Fluctuation theory for spectrally positive additive Lévy fields

HAL Id: hal-02427665

https://hal.archives-ouvertes.fr/hal-02427665

Preprint submitted on 3 Jan 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

fields

Loïc Chaumont, Marine Marolleau

To cite this version:

Loïc Chaumont, Marine Marolleau. Fluctuation theory for spectrally positive additive Lévy fields.

2020. �hal-02427665�

LOÏC CHAUMONT AND MARINE MAROLLEAU

Abstract. A spectrally positive additive Lévy field is a multidimensional field ob- tained as the sum X_t = X⁽¹⁾_t

1 + X⁽²⁾_t

2 +· · ·+ X^(d)_t

d , t = (t₁, . . . , t_d) ∈R^d+, where X^(j) =

t(X^1,j, . . . , X^d,j),j = 1, . . . , d, are dindependentR^d-valued Lévy processes issued from 0, such that X^i,j is non decreasing for i 6= j and X^j,j is spectrally positive. It can also be expressed as X_t = Xt1, where1 =^t(1,1, . . . ,1) and Xt = (X_t^i,j

j )_1≤i,j≤d. The main interest of spaLf’s lies in the Lamperti representation of multitype continuous state branching processes. In this work, we study the law of the first passage timesTrof such fields at levels −r, where r∈R^d+. We prove that the field{(Tr,XT_r),r∈R^d+} has sta- tionary and independent increments and we describe its law in terms of this of the spaLf X. In particular, the Laplace exponent of(T_r,XTr)solves a functional equation leaded by the Laplace exponent of X. This equation extends in higher dimension a classical fluctuation identity satisfied by the Laplace exponents of the ladder processes. Then we give an expression of the distribution of{(T_r,XTr),r∈R^d+}in terms of the distribution of{Xt,t∈R^d+} by the means of a Kemperman-type formula, well-known for spectrally positive Lévy processes.

1. Introduction

A spectrally positive, additive Lévy field (spaLf) is defined by X_t :=

d

X

j=1

X_t^i,j_j , i= 1, . . . , d

!

= X⁽¹⁾_t1 +· · ·+ X^(d)_t

d , t = (t₁, . . . , t_d)∈R^d+,

where X^(j) =^t(X^1,j, . . . , X^d,j), j = 1, . . . , d, are d independentR^d-valued Lévy processes such that X^i,j are non decreasing for i 6= j and X^j,j is spectrally positive. SpaLf’s can be considered as (non-trivial) extensions in higher dimension of spectrally positive Lévy processes and the purpose of this article is to develop fluctuation theory for such random fields. The particular pathwise features of spaLf’s allow us to define their first passage times T_r = (Tr⁽¹⁾, . . . , Tr^(d)) at multivariate levels −r ∈ R^d− as the smallest of the indices t = (t₁, . . . , t_d) satisfying X_t = −r in the usual partial order of R^d. The distribution of the variables (T_r,XTr), r ∈ R^d+ can then be related to the distribution of the field {X^t,t ∈ R^d+}, where X^t = (X_t^i,j

j )_1≤i,j≤d. In doing so we obtain some fluctuation-type identities in the general framework of multivariate stochastic fields. These results pro- vide an intrinsic motivation for the present study that can be considered in the line of several works on additive Lévy processes from Khoshnevisan and Xiao, see for instance [9].

Date: December 22, 2019.

2010Mathematics Subject Classification. 60G51.

Key words and phrases. Additive Lévy field, multivariate first hitting time, fluctuation theory, Kem- perman’s formula.

1

The original motivation comes from an extension of the Lukasiewicz-Harris coding of Bienaymé-Galton-Watson trees through downward skip free random walks. In [7], the au- thors proved that multitype Bienaymé-Galton-Watson trees can be coded by multivariate random fields

d

X

j=1

S_n^i,j

j, i= 1, . . . , d

!

, where S^(j) = (S^1,j, . . . , S^d,j), j = 1, . . . , d,

are d independent Z^d-valued random walks such that S^i,j are non decreasing for i 6= j and S^j,j is downward skip free. These random fields are the discrete time counterparts of spaLf’s which suggests the possibility of coding continuous multitype branching trees in an analogous way. It seems quite complicated to achieve such a result as the notion of continuous multitype tree is not clearly defined for general mechanisms. However, reducing the analysis to processes rather than trees, one may still consider the Lamperti representation which provides a pathwise relationship between branching processes and their mechanism. This representation can be extended to continuous time multitype branching processes by using spaLf’s. It was done in [5] for the discrete valued case and in [4] and [8] for the continuous one. More specifically, let Z = (Z⁽¹⁾, . . . , Z^(d)) be a continuous time multitype branching process issued from r ∈ R^d+. Then Z can be represented as the unique pathwise solution of the following equation,

(Z_t⁽¹⁾, . . . , Z_t^(d)) = r +

d

X

j=1

XR^1,jt

0Zs^(j)ds, . . . ,

d

X

j=1

XR^d,jt 0Zs^(j)ds

!

, t≥0,

where X^(j), j = 1, . . . , d, are Lévy processes as described above. Now recall that 0 is an absorbing state for Z. Then it follows from the above equation that the path of Z up to its first passage time at 0 is entirely determined by the path of the spaLf

{X_t,t∈R^d+}=







d

X

j=1

X_t^i,j_j

!

1≤i,j≤d

,t∈R^d+







up to its first passage time T_r at level−r. This fact which is plain in the case d= 1 will be proved in the general case in the upcoming paper [6], where extinction of continuous time multitype branching processes is characterized through path properties of spaLf’s.

The next section consists in an important preliminary lemma for deterministic paths whose aim is to prove the existence of first passage times of spaLf’s and to derive their first basic properties. Then in Section 3 we will turn our attention to the law of these first passage times. In particular we will prove that in analogy with the one dimensional case, their Laplace exponent is the inverse of the Laplace exponent of the spaLf. The situation for d ≥ 2 differs significantly from the one dimensional case as we first need to give necessary and sufficient conditions for the multivariate hitting times Tr to be finite on each coordinate, with positive probability, for all r ∈ R^d+. (When d = 1, this is equivalent to saying that the spectrally positive Lévy process is not a subordinator.) Another fundamental difference concerns the matrix valued fieldXTr which is simply equal to r on the set T_r < ∞, when d = 1. In Section 4 we will focus on the law of the field (T_r,XTr) and prove that its Laplace exponent solves a functional equation leaded by the Laplace exponent of the spaLf X. This equation, see (4.18) in Theorem 4.1 below, can be compared to the classical Wiener-Hopf factorization involving the ladder processes of spectrally positive Lévy processes. Then in Theorem 4.2 the distribution of(T_r,XTr)will

be fully characterized in terms of the distribution of the original stochastic fieldX, through an extension of Kemperman’s formula, see Corollary VII.3 in [3]. More specifically, our result states that the measure

P(T_r∈dt, X_t^i,j_j ∈dx_ij,1≤i, j ≤d) dr is the image of the measure

det(−(x_i,j)i,j∈[d]) t₁t₂. . . t_d

d

Y

j=1

P(X_t^i,j_j ∈dx_ij, i= 1, . . . , d)dt₁. . .dt_d,

through the mapping(t,(x_i,j)_i,j∈(d])7→(t,(x_i,j)_i,j∈[d],−(x_i,j)_i,j∈[d]·1), where1= (1,1, . . . ,1).

In order to prove it, we will use a similar identity recently obtained in [7] and [5] in the discrete time and space settings together with a discrete approximation.

2. A preliminary lemma in the deterministic setting

We use the notation R+ = [0,∞), R+ = [0,∞] and [d] ={1, . . . , d}, where d≥1 is an integer. For s = (s₁, . . . , s_d) and t = (t₁, . . . , t_d) ∈ R^d+, we write s ≤ t if s_i ≤ t_i for all i∈[d] and we writes<t if s≤tand there exists i∈[d]such that s_i < t_i.

Recall that a real valued function x : R+ → R is said to be càdlàg, if it is right continuous on R+ and has left limits on (0,∞). Such a function is said to be downward skip free if for all s≥0, x(s)−x(s−)≥0, where we set x(0−) =x(0). We also say that x has no negative jumps. We will use the notation x_t or x(t)indifferently.

Definition 2.1. We call E_d, the set of matrix valued functions x={(x^i,j_tj)i,j∈[d],t ∈R^d+} such that for all i, j, x^i,j is a càdlàg function and

(i) x^i,j₀ = 0, for all i, j ∈[d],

(ii) for all i∈[d], x^i,i is downward skip free,

(iii) for all i, j ∈[d] such that i6=j, x^i,j is non decreasing.

For s ∈ R

d

+, we denote by [d]_s the set of indices of finite coordinates of s, that is [d]_s = {i∈[d] :s_i <∞}. For i6=j, we set x^i,j(∞) =x^i,j(∞−) = lim

s→∞x^i,j(s).

Definition 2.2. Let x ∈ E_d and r = (r₁, . . . , r_d) ∈R^d+. Then s∈ R^d+ is called a solution of the system (r,x) if it satisfies

(2.1) (r,x) r_i+

d

X

j=1

x^i,j(s_j−) = 0, i∈[d]_s.

(In particular, s = (∞,∞, . . . ,∞) is always a solution of the system(r,x) since[d]s =∅.) Note that in (2.1) it is implicit that P

j∈[d]\[d]s

x^i,j(s_j−) < ∞, for all i ∈ [d]_s, although by definition s_j = ∞, for j ∈ [d]\[d]_s. The next lemma is a continuous time and space counterpart of Lemma 1 in [5]. The proof of the present result follows a similar scheme, however we need to perform it here as it requires more care.

Lemma 2.1. Let x∈ E_d and r = (r₁, . . . , r_d)∈R^d+.

1. There exists a solution s = (s₁, . . . , s_d) ∈ R^d+ of the system (r,x) such that any other solution t of (r,x) satisfies t ≥s. The solution s will be called the smallest solutionof the system (r,x).

2. Let s ands⁰ be the smallest solutions of the systems (r,x) and(r⁰,x), respectively.

If r⁰ ≤r, then s⁰ ≤s. Moreover if (r_n)_n≥0 is non decreasing with lim

n→∞r_n = r then the sequence (s_n)n≥0 of smallest solutions of (r_n,x) satisfies lim

n→∞s_n= s.

3. Letsbe the smallest solution of(r,x). Ifuis such that for all i∈[d]_u,

d

P

j=1

x^i,j(u_j−)

≤ −r_i, then u ≥ s. As a consequence, for all u ∈ R^d+ such that u < s, there is i∈[d] such that

d

P

j=1

x^i,j(u_j−)>−r_i.

4. The smallest solution s of (r,x) satisfies s_i = inf

t:x^i,i_t− = inf

0≤u≤si

x^i,i_u

, for all i∈[d]_s.

Proof. This proof is based on the observation that for each i∈[d], as a function oft, the term

d

P

j=1

x^i,j(t_j) has no negative jumps. Moreover, when t_i is fixed, it is non decreasing.

Let us setv⁽¹⁾_i =ri and for n≥1,

s⁽ⁿ⁾_i = inf{t :x^i,i_t− =−v_i⁽ⁿ⁾} and v⁽ⁿ⁺¹⁾_i =r_i+X

j6=i

x^i,j(s⁽ⁿ⁾_j −),

where inf∅ =∞. Set also s⁽⁰⁾ = 0 and note that [d]_s⁽⁰⁾ = [d]. Then since for i 6= j, the x^i,j’s are positive and non decreasing, we have

s⁽ⁿ⁾ ≤s⁽ⁿ⁺¹⁾ and [d]_s(n+1) ⊆[d]_s(n), n≥0. Let us set s^(∞) = lim

n→∞s⁽ⁿ⁾. Then s^(∞) is the smallest solution of the system (r,x) in the sense which is defined in Lemma 2.1. Indeed, let i ∈ [d]_s^(∞). By definition and since x^i,i has no negative jumps, for all n ≥ 1, x^i,i(s⁽ⁿ⁾_i −) = −v_i⁽ⁿ⁾. Moreover, since the processes t 7→ x^i,j(t−) are left continuous, lim

n→∞x^i,i(s⁽ⁿ⁾_i −) = x^i,i(s^(∞)_i −) and lim

n→∞v_i⁽ⁿ⁾ = r_i+P

j6=i

x^i,j(s^(∞)_j −). Hence (2.1) is satisfied for s^(∞), that isr_i+

d

P

j=1

x^i,j(s^(∞)_j −) = 0, for all i∈[d]_s^(∞). Now let t∈R^d+ satisfying (2.1), that is

(2.2) r_i+X

j6=i

x^i,j(t_j−) +x^i,i(t_i−) = 0, i∈[d]_t.

We can prove by induction thatt≥s⁽ⁿ⁾, for all n≥1. Firstly for (2.2) to be satisfied, we should have ti ≥inf{s :x^i,i(s−) =−ri}, for all i∈[d]t, hencet≥s⁽¹⁾. Now assume that t≥s⁽ⁿ⁾. Then [d]_t ⊆[d]_s(n) and from (2.2), for eachi∈[d]_t,

x^i,i(t_i−) = − r_i+X

j6=i

x^i,j(t_j−)

!

≤ − r_i+X

j6=i

x^i,j(s⁽ⁿ⁾_j −)

! .

Therefore t_i ≥ inf (

s:x^i,i(s−) =− r_i+P

j6=i

x^i,j(s⁽ⁿ⁾_j −)

!)

, so that t ≥ s⁽ⁿ⁺¹⁾ and the first assertion is proved.

Ifr⁰ ≤r, then one can easily prove by induction that, with obvious notation,s⁰⁽ⁿ⁾ ≤s⁽ⁿ⁾ for alln≥1and the first part of assertion2.follows. For the second part, sets⁰ := lim

n→∞s_n.

Then first part of assertion2.yieldss⁰ ≤s. Moreover, from the left continuity of the func- tionst7→x^i,j_t−,ri+

d

P

j=1

x^i,j(s⁰_j−) = 0,i∈[d]s⁰ hences⁰ is a solution of(r,x)and thuss⁰ = s.

Let u∈R^d+, such that

d

P

j=1

x^i,j(u_j−)≤ −r_i, for alli ∈[d]_u and set r_i⁰ =−

d

P

j=1

x^i,j(u_j−).

Since r⁰ ≥ r, it follows from 2. that the smallest solution s⁰ of the system (r⁰,x) is such thats⁰ ≥s. But sinceuis also a solution of(r⁰,x), 1. impliesu≥s⁰ and the first assertion of 3. follows. The second assertion of 3. is a consequence of the first one. Indeed, u <s implies that u≥s is not satisfied.

Assertion 4. follows from the above construction of s = s^(∞). Indeed, if there exists i∈[d]_s and t_i < s_i such thatx^i,i(t_i−)≤x^i,i(s_i−) then

(2.3) X

j6=i

x^i,j(s_j−) +x^i,i(t_i−)≤ X

j∈[d]

x^i,j(s_j−) =−r_i and for all k ∈[d]s\ {i},

(2.4) X

j6=i

x^k,j(s_j−) +x^k,i(t_i−)≤ X

j∈[d]

x^k,j(s_j−) = −r_k.

Then set for all k ∈ [d]_s, r_k⁰ = − P

j6=i

x^k,j(s_j−) +x^k,i(t_i−)

!

and for all k ∈ [d]\[d]_s, r⁰_k =r_k. Let s⁰ be the smallest solution of the system (r⁰,x). From part 2. of the present lemma, sincer⁰ ≥r,s⁰ ≥s. On the other hand, from (2.3), (2.4) and part 3. of the present lemma,s>(s₁, . . . , si−1, t_i, s_i+1, . . . , s_d)≥s⁰ which is a contradiction.

We emphasize that according to our definition, some of the coordinates of the smallest solution of the system (r,x)may be infinite.

3. Fluctuation theory for additive Lévy fields

Vectors ofR^d will be denoted byx = (x₁, . . . , x_d) and e_i = (0, . . . ,0,1,0, . . . ,0)will be the i-th unit vector of R^d+. We will denote by hx,yi, x,y ∈ R^d the usual scalar product on R^d and by |x| the euclidian norm of x. A matrix M = (m_i,j)i,j∈[d] ∈ M_d(R∪ {∞}) is said to be irreducible if for all i, j ∈ [d], there is a sequence i = i₁, i₂, . . . , i_n = j, for some n≥1, such that mi_k,i_k+1 6= 0, for all k= 1, . . . , n−1. For two matrices A and B of Md(R), with columnsa⁽¹⁾, . . . ,a^(d) and b⁽¹⁾, . . . ,b^(d), respectively, we define the following special product,

hhA, Bii=X

j∈[d]

ha^(j),b^(j)i.

A matrix A = (a_i,j)i,j∈[d] is called essentially nonnegative (or a Metzler matrix) if a_i,j is nonnegative whenever i 6= j. For instance, for any element x = {(x^i,j_tj)i,j∈[d], t ∈ R^d+} of the set E_d introduced at the previous section, the matrix xt = (x^i,j_tj)i,j∈[d] is essentially nonnegative for all tj ≥0.

In this work, we shall considerd independent Lévy processesX⁽¹⁾, . . . ,X^(d) onR^d+, such that with the notation X^(j) = ^t(X^1,j, . . . , X^d,j), for all j ∈ [d], the process X^j,j is a real spectrally positive Lévy process, that is it has no negative jumps, and for all i 6= j, the Lévy process X^i,j is a subordinator. We emphasize that the processesX^1,j, . . . , X^d,j are

not necessarily independent. Moreover, we do not exclude the possibility for a process X^i,j to be identically equal to 0. It is known, see Chap. VII, in [3], that the Lévy process X^(j) admits all negative exponential moments. We denote by ϕ_j its Laplace exponent, that is

E[e^{−hλ,X(j)t i}] =e^tϕj(λ), t≥0, λ= (λ₁, . . . , λ_d)∈R^d+.

Then from Lévy Khintchine formula and the above assumptions on X^(j), ϕ_j has the following form,

(3.5) ϕ_j(λ) =−

d

X

i=1

a_i,jλ_i+1

2q_jλ²_j − Z

R^d+

(1−e^−hλ,xi− hλ,xi1{|x|<1})π_j(dx), λ∈R^d+, where (a_i,j)_i,j∈[d] is an essentially nonnegative matrix, q_j ≥ 0 and π_j is a measure on R^d+

such that π_j({0}) = 0 and Z

R^d+

"

(1∧ |x|²) +X

i6=j

(1∧x_i)

#

π_j(dx)<∞.

Note that for allj ∈[d],ϕ_j is log-convex, i.e. the function logϕ_j is convex on(0,∞)^d. In particular, ϕ_j is a convex function. Moreover, for all i 6=j and λ₁, . . . , λi−1, λ_i+1, . . . , λ_d the function λ_i 7→ϕ_j(λ) is non increasing.

Let us now define the multivariate stochastic field X_t := X⁽¹⁾_t1 +· · ·+ X^(d)_t

d =

d

X

j=1

X_t^i,j

j

!

i∈[d]

, for t = (t₁, . . . , t_d)∈R^d+.

Then X:={X_t, t∈R^d+} is a particular case of additive Lévy field in the sense of [9]. Its law is characterized by the Laplace exponent ϕ:= (ϕ₁, . . . , ϕ_d), that is

E[e^−hλ,Xti] =e^ht,ϕ(λ)i, t, λ∈R^d+.

Such an additive Lévy field will be called a spectrally positive additive Lévy field (spaLf).

This terminology is justified by the results of this section which extend fluctuation the- ory for spectrally positive Lévy processes. Let us also introduce the field of essentially nonnegative matrices

{Xt,t∈R^d+}={(X_t^i,j_j )i,j∈[d],t∈R^d+}.

Note that the spaLf X can be defined as X_t=Xt·1, where1=^t(1,1, . . . ,1). Moreover, we emphasize that the spaLfX carries on the same information as the field of essentially nonnegative matrices {Xt,t ∈ R^d+}. For this reason, the terminology ’spaLf’ will refer indifferently to X or to X. Let r = (r₁, . . . , r_d) ∈ R^d+, since X ∈ E_d a.s., according to Lemma 2.1 there is almost surely a smallest solution to the system

(3.6) (r,X)

d

X

j=1

X_s^i,j_j− =−r_i, i∈[d]_s.

We will denote by T_r = (Tr⁽¹⁾, . . . , Tr^(d)) this solution and use the notation (3.7) T_r = inf{t :X_t− =−r}, with X_t− =

d

X

j=1

X_t^i,j_j−

!

i∈[d]

.

ThenT_r will be referred to as the (multivariate) first hitting time of level−rby the spaLf {X_t,t∈ R^d+}. Note that according to Lemma 2.1, some of the coordinates of T_r can be infinite.

Proposition 3.1. Let X be a spaLf and forr∈R^d+, letT_r be its first hitting time of level

−r as defined above. Then,

1. for all j ∈ [d] and r∈ R^d+, X^(j)

Tr^(j)− = X^(j)

Tr^(j)

a.s. on {Tr^(j) <∞}. In particular, for all i∈[d],

(3.8)

d

X

j=1

X^i,j

Tr^(j)− =

d

X

j=1

X^i,j

Tr^(j)

=−r_i a.s. on the set {Tr⁽ⁱ⁾ <∞}.

2. For all r⁰ ∈ R^d+ such that P(T_r⁰ ∈ R^d+) > 0, conditionally on {T_r⁰ ∈ R^d+}, the field {T_r+r⁰ −T_r⁰,r ∈ R^d+} has the same law as the field {T_r, r ∈ R^d+} and it is independent of the field {T_r, r≤r⁰}. In particular, for all r,r⁰ ∈R^d+,

(3.9) T_r+r^{0 (law)}= T_r+ ˜T_r⁰, where T˜_r⁰ is an independent copy of T_r⁰.

3. If P(T_r ∈ R^d+) > 0 for some r ∈ (0,∞)^d, then P(T_r ∈ R^d+) > 0 for all r ∈ R^d+. Under this condition, there is a mapping φ = (φ₁, . . . , φ_d) : R^d+ → (0,∞)^d such that

(3.10) E[e^−hλ,Tri] =e^{−hφ(λ),ri}, λ∈R^d+.

Moreover, the mapping φ is differentiable and each φ_i is a concave function.

Proof. The first assertion is a consequence of quasi-left continuity for Lévy processes.

Indeed, let us denote by (F_t^(j))t≥0 the natural filtration generated byX^(j) and setF∞^(j) = σ

S

t≥0

F_t^(j)

. Then for all t_j ≥0, the set {T_r^(j) ≤t_j}= [

u∈(Q∪{∞})d

uj=tj

(

∃s≤u :r_i+

d

X

k=1

X_s^i,k_k− = 0, i∈[d]_s )

belongs to the sigma-field G_t^(j)_j :=σ F_t^(j)_j ∪ S

i6=j

F∞⁽ⁱ⁾

!!

, so that Tr^(j) is a stopping time of the filtration (G_t^(j))t≥0. Moreover, since the processes X⁽ⁱ⁾, i ∈ [d] are independent, X^(j) is a Lévy process in the latter filtration. Now let us consider the sequence (T_rn)n≥1, where r_n = r−e_j/n. Then from part 2. of Lemma 2.1, Tr^(j)n is an increasing sequence of (G_t^(j))-stopping times and this sequence satisfies lim

n→∞Tr^(j)n = Tr^(j). Therefore from quasi- left continuity of X^(j), see Proposition I.7 in [3], X^(j)

Tr^(j)− = X^(j)

Tr^(j)

a.s. on {Tr^(j) < ∞}. It clearly implies (3.8).

In order to prove 2. it suffices to see that conditionally on {T_r⁰ ∈R^d+}, the stochastic field {X˜_t,t ∈ R^d+}= {X_Tr0+t+ r⁰,t ∈ R^d+} is independent of {X_t, t≤ T_r⁰} and has the same law as {X_t,t ∈ R^d+}. We conclude by noticing that T˜_r = inf{t : ˜X_t = −r} = T_r+r⁰ −T_r⁰.

Assertion3. follows from Lemma 2.1 and (3.9). Indeed, if there exists r∈(0,∞)^d such that P(T_r ∈R^d+)>0then from Lemma 2.1, for all¯r≤r, T_¯r ≤T_r a.s. and in particular, P(T_¯r ∈ R^d+) > 0. On the other hand, for all r⁰ ∈ (0,∞)^d, identity (3.9) implies that T_r^{0 (law)}= T⁽¹⁾_r(1) +...+T^(p)_r(p) where p ≥ 1, the r⁽ⁱ⁾’s are such that r⁽ⁱ⁾ ≤ r for all i ∈ [p], r⁽¹⁾ +...+ r^(p) = r⁰, and the T⁽ⁱ⁾’s are independent copies of T. As a consequence, we obtain P(T_r⁰ ∈ R^d+) =

p

Q

i=1

P(T⁽ⁱ⁾_r(i) ∈ R^d+) > 0. Now let us prove the second part of this assertion. Let r ∈ (0,∞)^d be such that P(T_r ∈ R^d+) >0 and let λ ∈ R^d+, then by (3.9), for all r⁰ ∈(0,∞)^d,

0< f(λ,r + r⁰) = E[e^{−hλ,Tr+r0i}]

= E[e^−hλ,Tri]E[e^−hλ,Tr0i] =f(λ,r)f(λ,r⁰).

Since f is continuous and f(λ,0) =P(T_r ∈R^d+)>0, this equation implies that f(λ,r) = e^{−hφ(λ),ri}, for some φ(λ) ∈ R^d. Furthermore take r = re_i, for some r > 0 and i ∈ [d], so that E[e^−hλ,Tri] = e^−rφi(λ). Then from right continuity, T_r > 0 almost surely, so that f(λ,r)<1, for allλ∈R^d+ such that λ >0and thusφ_i(λ)∈(0,∞). On the other hand it is plain from (3.10), theφ_j’s are concave functions for allj ∈[d]andφis differentiable.

Note that in (3.8), if for some j 6= i, Tr^(j) = ∞ with positive probability on the set {Tr⁽ⁱ⁾ < ∞}, then X^i,j ≡ 0, a.s. This is due to the fact that X^i,j are subordinators for i6=j, therefore either X^i,j ≡0 a.s. or X_∞^i,j =∞ a.s.

Let us emphasize the following direct consequence of Theorem 3.1, (3.11) P(T_r ∈R^d+) = e^−hφ(0),ri,

so that in particular P(T_r ∈ R^d+) = 1, for all r ∈ (0,∞)^d if and only if φ(0) = 0. Note also that Lemma 3.1 does not allow us a full description of the law of the d-dimensional stochastic field {T_r,r∈ R^d+}. This is the case only when d = 1. In particular for d ≥ 2, if r and r⁰ are not ordered, then we do not know the joint law of (T_r,T_r⁰). Moreover, looking at part 2. of Proposition 3.1, one is tempted to think that, when d≥2, the field {Tr, r ∈ R^d+} is a spaLf, but it is actually not the case. Indeed from the construction of this field, the processes {Trei, r ≥ 0}, i ∈ [d] are clearly not independent. However, it is easy to derive from Proposition 3.1, that each of these processes is a multivariate subordinator whose Laplace exponent is φ_i. The following result provides an expression of its Lévy measure. It is a consequence of further results (e.g. Theorem 4.2) and it will be proved at the end of this paper.

Corollary 3.1. Assume that P(T_r∈R^d+)>0for all r∈(0,∞)^d. Then for all i∈[d], the process {Trei, r ≥ 0} is a multivariate subordinator whose Laplace exponent is φi given in (3.10). Assume moreover for all j ∈[d], the j-th column X^(j)_t of the matrix Xt admits a density which is continuous on F₁ × F₂ × · · · ×F_d, where F_i = R+, for i 6= j and F_j = R. Define the matrix Xbt = (Xb_t^i,j_j )_i,j∈[d] by Xb_t^i,i_i =

d

P

j=1

X_t^i,j_j and Xb_t^i,j_j = X_t^i,j_j , i 6= j, and let p_t :M_d(R) →R be the density of Xbt. Then the Lévy measure of the multivariate subordinator {T_rei, r≥0} is given by

νi(dt) = Z

R^d(d−1)+

det(−x^i,i)

t₁. . . t_d pt(x⁰)Y

k6=j

dxkjdt, if d >1 and ν(dt) = pt(0)

t dt, if d= 1.

Here x^i,i is the matrix x= (x_i,j)i,j∈[d] given by x_i,i =−P

j6=i

x_i,j and x_i,j =x_i,j for i6=j to which line and column of indexihave been removed andx⁰ = (x⁰_i,j)_i,j∈[d], where x⁰_i,j =x_i,j, for i6=j and x⁰_i,i = 0.

We will now define a d-dimensional Lévy process whose law is obtained from the law of X^(j) through the Esscher transform associated to the martingale

(e^{−hµ(j),X(j)t i−tϕj(µ(j))})t≥0,

for any µ^(j) ∈R^d+. Recall that (F_t^(j))t≥0 denotes the natural filtration generated by X^(j). Then for t≥0 and A∈ F_t^(j), the law of this new Lévy process is defined by

P^µ

(j)(A) = E[1I_Ae^{−hµ(j),X(j)t i−tϕj(µ(j))}].

Let us now consider d independent Lévy processes X^µ(j),(j), j ∈ [d] with respective laws P^µ

(j). The Laplace exponent of X^µ(j),(j) is given by

ϕ^µ_j^(j)(λ) = ϕ_j(λ+µ^(j))−ϕ_j(µ^(j)), λ∈R^d+. Moreover, a new spaLf is obtained by setting

(3.12) X^µ_t := X^µ_t1^(1),(1)+· · ·+ X^µ_td^(d),(d), t = (t1, . . . , td)∈R^d+,

whereµ= (µ⁽¹⁾, . . . , µ^(d))∈M_d(R+)is the matrix whose columns are equal toµ^(j),j ∈[d].

Let us set F_t =σ{X_s, s ≤t} for all t∈ R^d+, then F_t =σ(F_t⁽¹⁾₁ ∪ F_t⁽²⁾₂ · · · ∪ F_t^(d)_d ) and the law of the spaLf X^µ is given by,

(3.13) P^µ(A) =E[1IAe^{−hhµ,Xtii−ht,¯ϕ(µ)i}], t∈R^d+, A∈ Ft,

where we have setϕ(µ) = (ϕ¯ ₁(µ⁽¹⁾), . . . , ϕ_d(µ^(d)))and we recall thathhµ,Xtii= P

j∈[d]

hµ^(j),X^(j)_tj i.

We will refer to (3.13) as the Esscher transform of the additive field X. The Laplace ex- ponent of X^µ is then

ϕ^µ(λ) := (ϕ^µ₁⁽¹⁾(λ), . . . , ϕ^µ_d^(d)(λ)), λ∈R^d+.

Let us denote by J_ϕ(λ), λ ∈ (0,+∞)^d, the transpose of the opposite of the Jacobian matrix of ϕ, that is

(3.14) J_ϕ(λ)_i,j =− ∂

∂λ_iϕ_j(λ), i, j ∈[d].

Recall that since all processes X^i,j, i, j ∈[d], are spectrally positive Lévy processes, their expectation is always defined and E[X₁^i,j] ∈ (−∞,∞]. Moreover ϕ is differentiable on (0,∞)^d and the partial derivatives ofϕ at 0 satisfyE[X₁^i,j] =−lim

λ→0

∂

∂λiϕ_j(λ). We will set

∂

∂λiϕ_j(0) := lim

λ→0

∂

∂λiϕ_j(λ), and

(3.15) J_ϕ(0)_i,j =− ∂

∂λ_iϕ_j(0) =E[X₁^i,j], i, j ∈[d]. Then let us consider the following hypothesis:

(H) The set D:={λ∈R^d+ :ϕ_j(λ)>0, j ∈[d]} is non empty.

This hypothesis implies in particular that none of the processes X^j,j,j ∈[d] is a subordi- nator but it is actually stronger as we will see later on. Moreover since allX^i,j,i6=j are subordinators, it is clear that actually D⊂(0,∞)^d.

Theorem 3.1. Let r = (r₁, . . . , r_d) ∈ R^d+ and let T_r = (Tr⁽¹⁾, . . . , Tr^(d)) ∈ R^d+ be the first hitting time of level −r by the spaLfX, then

1. T_r∈R^d+ holds with positive probability for some(and hence for all) r∈R^d+ if and only if (H) holds.

2. Suppose that (H) holds, then φ(λ) ∈ D, for all λ ∈ (0,∞)^d. Moreover, the mapping φ : (0,∞)^d → D is a diffeomorphism whose inverse corresponds to the mappingϕ:D→(0,∞)^d, that is

ϕ(φ(λ)) =λ , λ∈(0,∞)^d.

Proof. Assume that(H) holds, let µ∈D and let us consider the spaLf X^µ whose law is defined in (3.13). In the present case, µ also denotes the matrix whose each column is equal toµ. Then as already observed,µ∈(0,∞)^d, so that all the random variables X₁^µ,i,j are integrable and the mean matrix of X^µ is given by

E[X₁^µ,i,j] =− ∂

∂λ_iϕj(µ), i, j ∈[d].

It is actually the transpose of the opposite of the Jacobian matrix of ϕdenoted byJ_ϕ(µ) and defined in (3.14). Note that J_ϕ(µ) is an essentially nonnegative matrix so that from Lemma A.2 in [2], there is a real eigenvalue ρ^µ such that Re(ρ) < ρ^µ for all the other eigenvalues ρ. Moreover, since ϕ_j is a differentiable convex function and ϕ_j(0) = 0, one has

d

X

i=1

∂

∂λ_iϕ_j(µ)µ_i ≥ϕ_j(µ)>0,

so that from Theorem 3 of [1],J_ϕ(µ)^T, and thereforeJ_ϕ(µ), is a stable matrix in the sense of [1]. In particular, ρ^µ<0.

Let us first assume that J_ϕ(µ) is irreducible. Then from Lemma A.3 in [2], we can choose an eigenvector v^µ associated to ρ^µ such that v^µ_i >0, for all i∈[d]. From the law of large numbers of Lévy processes, we obtain

t→+∞lim t⁻¹X^µ_tvµ =ρ^µv^µ a.s.

Therefore, from part 3. of Lemma 2.1, {X^µ_t, t∈R^d+} reaches each levelαv^µ, withα < 0, almost surely. Then from the definition (3.13) of the law of X^µ, the field {X_t, t ∈ R^d+} reaches each level αv^µ, α < 0, with positive probability and since v_i^µ > 0, i ∈ [d], from part 2. of Lemma 2.1, it reaches each level−r∈R^d− with positive probability.

Now let us assume that J_ϕ(µ) is not irreducible that is there exists a permutation matrix P_σ and three matrices A₁, A₂ and B such that A₁ is of size 1≤p≤d−1and

P_σ⁻¹J_ϕ(µ)P_σ =



 A₁ 0

B A₂



.

In particular, for all(i, j)∈I×J whereI ={σ(1), ..., σ(p)} andJ ={σ(p+ 1), ..., σ(d)}, E[X₁^µ,i,j] = 0 that is X₁^µ,i,j = 0 a.s.

Therefore we can write for all r∈R^d+,

P(T^µ_r ∈R^d+) = P ∃t∈R^d+:∀i∈[d],

d

X

j=1

X^µ,i,j(t_j) =−r_i

!

= P ∃t∈R^d+:∀i∈I,X

j∈I

X^µ,i,j(t_j) = −r_i

and ∀i∈J,X

j∈J

X^µ,i,j(t_j) =− r_i+X

j∈I

X^µ,i,j(t_j)

!!

.

Let T^µ,I_r be the smallest solution of the system (r_I,X^I,µ), where we set r_I = (r_i)i∈I and X^I,µ= (X^µ,i,j)i,j∈I. Then conditioning on the event{T^µ,I_r ∈R^p+}, we obtain

P(T^µ_r ∈R^d+) =P(T^µ,J_r0 ∈R^d−p+ |T^µ,I_r ∈R^p+)P(T^µ,I_r ∈R^p+), where we have set r⁰ = r_i+P

j∈I

X^µ,i,j(T_r^µ,I,j)

!

i∈J

. Then T^µ,J_r0 is the smallest solution of the system(r⁰,X^J,µ)with X^J,µ = (X^µ,i,j)_i,j∈J. Thus if A₁ and A₂ are irreducible, then we derive from the previous case that P(T^µ,I_r ∈R^p+) = 1 and P(T^µ,J_r0 ∈R^d−p+ |T^µ,I_r ∈R^p+) = 1.

In other words, we have P(T^µ_r ∈R^d+) = 1and then P(T_r∈ R^d+)>0. On the other hand, if A₁ and/or A₂ are not irreducible, then we can repeat this argument.

Conversely, let us assume that T_r ∈R^d+ holds with positive probability for all r∈ R^d+

and let φ be the function defined in part 3 of Proposition 3.1. Let us show that for all λ∈(0,∞)^d, ϕ(φ(λ)) =λ, which implies in particular that φ(λ)∈ D. It follows from the independence and stationarity of the increments of the spaLf {X_t, t ∈ R^d+} that for all r,t, λ∈R^d+,

E[e^−hλ,Tri1I{t<Tr}] = Z

Cr

E[e^−hλ,Tri1I{t<Tr}|X_t = x]P(X_t ∈dx)

= Z

Cr

e^−hλ,tiE[e^−hλ,Tr+xi]P(Xt ∈dx)

= e^−hλ,tie^{−hr,φ(λ)i}

ehϕ(φ(λ)),ti− Z −r1

−∞

. . . Z −r_d

−∞

e^{−hx,φ(λ)i}P(X_t∈dx)

, where C_r is the union of all the sets E₁ × · · · ×E_d with at least one i ∈ [d] such that E_i =]−r_i,+∞[ and for the others j ∈[d],E_j =R. Then we derive the identity

(3.16)

1−e^hr,φ(λ)iE[e^−hλ,Tri1I{t<Tr}^c] =e^−hλ,ti

ehϕ(φ(λ)),ti− Z −r₁

−∞

. . . Z −r_d

−∞

e^{−hx,φ(λ)i}P(Xt ∈dx)

. Let r⁰,r⁰⁰ ∈ (0,∞)^d be such that r⁰ + r⁰⁰ = r, then from Proposition 3.1, T_r can be decomposed as T_r = T_r⁰ + ˜T_r⁰⁰, where T˜_r⁰⁰ is an independent copy of T_r⁰⁰. Moreover {t<T_r}^c⊂ {t<T_r⁰}^c∩ {t<T˜_r⁰⁰}^c, so that

E[e^−hλ,Tri1I{t<Tr}^c]≤E[e^−hλ,Tr0i1I{t<T_r0}^c]E[e^−hλ,Tr00i1I{t<T_r00}^c].

If the coordinates of r are integers, then applying this identity recursively, we obtain, (3.17) E[e^−hλ,Tri1I{t<Tr}^c]≤

d

Y

j=1

E[e^−hλ,Teji1I{t<T_ej}^c]^rj.

Then we can find t whose coordinates are sufficiently small so that for all j, E[e^−hλ,Teji1I_{t<Tej}^c]<E[e^−hλ,Teji] =e^−φj(λ).

Therefore lim

r→∞e^hr,φ(λ)iQd

j=1E[e^−hλ,Teji1I_{t<Tej}^c]^rj = 0 and from (3.17) we derive that the left member of (3.16) tends to 1, while the right member tends toe^−hλ,tiehϕ(φ(λ)),ti, which shows that ϕ(φ(λ)) =λ. This is true in particular for all λ∈(0,∞)^d and hence Dis not empty. This achieves the proof of both assertions 1.and 2.

From part 1. of Theorem 3.1, assuming(H)for a spaLfXensures thatXhits all negative levels in a finite time with positive probability. Whend= 1, this is simply assuming that the spectrally positive Lévy process we consider is not a subordinator.

Let us give an example of a 2-dimensional spaLf. Assume that, for j ∈ [2], the X^j,j’s are independent Brownian motions B^(j) with driftsa_j ∈R, that is X_t^j,j =B_t^(j)+a_jt and that for i6=j,X^i,j is a pure drift, that is X_t^i,j ≡a_ijt, a_ij ≥0. The Laplace exponents ϕ_j are then explicitly given by

ϕ_j(λ) = −λ_ia_ij −λ_ja_j+ 1

2q_jλ²_j, i6=j, λ∈R²+.

Assume q_j >0, j ∈[2]. After some calculus, we are able to explicit the set D defined in hypothesis (H). It is given by

D= (

λ∈R²+:λ₁ > a₁+p

∆₁(λ₂)

q₁ ∨0

!

and λ₂ > a₂+p

∆₂(λ₁)

q₂ ∨0

!) , where∆j(λi) = a²_j+ 2aijqjλi for allj ∈[2]and i6=j. Note that this set is not empty and so the assumption (H) holds. In particular, thanks to Theorem 3.1, the spaLf X reaches all the level −r ∈ R²− with positive probability and according to the second part of this theorem, we know that the mapping ϕ admits an inverse φ on the set D. This inverse φ= (φ₁, φ₂)is given by

φ_j(λ) = 1 qj

q

2q_jλ_j +a²_j + 2a_ijq_jφ_i(λ) + a_j qj

, j ∈[2], i6=j, λ∈R²+.

In order to carry on with the general study of the fluctuation of the spaLf X, we shall now give a characterization of the condition φ(0) = 0 in terms of the Jacobian matrix J_ϕ(0). As a first remark, note that if for somej ∈[d],J_ϕ(0)_j,j >0, then lim

t→+∞X_t^j,j = +∞

a.s. and hence the field{X_t,t∈R^d+}cannot reach all the levels−r∈R^d− with probability one. Thereforeφ(0)>0 whenever there isj such that J_ϕ(0)_j,j >0.

Recall that whenever the essentially nonnegative matricesJ_ϕ(λ), defined in(3.14) and (3.15) for λ ∈ [0,∞)^d have finite entries and are irreducible, according to the Perron- Frobenius theory, there are real eigenvalues ρ^λ with multiplicity equal to 1 and such that the real part of any other eigenvalue is less thanρ^λ, see Appendix A of [2]. We set ρ⁰ =ρ.