕ(u;t,T)
= E ΛkeiuΛT , k = 0,1,...,n.
ik ∂uk T Ft
h (cid:12) i
Now, let us recall that the n-th Bell polynomial(cid:12) B can also be written as
(cid:12) n
n
B (x ,...,x ) = B (x ,...,x ), (3.4)
n 1 n n,k 1 n−k+1
k=0
X
where B stands for th partial (n,k)-th Bell polynomial, i.e., B := 1 and
n,k 0,0
B (x ,...,x ):= n! x1 j1 x2 j1 xn−k+1 jn−k+1,
n,k 1 n−k+1
j !j ! j ! 1! 2! ··· (n k+1)!
X 1 2 ··· n−k+1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:18) − (cid:19)
where the sum runs over all sequences of non-negative indices such that j +j + = k
1 2
···
and j +2j +3j + = n. Using the Bell polynomials we have an expression for the
1 2 3
···
chain rule for higher derivatives:
dn n
f g = (f(k) g)B (g(1),...,g(n+k−1)),
dxn ◦ ◦ n,k
k=0
X
where the superscript denotes the correspondent derivative, i.e., f(k) := dk f and g(k) :=
dxk
dk g, which are assumed to exist. This expression is known as the Riordan’s formula —for
dxk
these results on Bell polynomials we refer to [6, 12, 16].
8
It remains to apply Riordan’s formula to f = exp and g = Ψ in order to get
dn n
ϕ = ϕB (Ψ(1),...,Ψ(n+k−1))= ϕB (Ψ(1),...,Ψ(n)),
dxn n,k n
k=0
X
where the last equivalence follows from (3.4).
3.2 Proof of Theorem 1.2
Fromthismomenton,weshallworkwithastrictlypositivepure-jumpLévyprocess(L )
t t≥0
having a Lévy measure ν satisfying
epzν(dz)<
ˆ ∞
(−ε,ε)
for every ε > 0 and certain p > 0. This condition implies in particular that (L ) have
t t≥0
moments of all orders, and the polynomials are dense in L2(dt ν). Notice that this
×
condition is always satisfied if the Lévy measure has compact support.
In other to prove the corollary we need the following.
3.2.1 Preliminaries on Malliavin calculus via chaos expansions
Let us now introduce basic notions of Malliavin calculus for Lévy processes which we shall
use as a framework. Here we mainly follow [7].
For every T > 0, let 2((dt ν)n) := 2(([0,T] R )n) be the space of determinisitic
LT × L × 0
functions such that
1
2
f := f2(t ,z ,...,t ,z )dt ν(dz ) dt ν(dz ) < ,
k kL2T((dt×ν)n) ˆ([0,T]×R0)n 1 1 n n 1 1 ··· n n ! ∞
andsuch that they arezero over k-diagonal sets, see[19, Remark 2.1]. The symmetrization
f˜of f is defined by
1
f˜(t ,z ,...,t ,z ) := f(t ,z ,...,t ,z ),
1 1 n n n! σ(1) σ(1) σ(n) σ(n)
σ
X
where the sum runs over all the permutations σ of 1,...,n . For every f in the subspace
{ }
of symmetric functions, 2((dt ν)n) := f 2((dt ν)n) : f = f˜ , we define the
LT × { ∈ L × }
n-fold iterated integral of f by
e
T t2
I (f) := n! f(t ,z ,...,t ,z )N(dt ,dz ) N(dt ,dz ).
n ˆ0 ˆR0···ˆ0 ˆR0 1 1 n n 1 1 ··· n n
e e
9
For constant values f R we set I (f ) := f . In these terms, the Wiener-Itô chaos
0 0 0 0
∈
expansion for Poisson random measures, due to [10], states that every -measurable
T
F
random variable F 2(P) admits a representation
∈ L
∞
F = I (f )
n n
n=0
X
viaaunique sequence of elements f 2((dt ν)n). Invirtueof this result, each random
n ∈ LT ×
field (Xt,z)(t,z)∈[0,T]×R0 has an expassion
e
∞
X = I (f (,t,z)), f (,t,z) 2((dt ν)n),
t,z n n n T
· · ∈ L ×
n=0
X
e
provided, ofcourse, thatX is -measureblewithE[X2 ]< forall(t,z)in[0,T] R .
t,z T t,z 0
F ∞ ×
Now we are in position to define the Skorohod integral and the Malliavin derivative.
Definition. The random field (Xt,z)(t,z)∈[0,T]×R0 belongs to Dom(δ) if
∞
2
(n+1)! f˜ <
n
L2((dt×ν)n) ∞
nX=0 (cid:13) (cid:13) T
(cid:13) (cid:13)
(cid:13) (cid:13)
and has Skorohod integral with respect to N
T e ∞
δ(X) = X N(δt,dz) := I (f˜ ).
ˆ ˆ t,z n+1 n
0 R0 n=0
X
e
Definition. Let D be the stochastic Sobolev space consisting of all -measureble ran-
1,2 T
F
dom variables F 2(P) with chaos expansion F = ∞ I (f ) satisfying
∈ L n=0 n n
∞ P
2
F := (n)n! f˜ < .
D n
k k 1,2 L2((dt×ν)n) ∞
Xn=1 (cid:13) (cid:13) T
(cid:13) (cid:13)
(cid:13) (cid:13)
For every F D its Malliavin derivative is defined as
1,2
∈
∞
D F := nI (f (,t,z)).
t,z n−1 n
·
n=0
X
Let us mention here that Dom(δ) 2(P dt ν), δ(X) 2(P), D 2(P) and
1,2
⊆ L × × ∈ L ⊂ L
DF 2(P dt ν).
∈ L × ×
We have the following theorems are central for the results below; for their proof and more
details we refer to [7] and references therein.
10