Some explicit Krein representations of certain subordinators, including the Catherine Donati-Martin, Marc Yor

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Catherine Donati-Martin, Marc Yor. Some explicit Krein representations of certain subordinators, including the Gamma process. 2005. ￿hal-00004438￿

HAL Id: hal-00004438 https://hal.archives-ouvertes.fr/hal-00004438 Preprint submitted on 14 Mar 2005

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∗ τt 1 −2 (law) α,0 (1.1) ( B α du, t 0) = (S ; t 0), | u| ≥ t ≥ Z0 and that all symmetric stable L´evy processes may be obtained in this man- 1 −2 ner, replacing the even power x α by ”symmetric” powers i.e.: σ(x) = ∗ 1 −2 1 | τ|t ds sgn(x) x α . For example, ( ; t 0) is a standard Cauchy process. | | π 0 Bs ≥ R 2 More generally, every asymmetric stable L´evy process may be represented in a similar manner from Brownian motion. Although in the sequel, we shall α,β also extend (1.1) to present in general (St ; t 0) in terms of Brownian ad- ∗ ≥ ditive functionals taken at (τt ; t 0), these presentations are not so simple, and we prefer to those the Krein≥ representations evoked in (1.c). (1.e) The rest of the paper is organized as follows. Our main Krein rep- resentation results are presented in Section 2. The full proofs are given in Section 3. The Brownian additive functional representations are discussed in Section 4. Finally, in Section 5, we also give some Krein representations of the symmetric L´evy processes on R (without Gaussian component) whose L´evy measures are given by: exp( β y ) ν˜ (dy)= − | | dy α,β y α+1 | | (0 α< 2, β> 0). ≤ (1.f) As an end to this introduction, let us point out that, the subordinators Sα,β we represent here, and their symmetric counterparts, are arguably the most studied and used among L´evy processes, and this, for the following rea- sons: for α> 0, Sα,β is obtained by Esscher transform (see [7, 26]) from the fundamental stable (α) subordinator, hence it ”retains” some scaling prop- erty, while the Gamma process ([6], [27], [28], [29], [30]) and the variance- gamma processes ([17], [16], [15]) have some fundamental quasi-invariance properties, which make them comparable, in some respect, to Brownian mo- tion with drift. (1.g) Finally, we refer the reader to a systematic compendium [5] of constants Cα related to various choices found in the literature of local times with respect to BES( α). We also intend in [4] to study a number of properties of the symmetrized− BES(α, β ) processes with a view towards a discussion of Krein ↓ representations for the variance-gamma processes.

2 Definitionofthe BES( α, β ) processes and − ↓ main Krein representation results

(2.1) We first recall that for 0 <α< 1, BES( α), which we call the Bessel process of index α, (or dimension d = 2(1 α−)) is the R -valued diffusion − − + 3 with infinitesimal generator 1 d2 1 2α d + − , 2 dx2 2x dx (−α) which is instantaneously reflecting at 0. We denote by Px its distribution on C(R , R ), where R (ω)= ω(t), = σ R ; s t and R = x. + + t Rt { s ≤ } 0 For the sequel, it will be convenient to introduce the parameter θ = √2β for (−α),β↓ β > 0. We may now define BES( α, β ) as the diffusion with law (Px ), − ↓ obtained by Girsanov1 transform from BES( α): − ˆ (−α),β↓ Kα(θRt) α (−α) (2.1) Px Rt = exp(Cα(2β) lt βt)Px Rt | Kˆα(θx) − | where Kˆ (x) = xαK (x), x 0, which satisfies Kˆ (0) = 2α−1Γ(α) and α α ≥ α (lt; t 0) is a choice of the local time at level 0 of R made so that the following≥ holds: β↓ Itˆo’s excursion measure nα (de) associated with BES( α, β ) together with this choice of local time may be described as follows: − ↓ α a) nβ↓(V (e) dv)= 2 Γ(α+1) exp( βv)dv, where V (e) denotes the lifetime α ∈ vα+1 − of the generic excursion e. b) Conditionally on V = v, the process (e(u),u v) is distributed as a Bessel bridge of index α, and length v. ≤ Note that this description is valid, in particular, for β = 0. The conditioned diffusions BES( α, β ), started at x > 0 and killed at T0 are described in [21]. See also Watanabe− ↓ [31, 32, 33]. We now discuss some immediate consequences of these choices: from (a), it follows that: ∞ dv E(−α),β↓(exp( λτ )) = exp u2αΓ(α + 1) (1 e−λv)e−βv 0 − u − vα+1 −  Z0  and elementary computations yield: Γ(α 1) E(−α),β↓(exp( λτ )) = exp u2αΓ(α + 1) − (λ + β)α βα 0 − u − α { − }   π = exp u2α (λ + β)α βα − sin(πα){ − }   1We shall also use the well-known general fact that such an absolute continuity rela- tionship extends with t replaced by any T on the set (T < ). ∞ 4 As a consequence of these computations, it follows that, on one hand: π (2.2) E(−α)(exp( λτ )) = exp u2α λα 0 − u − sin(πα)   and from (2.1) taken at x = 0 and t = τu, we can determine the constant Cα: π C = . α sin(πα) On the other hand,

(−α),β↓ lim E0 (exp( λτu)) = exp( u ln(λ + β) ln(β) ) α −→ 0 − − { − } 1 (2.3) = (1 + λ/β)u

0,β↓ Thus, assuming that we may represent the LHS of (2.3) as E0 (exp( λτu)) for the law P 0,β↓ of some diffusion BES(0, β ), instantaneously reflecting− at 0 ↓ 0, we will thus have obtained a Krein representation of the Gamma process with parameter β, i.e. the subordinator whose Laplace transform in λ, at time u, is given by (2.3). Theorem 2.1 1) For every β > 0, there exists a diffusion BES(0, β ) on ↓ R+, which is instantaneously reflecting at 0, and whose infinitesimal gener- ator on (0, ) is given by: ∞ 1 d2 1 K′ d + + 2β 0 ( 2βx) . 2 dx2 2x K dx  0  p p 2) For some choice of the local time at 0 of BES(0, β ), the corresponding Itˆomeasure of excursions may be described as: ↓ a) nβ↓(V (e) dv)= 1 exp( βv)dv 0 ∈ v − b) Conditionally on V = v, the process (e(u),u v) is distributed as a ≤ Bessel bridge of dimension 2, and length v. 3) Let x> 0, the following relation holds:

(0),β↓ K0(θRt) (0) (2.4) Px Rt∩(t

5 Note: The following relation, where we have extended (1/K )to [0, [, with 0 ∞ (1/K0)(0) = 0, is equivalent to (2.4):

(0) K0(θx) (0),β↓ (2.5) Px Rt = exp(βt)Px Rt . | K0(θRt∧T0 ) | The following theorem exhibits an absolute continuity relationship between (0),(β+γ)↓ (0),β↓ the laws Px and Px from which several important Laplace trans- forms may be immediately obtained.

Theorem 2.2 Let β,γ > 0, and x> 0; then, there is the relationship: (2.6)

K0( 2(β + γ)Rt) K (√2βx) γ P (0),(β+γ)↓ = 0 (1+ )lt exp( γt)P (0),β↓ . x Rt √ β x Rt | Kp0( 2βRt) K0( 2(β + γ)x) − | p In particular, for x = 0 and t being replaced by τl, formula (2.6) simplifies as:

(0),(β+γ)↓ K0( 2(β + γ)Rt) γ lt (0),β↓ (2.7) P0 Rt = (1 + ) exp( γt)P0 Rt . | Kp0(√2βRt) β − | and

(0),(β+γ)↓ γ l (0),β↓ (2.8) P R =(1+ ) exp( γτl)P R . 0 | τl β − x | τl

Note, also from (2.6), that the measure:

√ K0( 2βx) −lt (0),β↓ β exp(βt)Px Rt K0(√2βRt) | does not depend on β. Here are some consequences of (2.6), (2.7), (2.8):

- As a consequence of (2.6), one obtains the Laplace transform of (0),β↓ T = inf t; R =0 under Px : 0 { t }

(0),β↓ K0( 2(β + γ)x) (2.9) Ex (exp( γT0)) = − Kp0(√2βx)

6 2 In fact, T0 is distributed as a GIG(0; x, √2β) variable, i.e. with den- sity: 1 1 exp( (x2/t +2βt))dt. 2K0(x√2β) −2 As a consequence of (2.8), we obtain our main result:

(0),β↓ Corollary 2.3 (τl; l 0) is, under P0 , a gamma process with parameter β, i.e. (βτ : l 0) is≥ a standard Gamma process: l ≥ (0),β↓ 1 (2.10) E0 (exp( γτl)) = γ l for γ 0. − (1 + β ) ≥

3 Esscher and Girsanov transforms

This section is devoted to the proofs of the assertions contained in Section 2. (3.1) To begin with, we explain how, once we have obtained a Krein repre- sentation for a subordinator (St), we can obtain a related one for a second subordinator (S˜t) whose law is an Esscher transform of that of (St). We consider a positive diffusion (X ; t 0) with 0 as an instantaneous reflecting t ≥ boundary. We denote by L a choice of its local time at 0 and (τ : l 0) t l ≥ the corresponding inverse local time. We are looking for a diffusion X˜ on R+ with laws P˜x such that:

(3.1) P˜ F = exp(ψ(a)u aτ )P F 0| τu − u 0| τu for a> 0, where ψ denotes the Laplace exponent of the subordinator (τ ; u u ≥ 0). From (3.1), the inverse local timeτ ˜ of X˜ is the Esscher transform τ (a) of τ defined by: E [exp( (λ + a)τ )] (3.2) E [exp( λτ (a))] = 0 − u . 0 − u E [exp( aτ )] 0 − u The L´evy measure m(a)(dx) of τ (a) is related to the L´evy measure m(dx) of τ by m(a)(dx) = exp( ax)m(dx) (see [7], [26], Chapter VII, Section 3c). − To define P˜x, we need to compute the martingale

M = E[exp( aτ ) ], s τ s − t |Fs ≤ t 2For many references to, and applications of, GIG variables, see Matsumoto-Yor [18]

7 M = exp( as)E exp( aτ − ) s − Xs(ω) − t Ls(ω) Now, we have:  

E [exp( aτ )] = E [exp( aT )] E [exp( aτ )] x − v x − 0 0 − v and therefore: Ms = ϕa↓(Xs) exp(ψ(a)Ls as) M0 − where ϕa↓ denotes the function defined by

(3.3) ϕ ↓(x)= E [exp( aT (X))] (ϕ ↓(0) = 1) a x − 0 a where T (X) is the first hitting time of 0 by X. More generally, for x 0, 0 ≥ we define the law P˜x via the absolute continuity relation:

˜ ϕa↓(Xt) (3.4) Px Ft = exp(ψ(a)Lt at)Px Ft . | ϕa↓(x) − |

The generators L and L˜ of X and X˜ respectively are linked by:

ϕ′ (x) d L˜ = L + a↓ . ϕa↓(x) dx

(3.2) Examples:

(−α) 1. Px = Px , 0 <α< 1. Then, π ϕ (x)= c Kˆ (√2ax), ψ(a)= (2a)α a↓ α α sin(πα)

(−α),a↓ and P˜x = Px .

(−α),β↓ 2. Px = Px , 0 <α< 1, β > 0. Then,

Kˆ ( 2(β + a)x) π ϕ (x)= c′ α ; ψ(a)= (2a +2β)α (2β)α a↓ α ˆ sin(πα) Kpα(√2βx) { − }

(−α),(β+a)↓ and P˜x = Px .

8 From [3] and [20], we know that the inverse local time of BES( α) is a stable subordinator of index α. It follows from subsection (3.1) and− Example 1 above that the inverse local time of BES( α, β ) is a subordinator Sα,β. The description of Itˆo’s excursion measure− of BES↓ ( α, β ) follows from the description of the excursion measure for BES( α−) given↓ in [3] and the − Esscher transform. Note that we have not chosen the same normalisation for the local time as in [3]; we have the following relation between the two local time processes (from (2.2) and subsection 3.2 in [3]): (2αΓ(α))2 lBY = lDY t 2(1 α) t − BY where lt denotes the local time considered in [3]. Note that there is a (−1/2) mistake in [3] after (3.i) due to the identification of τt (the inverse local time of the reflected Brownian motion) with τt (for the Brownian motion) instead of τt/2 and the correct formula (see p.45, after (3.i)) is

E[exp( kτ (−ν) )] = exp( tc 2ν−1kν). − tc(1/2,2ν) − ν

(3.3) Proof of Theorems 2.1 and 2.2 1) We consider the diffusion X on [0, [ with generator ∞ 1 d2 1 K′ d + + 2β 0 ( 2βx) . 2 dx2 2x K dx  0  p p A pair (s(x), m(dx)) of scale function and speed measure is given by: x dy 2 s(x)= 2 , m(dx)=2xK0 ( 2βx)dx. 0 yK0 (√2βy) Z p 1 Note that K0(y) 0 ln(2/y), thus 2 is integrable in 0 and s(0) = 0. ∼ yK0 (√2βy) In order to obtain the behavior of X at the boundary point 0, we shall apply the criterium of Rogers-Williams [25, V.51] to the diffusion in natural scale Y = s(X). The speed measure of Y is given by: 2 m (dx)= dx. Y (s′ s−1(x))2 ◦ Then, xm (dx) < implying that T (Y ) < P p.s. for all x > 0, 0+ Y ∞ 0 ∞ x and m(dx) < ensuring that 0 is reflecting. 0+R ∞ R 9 The description of the Itˆomeasure follows from that of BES( α, β ) letting α 0. − ↓ −→ The absolute continuity relation (2.4) is given in Pitman-Yor [21]. We can also obtain it from (2.1) and:

2α (+α) Rt∧T0 (−α) P R = P R , x> 0. x | t x x | t   Thus, 2α ˆ Rt∧T0 (−α),β↓ Kα(θRt) (+α) P R = exp( βt)P R x x | t ˆ − x | t   Kα(θx) and letting α 0, −→ (0),β↓ K0(θRt) (0)  1(t

K0( 2(β + a)x) a ϕa↓(x)= C ; ψ(a) = ln(1+ ). Kp0(√2βx) β The other formulas follow easily. 

4 Brownian representations of the subordi- nators Sα,β

We keep the same normalisation as in Section 2, i.e. the Laplace transform of a is given by (2.2) Proposition 4.1 1. The stable subordinator Sα can be represented by the Brownian additive functional (see [3])

α S = Aα(τ Γ(α)2t ) t − α2α 1 where s 1 −2 (4.1) A (s) := B α du, s 0, α | u| ≥ Z0 and τt is the inverse local time of B.

10 2. The subordinator Sα,1 can be represented as:

τt (4.2) h ( B ) dr. α | r| Z0 −1 2 4 −1 where hα(x)=2(sα (2x)) Kα(sα (2x)) and x dy (4.3) s (x)= α yK2(y) Z0 α 3. The Gamma process S0,1 has the following representation:

τt (4.4) h( β ) du ; t 0 | u| ≥ Z0  with

−1 2 4 −1 (4.5) h(x)=2(G (2x)) K0 (G (2x)), where G−1 is the inverse of the increasing function G given by: z dx (4.6) G(z)= xK2(x) Z0 0 Sketch of Proof: 1)The first point has been obtained in [3]. More precisely, λ (4.7) E[exp( A (τ ))] = exp( tc λα), λ 0, − 2 α t − α ≥ where π αα 2 c = . α α sin(πα) Γ(α)   2) By a Girsanov transform, we can find a diffusion X such that the subor- dinator τt(X) X 1 −2 A (τ ) := X α du α t | u| Z0 is a Esscher transform of the stable subordinator Aα(τt) and thus is dis- tributed as Sα,1(up to a constant). Then, we write the diffusion in natural scale sα(X) as a time change Brownian motion. We can also start from BES(α, 1 ) and use a time change method. ↓ 11 3) By an application of Itˆo’s formula, we can prove that:

t G−1(2 B )= Y 2h( B )ds | t| | s| Z0  (law) where Y is a BES(0, 1 ) process from which we can deduce that τt(Y ) = τt(B) ↓ 0 2h( Bs )ds where τt(Y ) is the inverse diffusion local time of Y , defined with the speed| | measure m (dy)=2yK2(y)dy. R Y 0 5 Some complements

5.1 The case α =1/2 Let us go back to the representation (4.2) for the subordinator with L´evy measure k′ exp( y ) dy . In the case α =1/2, (4.2) becomes: α − 2 y1+α τt dr (5.1) . (1+2 β )2 Z0 | r| We now explain how this representation may be reduced to the more classical one, as given in (5.2) below. For this purpose, we recall that the stable subordinator of index 1/2 can also be realized as the distribution of (T ; t 0) t ≥ where Tt denotes the first hitting time of t by a Brownian motion, starting from 0. We have the same interpretation for its Esscher transform. Indeed, t sgn(B )dB t ds ln(1 + 2 B ) = 2 s s + L | t| (1+2 B ) − (1+2 B )2 t Z0 | s| Z0 | s|  t ds = 2 β(1) + L − (1+2 B )2 t  Z0 | s|   where (β(1)(u); u 0) is a Brownian motion with drift 1. From Skorokhod’s≥ lemma, we derive: t dr L = sup β(1); s t s ≤ (1+2 B )2  Z0 | r|  Therefore, we obtain: τl dr (5.2) = inf u; β(1) = l := T (1). (1+2 B )2 { u } l Z0 | r| 12 (1) y dy Thus, from (5.1), (Tl , l 0) is a subordinator with L´evy measure C exp( 2 ) y3/2 and Laplace transform ≥ −

λ2 E exp( T (1)) = exp( l(√λ2 +1 1)). − 2 l − −   Of course, the above Laplace transform could also have been computed from the Laplace transform of Tl and the Cameron-Martin formula.

5.2 Symmetric L´evy processes From the Brownian representation of the Gamma process, we can give a Brownian representation for the symmetric Gamma process (or ) distributed as (γ1(t) γ2(t); t 0) where γ1 and γ2 are two independent gamma processes. More− generally,≥ we have:

Proposition 5.1 Let S1 and S2 be two independent subordinators with Brow- nian representations:

τt (law) (S (t); t 0) = dsϕ ( B ); t 0 , i =1, 2. i ≥ i | s| ≥ Z0 

Then, we have the Brownian representation for (S1(t) S2(t); t 0) as follows: − ≥

τ (law) 2t (5.3) (S (t) S (t); t 0) = dsϕ(B ); t 0 1 − 2 ≥ s ≥ Z0  ϕ (x),x> 0 where ϕ(x)= 1 . ϕ ( x),x< 0  − 2 − Proof: It suffices to write

τt τt τt dsϕ(B )= ds1 ϕ (B ) ds1 ϕ ( B ). s (Bs>0) 1 s − (Bs<0) 2 − s Z0 Z0 Z0 Then, we use the representation

+ (+) − (−) (5.4) B = β s 1 du, B = β s 1 du s | | 0 (Bu>0) s | | 0 (Bu<0) R R

13 where β(+) and β(−) are (as a consequence of Knight’ s theorem) two indepen- dent Brownian motions3 ; see, e.g., proofs of the arc sine law for Brownian motion, inspired from D. Williams [34] (see [9], [19]). Then, we can write

(+) (−) τt Aτt Aτt dsϕ(B )= dhϕ ( β(+) ) dhϕ ( β(−) ) s 1 | h | − 2 | h | Z0 Z0 Z0 where τt τt (+) (−) Aτt = ds1(Bs>0)ds; Aτt = ds1(Bs<0)ds, Z0 Z0 and moreover, from (5.4), we also learn that:

(+) (+) (−) (−) Aτt = τt/2(β ), Aτt = τt/2(β ).

Finally, we have obtained the result. 

5.3 The Itˆomeasure of BES( α, β ) − ↓ The description of the Itˆomeasure given in Section 2 relies upon one of the descriptions of the Itˆomeasure nα of BES( α) given in [3]. There is a second description of n conditionally to the maximum− of the excursion.

β↓ Proposition 5.2 a) Under nα , the distribution of M satisfies:

β↓ α Kα(√2βx) (5.5) nα (M x) = 2(2β) ≥ Iα(√2βx) b) Conditionally on M = x, the maximum is attained at a unique time R and the processes (et;0 t R) and (eV −t, 0 t V R) are two independent BES(α, β ) processes,≤ ≤ starting from 0, stopped≤ ≤ at− their first hitting time T ↑ x of x.

β↓ Proof: a) From the description of nα given in section 2, we have: ∞ exp( βv) nβ↓[f(M 2)]=2αΓ(α + 1) − Π(α)[f(M 2)]dv α vα+1 v Z0 3 (+) However, we emphasize that, knowing B, only the reflected Brownian motions βh (−) (+) (−) | | and βh are accessible, and not β and β . | |

14 (α) where Πv is the distribution of a Bessel bridge of index α and length v. ∞ exp( βv) nβ↓[f(M 2)]=2αΓ(α + 1) − Π(α)[f(vM 2)]dv α vα+1 1 Z0 Now, from [22, Theorem 3.1]

(α) ˜ ˜ −2α Π1 [φ(r)] = cαE[φ(R)(M) ] where the process R˜ is defined by

−1/2 R˜ =(T + Tˆ) Y ˆ , 0 t 1 t t(T +T ) ≤ ≤ and Y is the process connecting the paths of two independent BES(2+2α) processes R on [0, T ] (first hitting time of 1) and Rˆ on [0, Tˆ] back to back, i.e. Rt t T Yt = ≤ Rˆ ˆ T t T + Tˆ  T +T −t ≤ ≤ Then, −1/2 M˜ := sup R˜s =(T + Tˆ) s≤1 α and cα =2 Γ(α + 1). It follows that: v Π(α)[f(vM 2)] = c E f (T + Tˆ)α . 1 α ˆ  T + T   Thus,

∞ exp( βw(T + Tˆ)) nβ↓[f(M 2)] = 2αΓ(α + 1)c E − f(w)dw α α wα+1 "Z0 # ∞ α f(w) ˆ = 2 Γ(α + 1)cα α+1 E exp( βw(T + T )) dw 0 w − Z h 2 i ∞ f(w) (√2βw)α = [2αΓ(α + 1)]−1c dw α wα+1 I (√2βw) Z0  α  ∞ f(w) = (2β)α dw wI2(√2βw) Z0 α ∞ f(y2) = 2(2β)α dy yI2(√2βy) Z0 α 15 Therefore, we obtain 1 nβ↓(M dy) = 2(2β)α dy α 2 ∈ yIα(√2βy) hence: ∞ 1 nβ↓(M x) = 2(2β)α dy α ≥ yI2(√2βy) Zx α K (√2βx) = 2(2β)α α . Iα(√2βx)

We also refer to [24] for related computations. As a verification, we can let β 0 to obtain: −→ 1 n (M x)=22αΓ(α)Γ(α + 1) . α ≥ x2α

This agrees with the description of the distribution of M undern ˆα given in Biane-Yor [3]: 1 nˆ (M x)= α ≥ x2α and from our normalisation given in Section 2,

2α nα =2 Γ(α)Γ(α + 1)ˆnα. b) We refer to [21] for the definition of Bessel processes with drift BES(α, β ). β↓ ↑ The description of nα , conditionally to M, follows from the description of n (see [3]), the relation nβ↓(de) = exp( βV (e))n (de) and α α − α (α),β↑ −α (α) P F = C x I (θx) exp( βT )P F 0 | Tx α α − x 0 | Tx (see Proposition 3.1 in [21]). 

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