Looking for martingales associated to a self-decomposable law

We construct martingales whose 1-dimensional marginals are those of a centered self-decomposable variable multiplied by some power of time t . Many examples involving quadratic functionals of Bessel processes are discussed.


Introduction, Motivation 1.Notation
We first introduce some notation which will be used throughout our paper.
If A and B are two random variables, A d = B means that these variables have the same law.
If (X t , t ≥ 0) and (Y t , t ≥ 0) are two processes, (X t ) (1.d) = (Y t ) means that the processes (X t , t ≥ 0) and (Y t , t ≥ 0) have the same one-dimensional marginals, that is, for any fixed t, X t d = Y t .
If (X t , t ≥ 0) and (Y t , t ≥ 0) are two processes, (X t ) (d) = (Y t ) means that the two processes are identical in law.
All random variables and processes which will be considered are assumed to be real valued.

PCOC's
In a number of applied situations involving randomness, it is a quite difficult problem to single out a certain stochastic process (Y t , t ≥ 0), or rather its law, which is coherent with the real-world data.
In some cases, it is already nice to be able to consider that the one-dimensional marginals of (Y t ) are accessible.The random situation being studied may suggest, for instance, that: (i) there exists a martingale (M t ) such that (this hypothesis may indicate some kind of "equilibrium" with respect to time), (ii) there exists H > 0 such that (there is a "scaling" property involved in the randomness).
It is a result due to Kellerer [14] that (i) is satisfied for a given process (Y t ) if and only if this process is increasing in the convex order, that is: it is integrable (∀t ≥ 0, E[|Y t |] < ∞), and for every convex function ϕ : R −→ R, is increasing.
In the sequel, we shall use the acronym PCOC for such processes, since, in French, the name of such processes becomes: Processus Croissant pour l'Ordre Convexe.
A martingale (M t ) which has the same one-dimensional marginals as a PCOC is said to be associated to this PCOC.Note that several different martingales may be associated to a given PCOC.We shall see several striking occurrences of this in our examples.We also note that Kellerer's work [14] does not contain a constructive algorithm for a martingale associated to a PCOC.
On the other hand, Roynette [23] has exhibited two large families of PCOC's, denoted by (F 1) and (F 2): The family (F 1) consists of the processes 1 t t 0 N s ds, t ≥ 0 , and the family (F 2) consists of the processes t 0 (N s − N 0 ) ds, t ≥ 0 , where (N s ) above denotes any martingale such that:

Self-decomposability and Sato processes
It is a non-trivial problem to exhibit, for either of these PCOC's, an associated martingale.We have been able to do so concerning some examples in (F 1), in the Brownian context, with the help of the Brownian sheet ( [9]), and in the more general context of Lévy processes, with the help of Lévy sheets ( [10]).Concerning the class (F 2), note that, considering a trivial filtration, it follows that (t X ), where X is a centered random variable, is a PCOC.Even with this reduction, it is not obvious to find a martingale which is associated to (t X ).= U 1 .This process (U t ), which is unique in law, will be called the H-Sato process associated to Y 1 .Clearly, then We note that the self-decomposability property has also been used in Madan-Yor [18, Theorem 4,Theorem 5] in a very different manner than in this paper, to construct martingales with onedimensional marginals those of (t X ).

Examples
We look for some interesting processes in the class (S), in a Brownian framework.
Example 1 A most simple example is the process: and the RHS is a centered (3/2)-Sato process.Moreover the process (Y t ) obviously belongs to the class (F 2).

Example 2
The process and more generally the process where (R N (s)) is a Bessel process of dimension N > 0 starting from 0, belongs to the family (F 2) and is 2-self-similar.We show in Section 4 that the centered 2-Sato process: where ( s ) is the local time in 0 of the Brownian motion B, and Example 3 We extend our discussion of Example 2 by considering, for N > 0 and K > 0, the process: Then, in Section 5, a centered (2/K)-Sato process (and hence a martingale) associated to the PCOC V N ,K may be constructed from the process of first hitting times of a perturbed Bessel process R K,1− N 2 as defined and studied first in Le Gall-Yor [16; 17] and then in Doney-Warren-Yor [6].We remark that, if 0 < K < 2, then the process

Example 4
In Section 6, we generalize again our discussion by considering the process N is a PCOC to which we are able to associate two very different martingales.The first one is purely discontinuous and is a centered 1-Sato process, the second one is continuous.The method of proof is based on a Karhunen-Loeve type decomposition (see, for instance, [5] and the references therein, notably Kac-Siegert [13]).For this, we need to develop a precise spectral study of the operator K (µ) defined on L 2 (µ) by :

Organisation of the paper
We now present more precisely the organisation of our paper: in Section 2, we recall some basic results about various representations of self-decomposable variables, and we complete the discussion of Subsection 1.3 above; in Section 3, we consider the simple situation, as in Subsection 1.3, where Y t = R 2 N (t), for R N a Bessel process of dimension N starting from 0; -the contents of Sections 4, 5, 6 have already been discussed in the above Subsection 1.4.
We end this introduction with the following (negative) remark concerning further selfdecomposability properties for squared Bessel processes: indeed, it is well-known, and goes back to Shiga-Watanabe [25], that R 2 N (•), considered as a random variable taking values in C(R + , R + ) is infinitely divisible.Furthermore, in the present paper, we show and exploit the selfdecomposability of (0,∞) R 2 N (s) dµ(s) for any positive measure µ.It then seems natural to wonder about the self-decomposability of R 2 N (•), but this property is ruled out: the 2-dimensional vectors ) are not self-decomposable, as an easy Laplace transform computation implies.

Sato processes and PCOC's 2.1 Self-decomposability and Sato processes
We recall, in this subsection, some general facts concerning the notion of self-decomposability.We refer the reader, for background, complements and references, to Sato [24, Chapter 3].
A random variable X is said to be self-decomposable if, for each u with 0 < u < 1, there is the equality in law: for some variable X u independent of X .On the other hand, an additive process (U t , t ≥ 0) is a stochastically continuous process with càdlàg paths, independent increments, and satisfying U 0 = 0.An additive process (U t ) which is H-self-similar for some H > 0, meaning that, for each c > 0, = (c H U t ), will be called a Sato process or, more precisely, a H-Sato process.
The following theorem, for which we refer to Sato's book [24, Chapter 3, Sections 16-17], gives characterizations of the self-decomposability property.
Theorem 2.1.Let X be a real valued random variable.Then, X is self-decomposable if and only if one of the following equivalent properties is satisfied:
2) There exists a Lévy process 3) For any (or some) H > 0, there exists a H-Sato process In 2) (resp.3)) the Lévy process (C s ) (resp. the H-Sato process (U t )) is uniquely determined in law by X , and will be said to be associated with X .We note that, if X ≥ 0, then the function h vanishes on (−∞, 0), (C s ) is a subordinator and (U t ) is an increasing process.
The relation between (C s ) and (U t ) was made precise by Jeanblanc-Pitman-Yor [11, Theorem 1]: ) is a H-Sato process, then the formulae: define two independent and identically distributed Lévy processes from which (U t , t ≥ 0) can be recovered by: In particular, the Lévy process associated with the self-decomposable random variable U 1 is s/H , s ≥ 0.

Sato processes and PCOC's
We recall (see Subsection 1.2) that a PCOC is an integrable process which is increasing in the convex order.On the other hand, a process (V t , t ≥ 0) is said to be a 1-martingale if there exists, on some filtered probability space, a martingale 2, the converse holds true (Kellerer [14]).
The following proposition, which is central in the following, summarizes the method sketched in Subsection 1.3.
Then the process is a PCOC, and an associated martingale is where (U t ) denotes the H-Sato process associated with Y 1 according to Theorem 2.1.
3 About the process (R 2 N (t), t ≥ 0) In the sequel, we denote by (R N (t), t ≥ 0) the Bessel process of dimension N > 0, starting from 0.

Self-decomposability of R 2 N (1)
As is well-known (see, for instance, Revuz-Yor [22, Chapter XI]) one has In other words, where, for a > 0, γ a denotes a gamma random variable of index a.Now, the classical Frullani's formula yields: Then, R 2 N (1) satisfies the property 1) in Theorem 2.1 with and it is therefore self-decomposable.
The process 3, the process is a PCOC, and an associated martingale is where (U N t ) denotes the 1-Sato process associated with R 2 N (1) by Theorem 2.1.We remark that, in this case, the process (V N t ) itself is a continuous martingale and therefore obviously a PCOC.In the following subsections, we give two expressions for the process (U N t ).As we will see, this process is purely discontinuous with finite variation; consequently, the martingales (V N t ) and (M N t ), which have the same one-dimensional marginals, do not have the same law.

Expression of (U N t ) from a compound Poisson process
We denote by (Π s , s ≥ 0) the compound Poisson process with Lévy measure: This process allows to compute the distributions of a number of perpetuities ∞ 0 e −Λ s dΠ s where (Λ s ) is a particular Lévy process, independent of Π; see, e.g., Nilsen-Paulsen [20].In the case Λ s = r s, the following result seems to go back at least to Harrison [8].
Proof a) First, recall that for a subordinator (τ s , s ≥ 0) and f : R + −→ R + Borel, there is the formula: where Φ is the Lévy exponent of (τ s , s ≥ 0).Consequently, a slight amplification of this formula is: Then, as a consequence of the previous formula with µ = 2, f (s) = λ e −s , A = N /2, we get: Consequently, which proves the result.

Expression of (U N t ) from the local time of a perturbed Bessel process
There is by now a wide literature on perturbed Bessel processes, a notion originally introduced by Le Gall-Yor [16; 17], and then studied by Chaumont-Doney [3], Doney-Warren-Yor [6].We also refer the interested reader to Doney-Zhang [7].
We first introduce the perturbed Bessel process (R 1,α (t), t ≥ 0) starting from 0, for α < 1, as the nonnegative continuous strong solution of the equation where L t (R 1,α ) is the semi-martingale local time of R 1,α in 0 at time t, and It is clear that the process R 1,0 is nothing else but the Bessel process R 1 (reflected Brownian motion).We also denote by T t (R 1,α ) the hitting time: We set Finally, in the sequel, we set Proposition 3.2.For any α < 1, the process (L T t (R 1,α ), t ≥ 0) is a 1-Sato process, and we have

Proof
By the uniqueness in law of the solution to the equation ( 1), the process R 1,α is (1/2)-self-similar.
As a consequence, the process On the other hand, the pair the fact that (L T t (R 1,α ), t ≥ 0) is an additive process follows from the strong Markov property.
Finally, we need to prove: For the remainder of the proof, we denote R 1,α N by R, and From this formula, we learn that the martingale u∧T t 0 exp(−λ L s ) dB s , u ≥ 0 is bounded; hence, by applying the optional stopping theorem, we get: we obtain: Therefore, ], which proves the desired result.

A class of Sato processes
Let ( t , t ≥ 0) be the local time in 0 of a linear Brownian motion (B t , t ≥ 0) starting from 0. We denote, as usual, by (τ t , t ≥ 0) the inverse of this local time: Then the process A ( f ) defined by: is an integrable additive process.Furthermore,

Proof
Assume first that f is nonnegative.Then, By the theory of excursions (Revuz-Yor [22, Chapter XII, Proposition 1.10]) we have where n denotes the Itô measure of Brownian excursions and V ( ) denotes the life time of the excursion .The entrance law under n is given by: n( s ∈ dx; s < V ( )) = (2πs 3 ) −1/2 |x| exp(−x 2 /(2s)) dx. Therefore The additivity of the process A ( f ) follows easily from the fact that, for any t ≥ 0, (B τ t +s , s ≥ 0) is a Brownian motion starting from 0, which is independent of τ t (where ( u ) is the natural filtration of B).
Corollary 4.1.1.We assume that f is a Borel function on R + × R + satisfying (2) and which is mhomogeneous for m > −2, meaning that Then the process A ( f ) is a (m + 2)-Sato process.

Proof
This is a direct consequence of the scaling property of Brownian motion.

A particular case
Let N > 0. We denote by A (N ) the process

By Proposition 4.1, (A (N )
t ) is an integrable process and We now consider the process Y N defined by Theorem 4.2.The process A (N ) is a 2-Sato process and The proof can be found in Le Gall-Yor [17].Nevertheless, for the convenience of the reader, we give again the proof below.A more general result, based on Doney-Warren-Yor [6], shall also be stated in the next section.

Proof
In this proof, we adopt the following notation: (B t ) still denotes a standard linear Brownian motion starting from 0, S t = sup 0≤s≤t B s and σ t = inf{s; B s > t}.Moreover, for a < 1 and t ≥ 0, we set: Consequently, −a S t = α M t (H a ), with α = −a/(1 − a).

Proof
Since a < 1, we have, for 0 ≤ s ≤ t, Moreover, there exists s t ∈ [0, t] such that B s t = S t and therefore S s t = S t .Hence, B s t − a S s t = (1 − a) S t .

Proof
a) Since Z a t is a time of increase of the process we get: D a t = H a Z a t ≥ 0.Moreover, since the process (H a ) + is obviously Z a -continuous, the process D a is continuous.b) By Tanaka's formula,

Identification of the Sato process associated to Y N ,K
We denote, for N > 0 and K > 0, by Y N ,K the process: We also recall the notation: Theorem 5.2.The process

Proof
In the following proof, we denote R K,α N simply by R, and we set T t and M t for, respectively, T t (R) and M t (R).
The first part of the statement follows from the (1/2)-self-similarity of R and from the strong Markovianity of (R, M ), taking into account that, for any t ≥ 0, By occupation times formula, we deduce from 1) in Theorem 5.1, Using then 2) in Theorem 5.1, we obtain: By change of variable, the last integral is equal to Y N ,K (1), and hence, The final result now follows by self-similarity.
Corollary 5.2.1.The process is a PCOC, and an associated martingale is which is a centered (2/K)-Sato process.
Finally, we have proven, in particular, that for any ρ > −2 and any N > 0, the random variable is self-decomposable.This result will be generalized and made precise in the next section, using completely different arguments.
6 About the random variables R 2 N (s) dµ(s) In this section, we consider a fixed measure µ on R * + = (0, ∞) such that

Spectral study of an operator
We associate with µ an operator K (µ) on E = L 2 (µ) defined by where ∧ denotes the infimum.

Proof
As a consequence of the obvious inequality: and therefore K (µ) is a Hilbert-Schmidt operator.
On the other hand, denoting by (•, •) E the scalar product in E, we have: where B is a standard Brownian motion starting from 0. This entails that K (µ) is nonnegative symmetric.
Lemma 6.2.Let λ ∈ R. Then λ is an eigenvalue of K (µ) if and only if λ > 0 and there exists f ∈ L 2 (µ), f = 0, such that: ii) f admits a representative which is absolutely continuous on R + , f admits a representative which is right-continuous on R * + ; (In the sequel, f and f respectively always denote such representatives.)

Proof
Let f ∈ L 2 (µ) and g = K (µ) f .We have, for µ-a.e.t > 0, Thus g admits a representative (still denoted by g) which is absolutely continuous on R + and g(0) = 0.Moreover, g admits a representative which is right-continuous on R * + and is given by: In particular Hence: lim t→∞ g (t) = 0.
Besides, (7) entails: Consequently, 0 is not an eigenvalue of K (µ) and the "only if" part is proven.
We note that, since 0 is not an eigenvalue of K (µ) , K (µ) is actually a positive symmetric operator.On the other hand, by the previous proof, the functions f ∈ L 2 (µ), f = 0, satisfying properties i),ii),iii) in the statement of Lemma 6.2, are the eigenfunctions of the operator K (µ) corresponding to the eigenvalue λ > 0.

Proof
This is a direct consequence of (8).
Lemma 6.4.Let f 1 and f 2 be eigenfunctions of K (µ) with respect to the same eigenvalue.Then, Proof By ( 5), ( f 1 f 2 − f 1 f 2 ) = 0 in the sense of distributions on R * + .By right-continuity, there exists C ∈ R such that Letting t tend to ∞, we deduce from Lemma 6.3 that C = 0. Lemma 6.5.Let f be a solution of (5) with λ > 0, and let a > 0. We assume as previously that f (resp.f ) denotes the representative which is absolutely continuous (resp.right-continuous) on R * + .If f (a) = f (a) = 0, then, for any t ≥ a, f (t) = 0.

Proof
This lemma is quite classical if the measure µ admits a continuous density with respect to the Lebesgue measure (see, for instance, [4]).The proof may be easily adapted to this more general case.
We are now able to state the main result of this section.Theorem 6.6.The operator K (µ) is a positive symmetric compact operator whose all eigenvalues are simple, i.e. the dimension of each eigenspace is 1.

Proof
It only remains to prove that the eigenvalues are simple.For this purpose, let λ > 0 be an eigenvalue and let f 1 and f 2 be eigenfunctions with respect to this eigenvalue.Let a > 0 with µ({a}) = 0.By Lemma 6.4, Hence, there exist c 1 and c 2 with c 2 1 + c 2 2 > 0 such that, setting f = c 1 f 1 + c 2 f 2 , we have By Lemma 6.5, f (t) = 0 for any t ≥ a. But, since µ({a}) = 0, f is also left-continuous at a.Then, we may reason on (0, a] as on [a, ∞) and therefore we also have f (t) = 0 for 0 < t ≤ a. Finally, which proves the result.
In the following, we denote by λ 1 > λ 2 > • • • the decreasing (possibly finite) sequence of the eigenvalues of K (µ) .Of course, this sequence depends on µ, which we omit in the notation.The following corollary plays an essential role in the sequel.

Examples
In this subsection, we consider two particular types of measures µ.By the previous study, the sequence of eigenvalues of K (µ) is finite if and only if the space L 2 (µ) is finite dimensional, that is if µ is of the above form.In this case, the eigenvalues of K (µ) are the eigenvalues of the matrix (m i, j ) 1≤i, j≤n with m i, j = a i a j t i∧ j .
In particular, by the previous study, such a matrix has n distinct eigenvalues, which are > 0.

Representation of B 2 s dµ(s)
We again consider the general setting defined in Subsection 6.1, the notation of which we keep.
In this subsection, we study the random variable The use of the operator K (µ) and of its spectral decomposition in the type of study we develop below, is called the Karhunen-Loeve decomposition method.It has a long history which goes back at least to Kac-Siegert [12; 13].We also refer to the recent paper [5] and to the references therein.

Proof
We deduce from Corollary 6.6.1, by the Bessel-Parseval equality, that: Taking expectations, we get We set, for n ≥ 1, Then (Γ n , n ≥ 1) is a Gaussian sequence and where δ n,m denotes Kronecker's symbol.Hence, the result follows.

Proof
This is a direct consequence of the previous theorem, taking into account that, if Γ is a normal variable, then

Proof
We saw in Subsection 3.1 that R 2 N (1) satisfies the property 1) in Theorem 2.1 with and it is therefore self-decomposable.Using then the representation (10) of Y (µ) N , we obtain the desired result.
As a consequence, following Bondesson [1], we see that Y (µ) N is a generalized gamma convolution (GGC) whose Thorin measure is the discrete measure: δ 1/2λ n .

Particular case
We consider here, as in Section 5, the particular case: It is known (see Kent [15] and, for instance, Borodin-Salminen [2, formula 2.0.1, p. 387]) that where I ν denotes the modified Bessel function: .
, (B t ) denoting a standard linear Brownian motion starting from 0. (The strong solution property has been established in Chaumont-Doney [3].)