J Theor Probab DOI 10.1007/s10959-016-0719-z
Bouncing Skew Brownian Motions Arnaud Gloter1 · Miguel Martinez2
Received: 29 October 2015 / Revised: 9 April 2016 © Springer Science+Business Media New York 2016
Abstract We consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. In Gloter and Martinez (Ann Probab 41(3A):1628–1655, 2013), the evolution of the distance between the two processes, in local timescale and up to their first hitting time, is shown to satisfy a stochastic differential equation with jumps driven by the excursion process of one of the two skew Brownian motions. In this article, we show that the distance between the two processes in local timescale may be viewed as the unique continuous Markovian self-similar extension of the process described in Gloter and Martinez (2013). This permits us to compute the law of the distance of the two skew Brownian motions at any time in the local timescale, when both original skew Brownian motions start from zero. As a consequence, we give an explicit formula for the entrance law of the associated excursion process and study the Markovian dependence on the skewness parameter. The results are related to an open question formulated initially by Burdzy and Chen (Ann Probab 29(4):1693–1715, 2001). Keywords Skew Brownian motion · Self-similar process · Local time · Excursion process · Entrance law Mathematics Subject Classification (2010) 60G18 · 60J55 · 60J65
B
Arnaud Gloter
[email protected] Miguel Martinez
[email protected]
1
Laboratoire de Mathématiques et Modélisation d’Evry, UMR CNRS 8071, Université d’Evry-Val-d’Essonne, ENSIIE, 23 bd de France, Evry Val d’Essonne, France
2
Laboratoire d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050, Université de Marne-la-Vallée, 5 boulevard Descartes, Cité Descartes - Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France
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1 Introduction 1.1 Presentation of the Problem Consider (Bt )t≥0 a standard Brownian motion on some filtered probability space (, F, (Ft )t≥0 , P) where the filtration satisfies the usual right continuity and completeness conditions. Recall that the skew Brownian motion X x,β may be defined as the solution of the stochastic differential equation with singular drift coefficient, x,β
Xt
= x + Bt + β L 0t (X x,β ),
(1)
where β ∈ (−1, 1) is the skewness parameter, x ∈ R, and L 0t (X x,β ) is the symmetric local time at 0: t 1 L 0t (X x,β ) = lim 1[−ε,ε] (X sx,β )ds. ε→0 2ε 0 It is known that a strong solution of Eq. (1) exists, and pathwise uniqueness holds as well (see [3,11]). The skew Brownian motion is an example of a one-dimensional diffusion partially reflected at zero. It finds applications in the fields of stochastic modeling (see [15] and references therein) and of numerical simulations ([14]), especially as it is deeply connected to diffusion processes with non-continuous coefficients (see, e.g., [13]). The structure of the diffusion flow of a reflected, or partially reflected, Brownian motion has been the subject of several works (see, e.g., [2,4]). The longtime behavior of the distance between reflected Brownian motions with different starting points has been largely studied too (see, e.g., [6,7]) and may be viewed as a special case at the extremes (β = ±1). Actually, a quite intriguing fact about solutions of (1) is that they do not satisfy the usual flow property of differential equations, which prevents two solutions with different initial positions to meet in finite time. Indeed, it is shown in [2] that, when x,β −1 < β1 ≤ β2 < 1 and x > 0, almost surely, the path t → X t 2 remains above the 0,β path t → X t 1 , and both paths meet at an almost surely finite random time. The law of the values of the local times of these processes at the first hitting time is computed β2 < β1 < β2 < 1 in [9]. Moreover, it is shown in [5] that in the special case 0 < 1+β 2 x,β
0,β
and x > 0, the two paths t → X t 2 and t → X t 1 reflect on each other. This means that, almost surely, for every t0 > 0, there exists t > 0 such that x + β2 L 0t (X x,β2 ) = x,β 0,β β1 L 0t (X 0,β1 ) and so X t 2 = X t 1 at infinitely many times (see [5] Theorem 1.4 (iii)). Since the word “reflection” is widely used in the literature in somewhat different context, we prefer to say here that the two paths “bounce” on each other. x,β ,β The object of study of this paper is the “difference process” {Z t 1 2 = x,β X τ (X2 0,β1 ) : t ≥ 0} (where τt (X 0,β1 ) stands for the inverse local time of X 0,β1 ), t which measures the distance (looked at a proper local timescale) of two skew Brownian motions with different skewness parameters. In [9], this process is studied up to its first hitting time at 0 and the law of the hitting time is computed. In this article, we study the “difference process” after its first hit at zero in the β2 < β1 < β2 < 1). reflecting case (0 < 1+β 2
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In fact, we show that Z x is a self-similar process, so that its study may be investigated using the general theory of positive self-similar processes. In the language of this theory, one of the main results of this paper asserts that the difference process after its first hitting time at zero U behaves as the unique positive self-similar Markovian extension of the difference process killed when it hits zero that leaves zero continuously. The dynamic of the killed process is given by Theorem 1. In particular, we show in this paper that the description of the difference process given by (7) and valid before U is also true after U . So that Z x appears as a solution of some jump S.D.E. driven by the excursion process of X β1 (see Theorem 5). Using this description of the difference process Z x on R+ permits to retrieve an Itô-Dynkin formula. Our second main result is the computation of the law of the marginal Z 10 combining the Itô-Dynkin formula and analysis tools (namely the Mellin transform). The result is given in Theorem 6. Once this study is achieved, we provide two applications. A first application concerns the description of the excursions of the difference process with the calculation of the associated entrance law. A second application is the characterization of the 0,β inhomogeneous Markovian behavior of the process β → X τ 0 (B) on (0, 1) ∩ Q with 1
the computation of its transition mechanism. This gives a partial answer to an open question first formulated by C. Burdzy and Z.Q. Chen in [5] (see [5], Open Problem 1.9).
1.2 Organization of the Paper The paper is organized as follows. We need first to make some recalls concerning what we have decided to call “bouncing skew Brownian motions.” In the following introductory section, we recall the construction of excursions of the skew Brownian motion from those of a Brownian motion, and we introduce the object of study of this paper, namely the “difference x,β ,β x,β process” {Z t 1 2 = X τ (X2 0,β1 ) : t ≥ 0}. t The first part of this paper (Sect. 2) entitled “Bouncing skew Brownian motions on the real line” is devoted to the proof of our main results, which provide a description of the bouncing skew Brownian motions on the whole real line, after its first hit at zero. This part is decomposed in various subsections. We begin to show that the difference process is a self-similar process extending the killed process studied in [9] to the whole time strip [0, ∞) (see Proposition 3). In particular, the difference process itself admits an excursion process and a local time at 0 related through the master formula of Markovian exit systems. Based on the theory of Markovian extensions for selfsimilar processes (see, e.g., [8] and the references therein), we identify the underlying Lévy process given via the Lamperti transform and prove that Cramer’s condition is satisfied for the killed difference process. As a consequence, there exists a unique Markovian extension of the killed difference process that leaves zero continuously. We then prove our first main result (Theorem 4) : namely that the extension of the killed difference process to the whole strip that leaves 0 continuously is identified as the difference process of bouncing skew Brownian motions on the whole line
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itself. Consequently, the difference process appears as a solution of some jump S.D.E. driven by the excursion process of X β1 (see Theorem 5). Then, we manage to compute 0,β ,β explicitly the density of the difference process Z t 1 2 starting from the origin 0 with the help of an Itô-Dynkin formula adapted to our case. In the second part of the paper (Sect. 3), we present two applications of the previous theoretical results. We give a brief study of the excursion process related to the difference process. Namely, we compute the entrance law of the excursion process associated with the difference process and provide a last exit decomposition. We conclude this part with the study of the Markovian dependence of the skew Brownian motion w.r.t the skewness parameter β. We show that the previous results permit to give a first answer to an open question formulated initially by Burdzy and Chen [5]. In particular, we give the form of the inhomogeneous generator of the Markovian process with “time” β (see Sect. 3.2.2). Let us emphasize that these applications are obtained by direct computations, mainly using the explicit formula for the density of the difference process starting from the origin computed previously , so that the reader might want to skip the first part of this paper, keeping only formula (51) of Theorem 6 in mind, and go directly to the second part (Sect. 3) . 1.3 Bouncing Skew Brownian Motions Let x ≥ 0. Consider the two skew Brownian motions, x,β2
= x + Bt + β2 L 0t (X t
0,β Xt 1
0,β Bt + β1 L 0t (X t 1 ),
Xt
x,β2
=
),
(2) (3)
constructed on the same probability space (, F, Px ) that supports their common driving standard Brownian motion (Bt )t≥0 . When x = 0, we will simply write P instead of P0 . The main issue of this paper is to study the c.a.d.l.a.g. process defined as Z ux,β1 ,β2 = X τ
x,β2 0,β1 ) , u (X
(4)
where τu (X 0,β1 ) is the inverse of the local time of X 0,β1 , given by τu (X 0,β1 ) = inf{t ≥ 0 | L 0t (X 0,β1 ) > u}. 0,β
x,β ,β
x,β
0,β
Note that, since X τ (X1 0,β1 ) = 0, we have Z u 1 2 = X τ (X2 0,β1 ) − X τ (X1 0,β1 ) . u u u Since throughout this note the parameter β1 is associated with the process starting from 0 and β2 to the one starting from x, we will, from now on, suppress the dependence upon the skewness parameter and write X 0 , X x , Z x for X 0,β1 , X x,β2 , Z x,β1 ,β2 when no confusion is possible.
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Let us state our assumption h on β1 , β2 that will be used throughout this paper : h :
0<
β2 < β1 < β2 < 1. 1 + 2β2
(5)
In [5] Theorem 1.4 (iii), it is proved that, almost surely, for every t0 > 0, there exists t > 0 such that x + β2 L 0t (X x ) = β1 L 0t (X 0 ) and so X tx = X t0 at infinitely many times (see [5] Theorem 1.4 (iii)). In turn, this implies that the positive process Z x hits zero at infinitely many times. This justifies that, we will call below Z x the “difference process,” which in some manner describes how both skew Brownian motions “bounce” on one another. 1.3.1 Excursions of a Skew Brownian Motion Consider X 0,β a skew Brownian motion starting from 0 and introduce the inverse of its local time τu (X 0,β ) = inf{t ≥ 0 | L 0t (X 0,β ) > u}. Recall that the excursion process 0,β (eu )u>0 associated with X 0,β is eu (r ) = X τ (X 0,β )+r , for r ≤ τu (X 0,β ) − τu− (X 0,β ). u− The Poisson point process (eu )u>0 takes values in the space C0→0 of excursions. For e ∈ C0→0 , we denote R(e) the lifetime of the excursion and recall that by definition e does not hit zero on (0, R(e)), and e(r ) = 0 for r ≥ R(e). If we denote nβ the excursion measure of the X 0,β , we have the formula, for any Borel subset A of C0→0 , nβ (A) =
(1 + β) (1 − β) n|B.M| (A) + n|B.M| (−A) 2 2
(6)
where n|B.M| is the excursion measure for the absolute value of a Brownian motion. 1.3.2 Recalls on the Difference Process Up to Its First Hit at Zero In this paragraph, we make some recalls on known facts concerning the “distance process” derived in [9]. Up to its first hit at zero, the difference process is solution to a stochastic differential equation with jumps, driven by the excursion Poisson process of X 0 . Let us introduce (eu )u>0 the excursion process associated with X 0,β1 , 0,β1 0,β1 )+r , u− (X
eu (r ) = X τ
for r ≤ τu (X 0,β1 ) − τu− (X 0,β1 ).
The Poisson point process (eu )u>0 takes values in the space C0→0 of excursions with finite lifetime, endowed with the usual uniform topology. Remember that nβ1 stands for the excursion measure associated with X 0,β1 . 0,β 0,β Let us define T = inf{t ≥ 0 | X t 1 = X t 2 } ∈ [0, ∞] and U = L 0T (X 0,β1 ). Since X x,β2 and X 0,β1 are driven by the same Brownian motion, it is easy to see that they can only meet when X 0,β1 = 0. As a consequence, we have U = inf{u ≥ 0 | Z ux,β1 ,β2 = 0} ∈ [0, ∞], and Z x,β1 ,β2 > 0 on [0, U ).
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The description of Z x up to time U given in [9] is the following. Theorem 1 (See [9] Theorem 1) Assume x > 0 and 0 < β1 , β2 < 1. Almost surely, we have for all t < U , Z tx = x − β1 t +
x β2 (Z u− , eu ),
(7)
0
where : (0, ∞) × C0→0 → [0, ∞) is a measurable map. For h > 0, we can describe the law of e → (h, e) under nβ1 by 1+β a − 2β22 1 − β1 1+ , ∀a > 0. nβ1 ((h, e) ≥ a/β2 ) = 2h h
(8)
Corollary 1 Assume x > 0 and 0 < β1 , β2 < 1. We have for all t < U , Z tx = x − β1 t +
[0,t]×(0,∞)
aμ(du, da),
(9)
where μ(du, da) is the random jumps measure of Z x,β1 ,β2 on [0, U ) × (0, ∞). The x,β ,β compensator of the measure μ(du, da) is du × ν(Z u− 1 2 , da) with a −γ κ 1 + 1{a>0} da h2 h
ν(h, da) = where κ =
(1−β1 )(1+β2 ) 4β2
and γ =
(10)
1+3β2 2β2 .
Remark 1 Theorem 1 fully details the dynamic of the “distance process” before it (possibly) reaches 0. The “distance process” decreases with a constant negative drift and has positive jumps. Moreover, the value of a jump at time u is a function of the x,β ,β level Z u− 1 2 and of the excursion eu . The image of the excursion measure under this function, with a fixed level h > 0, is given by the explicit expression (8).
Remark 2 For 0 ≤ x ≤ x , it is possible to construct X 0,β1 , X x,β2 and X x ,β2 on the
same probability space. We have X 0,β1 ≤ X x,β2 ≤ X x ,β2 almost surely. By time change, we also have
(11) Z x,β1 ,β2 ≤ Z x ,β1 ,β2 . x,β ,β
Note that for any 0 ≤ t ≤ 2βx 1 , it is easy to show that Z t 1 2 has the same moments as the law of the jumps characterized by ν. Moreover, (11) ensures that this property extends for all t ≥ 0. In particular, we show that E for any ξ ∈ [1, 23 +
123
1 2β2 ).
x,β1 ,β2 ξ −1
Zt
< +∞
(12)
J Theor Probab
Let us introduce, for u ≥ 0, the sigma field, Gu = Fτu .
(13)
With these notations, the process (Z ux )u≥0 is (Gu )u≥0 adapted. Proposition 1 (See [9] Proposition 2) Assume x > 0 and 0 < β1 < β2 < 1. Almost surely, one has the representation for all t < U
L 0τt (X x ) =
(X τxu− , eu ),
(14)
0
where : (0, ∞) × C0→0 → [0, ∞) is the measurable map defined in Theorem 1. Moreover, we may show that almost surely if X τxu− > 0, L 0τu (X x ) − L 0τu− (X x ) = (X τxu− , eu ),
for all u with τu − τu− > 0.
(15)
Concerning the law of U , it is given by the following result. Theorem 2 (See [9] Theorem 3) Assume h-(5) and x > 0. 0,β x,β Then, the hitting time T = inf{t > 0 | X t 1 = X t 2 } is almost surely finite. 0 0,β 1 ), then the law of U has the density Denote U = L T (X ξ −2 1−3β1 2β1 x 1− 1[ βx ,∞) (u)du 1 β1 u 2β1 (16)
1 = 1 − 1 . where b(a, b) = 0 u a−1 (1 − u)b−1 du = (a) (b) and ξ (a+b) 2β1 2β2 β1 pU (x, du) = 1−β1 x b(1 − ξ , ) 1
Hence,
x β1 U
β1 u x
1 is distributed as a beta random variable B(1 − ξ , 1−β 2β1 ).
1.3.3 A Reminder of Notations and the Value of Constants Involved in the Computations We recall here the value of constants that will be thoroughly used in this article. 1. β1 ∈ (−1, 1), β2 ∈ (−1, 1) satisfy assumption h : 0 < 2. κ :=
(1−β1 )(1+β2 ) 4β2
; γ :=
(1+3β2 ) 2β2
;
ξ
:=
1 2β1
−
1 2β2
β2 1+2β2
< β1 < β2 < 1.
; θ := (1 − ξ ).
2 Bouncing Skew Brownian Motions on the Real Line In the next paragraph, we state the first result of this paper, which gives a first description of the “distance process” after it reaches 0.
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2.1 The Difference Process as a Jump Process Reflected at Zero In this section, we describe the difference process as a reflected stochastic differential equation with jumps. For an account concerning reflected SDE with jumps (in the case where the jump measure is finite), we mention [10]. However, we will see that the description of Z using standard methods as in [10] fails to describe precisely the behavior of the process at the boundary because of the degeneracy of the jumping measure at the boundary 0. Proposition 2 Assume h-(5) and x > 0. We have for all t ≥ 0, Z tx = x − β1 t + β2
x x >0 (Z I Z u− u− , eu ) + β2
u
u
x =0 I Z u−
τu
τu−
Ieu (s)<0 dL 0s X x
t
+ β1
I Z sx =0 ds = x − β1 t + 0
[0,t]×[0,∞)
a μ(du, ˜ da) + K t ,
(17)
where 1. Z tx t≥0 is an (Gt )-adapted process with values in [0, ∞). 2. μ(du, ˜ da) is the random measure of Z x on [0, ∞) × [0, ∞) with compensator x , da) with given by du × ν˜ (Z u− ν˜ (h, da) = 1{h>0}
a −γ κ 1 + 1{a>0} da. h2 h
(18)
3. (K t )t≥0 is an (Gt )-adapted process, null at t = 0 and such that 0
t
Z sx dK s = 0.
(19)
Remark 3 Note that in (9), the compensator of the random measure μ(du, da) described in (10) is not well defined for h = 0. Contrary to the random measure μ(du, da) in (9), the random measure μ(du, ˜ da) of (17) is now defined on the whole strip [0, ∞) × [0, ∞). This explains the difference between Eqs. (9) and (17). Proposition 2 is an attempt to study the difference process in the framework provided by the theory of reflected stochastic differential equation with jumps. Unfortunately, the non-differentiable character of (K t ) (due to the explosion of the measure describing the jumps near the boundary) does not permit to apply Itô’s formula directly. Before turning to the proof of Proposition 2, we need to clear out various preliminary results.
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Lemma 1 Assume h-(5) and x > 0. We have that β1 0
t
t I X sx =0 dL 0s X 0 = β2 I X s0 =0 dL 0s X x .
(20)
0
Proof Since X t0 − X tx t≥0 is a process with bounded variations, its local time is null. Thus, applying the Itô-Tanaka formula (for the symmetric sign function satisfying sgn(0) = 0), we have + X tx ∨ X t0 = X t0 − X tx + X tx t 1 I X s0 >X sx + I X s0 =X sx d X t0 − X tx + X tx = 2 0 t 0 x = x + Bt + β2 L t X + β1 I X s0 >X sx dL 0s X s0 0 t I X s0 >X sx dL 0s X sx − β2 0 t t x 1 0 0 0 β1 + . I X sx =X s0 dL s X − β2 I X s0 =X sx dL s X 2 0 0 But from the comparison principle for skew Brownian motions, we have that I X s0 >X sx = 0, so that + + X tx X tx ∨ X t0 = X tx = X t0 − X tx = x + Bt + β2 L 0t X x t t x 1 0 0 0 β1 + , I X sx =0 dL s X − β2 I X s0 =0 dL s X 2 0 0
so that we have necessarily (20). Lemma 2 Assume h-(5) and x > 0. (Z tx )t≥0 is continuous at time U . Proof This a consequence of [12] Lemma 3.2 p. 213.
Proof (proof of Proposition 2) Let e ∈ C0→0 and for any fixed h > 0 consider e → Xˆ h (e) the mapping constructed in [9], which gives a possible solution of Xˆ sh (e) = h + e(s) + β2 (h, e) , with
1 ε0+ 2ε
t
(h, e) = lim
0
I Xˆ h (e)∈(−ε,ε) ds. s
(21)
(22)
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It is shown in [9] that the solution of the above Eq. 21 is well defined for nβ1 -a-e excursion e ∈ C0→0 . From (20), we deduce X tx = x + X t0 − β1 L 0t X 0 + β2 L 0t X x t t = x + X t0 − β1 L 0t X 0 + β2 I X s0 <0 dL 0s X x + β2 I X s0 =0 dL 0s X x 0 0 t t = X t0 + x − β1 L 0t X 0 + β2 I X s0 <0 d L 0s X x + β1 I X sx =0 dL 0s X 0 . 0
0
Note that the measure I X s0 <0 d L 0s (X x ) is singular w.r.t. the measure d L 0s X 0 because the ladder only increases on the set {s ≥ 0 : X s0 = 0}. Thus, combining the results of Proposition 1, of Lemma 2, and the Eq. (15),
τt I X s0 <0 dL 0s X x + β1 I X sx =0 dL 0s X 0 0 0 t τu x 0 = x − β1 t + β2 Ieu (s)<0 dL s X + β1 I X τxs =0 ds
Z tx = x − β1 t + β2
τt
τu−
u
= x − β1 t + β2 + β2
x >0 I Z u−
u
x =0 I Z u−
u
τu−
= x − β1 t + β2
0
L 0τu
X
x
−
L 0τu−
Ieu (s)<0 dL 0s X x + β1
x x >0 (Z I Z u− u− , eu ) + β2
u
t
+ β1 0
X
t 0
x
I X τxs =0 ds. x =0 I Z u−
τu
τu−
u
Ieu (s)<0 dL 0s X x
I Z sx =0 ds.
Hence, we may set K t := β2
u
I
x =0 Z u−
τu
τu−
Ieu (s)<0 dL 0s
X
x
+ β1
t
0
The description given at Corollary 1 gives the announced result.
I Z sx =0 ds.
2.2 The Difference Process as a Self-Similar Extension of the Killed Difference Process In this section, we show that the difference process is self-similar and extends positively the killed difference process.
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2.2.1 Self-Similarity of the Difference Process Remember that U := inf{t > 0 : X τ (X2 0,β1 ) = 0}. x,β t
x,β = X τ (X2 0,β1 ) the process killed when it first reaches 0. We ∗ t t≤U t≤U ∗ may extend Z t†,x on the whole time line [0, ∞) to a process that we still note t≤U ∗ abusively Z t†,x such that 0 is a trap for Z †,x . t≥0
Let Z t†,x
We have
Z t†,x = x − β1 t +
†,x β2 (Z u− , eu ),
t < U
0
= x − β1 t +
[0,t]×(0,∞)
aμ(du, da),
t < U .
Proposition 3 Assume h-(5) and x > 0. The process Z tx t≥0 is a positive Markov 1-self-similar recurrent extension of Z t†,x . t≥0
Proof First, let us prove that Z t†,x
t≥0
is a self-similar process with index 1.
Indeed, let x > 0, c > 0. Note that
C0→0
β2 (h, e)nβ1 (de) =
∞
aν(h, da) = β2
0
1 − β1 1 − β2
does not depend on h. Thus, −1 −1 −1 c−1 Z c†,x t = c x − c β1 ct + c
†,x β2 (Z u− , eu ),
t ≤ c−1 U
0
β2 − β1 †,x = c−1 x + t + c−1 β2 (Z u− , eu ) 1 − β2 0
C0→0
β2 − β1 †,x = c−1 x + t+ β2 c−1 (Z cv− , ecv ) 1 − β2 0
0 C0→0 :=n˜ β1
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The process Z .c
−1 x
:= c−1 Z c†,x . is solution of the equation :
y
Z t = y − β1 t +
β2 c−1 (cZ v− , ecv ) y
0
0, n˜ β1 c−1 (ch; e) ≥ a = c nβ1 c−1 (ch; e) ≥ a
1+β2 β2 ca − 2β2 1 − β1 1+ =c 2ch ch 1+β2 β2 a − 2β2 1 − β1 1+ = . 2h h Consequently, we see that of c−1 the “law” (ch; e) under n˜ β1 is given by ν(h; da). −1 †,c x †,x −1 The processes Z t and c Z ct share the same infinitesimal generator, t≥0
t≥0
−1 start from the same point y = c x, and 0 is a trap for them both : this ensures that †,x is a 1-self-similar process. Zt t≥0
Second, we have that 1 1 c 2ε
c2 t
0
I[−ε,ε] (X s0 )dX 0 s
t c = I[−ε,ε] (X c02 u )du 2ε 0 t c 1 1 = I[− εc , εc ] ( X c02 u )d X c02 . u 2ε 0 c c
from which we deduce that 1 0 L 2 (X 0 ) = L 0t c ct
1 0 X 2 c c.
and
X t0 , L 0t (X 0 )
t≥0
∼
1 0 X 2 , L0 c ct t
1 0 X 2 c c.
. t≥0
Moreover, since X 0 and X x are driven by the same Brownian motion, we have 1 0 1 cx 0 1 0 X c2 t , X c2 t , L t X c2 . X t0 , X tx , L 0t (X 0 ) ∼ . t≥0 c c c t≥0
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Now, let Y := 1c X c02 . . We have that c2 τt0 (Y ) = c2 inf u > 0 : L 0u (Y ) > t = inf u > 0 : L 0u/c2 (Y ) > t = inf u > 0 : L 0u (cY./c2 ) > ct = τct0 (cY./c2 ) = τct0 (X 0 ). From the above, we deduce :
Z tx t≥0
=
=
∼
X τx 0 (X 0 ) t t≥0 1 cx X 0 c τct ( X 0 )
t≥0
1 cx X c c2 τt0 1c X c02 .
t≥0
1 cx Z = . c ct t≥0
Consequently, Z tx t≥0 is a 1-self-similar process, and it is an extension of the 1-self-similar process Z t†,x . t≥0 As mentioned in the introduction, the difference process Z tx t≥0 hits zero infinitely many times, ensuring that it extends Z t†,x recurrently.
t≥0
Corollary 2 From [20] p. 551 or [12] p. 220, we deduce
t
0
I Z sx =0 ds = 0,
∀t > 0, Px − a.s.
(23)
In particular, from (17), we arrive at the description Z tx = x − β1 t + β2
u
x x >0 (Z I Z u− u− , eu ) + β2
u
x =0 I Z u−
τu
τu−
Ieu (s)<0 d L 0s X x . (24)
2.2.2 Master Formula for Excursions of the Difference Process Note that because 0 is a regular point, there exists a local time 0t (Z ) for the difference process (Z t ) at 0. This is a positive continuous additive functional of the difference process (Z t ), increasing only on the visiting set {t ≥ 0 : Z t = 0}. Such 0t (Z ) 0 is uniquely determined up to a multiplicative constant. We normalize t (Z ) so that
∞ E 0 e−t d0t (Z ) = 1. Let us introduce also the inverse local time ςt0 of 0t (Z ) and defined by ςt0 := inf s > 0 : 0s (Z ) > t , t ≥ 0.
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J Theor Probab
Let M denote the closure of the zero set {t ≥ 0 : Z t = 0}, and let G denote the c set of strictly positive left endpoints of the maximal intervals components of M . We may associate with the excursions (e s ) of the difference process (Z t ) away from 0 0 the predictable exit system n, (Z ) , where n is a σ -finite measure on (, G ∗ ) (G ∗ denotes the universal completion of G 0 ) and such that if (θs ) denotes the usual shift operators on , we have the master formula Ex
∞
Vs .F ◦ θs = n(F).Ex 0
s∈G
Vs d0s (Z )
(25)
for any predictable positive process V and bounded G ∗ measurable functional F. For an arbitrary excursion e of the difference process, let us denote R(e) := inf{t > 0 : et = 0}. We will sometimes need a slight extension of (25), namely ⎡ 0⎣
E
⎤ Vs .F(s, e )⎦ = E0
∞
s
0
s∈G
Vs n (F(s, .)) d0s (Z )
(26)
holding for any measurable F : (R × C0→0 , B(R) ⊗ U) → R (where U stands for the Borel σ -algebra of C0→0 ). 2.2.3 Continuous Extension Leaving Zero Continuously: Cramer’s Condition Let Z˜ t
t≥0
denote an arbitrary self-similar extension of Z t†,x
with values in
t≥0 0 Z˜
by a similar [0, ∞) associated with an excursion measure n˜ and a local time formula as (25). is a (the) 1-self-similar extension of Definition 1 We say that the extension Z˜ t t≥0 Z t†,x that leaves 0 continuously if t≥0
n˜ Z˜ 0 > 0 = 0.
(27)
We refer to the criterion stated in Theorem 1 of [8] (see also [17]) in terms of the underlying Lévy process named Cramer’s condition. by the Lamperti So let us introduce the Lévy process associated with Z t†,x t≥0
transformation (see, e.g., [8] p. 233). For this, we consider the continuous additive functional A x defined by Atx =
123
t 0
1 Z s†,x
ds,
t ≥0
J Theor Probab
and its right continuous inverse ξtx t≥0 defined by ξtx = inf(s > 0 : Asx > t),
t ≥ 0.
Then, according to [12], the [−∞, +∞]-valued process Hu := ln Z ξ†x , u
u≥0
is a Lévy process. This process is referred to as the Lévy process underlying Z †,x . The following result gives a criterion for the existence of a self-similar extension that leaves 0 continuously. Theorem 3 (Cramer’s condition) 1. The 1-self-similar Markov process Z † admits a (1 -) self-similar recurrent extension that leaves 0 continuously if and only if there exists θ ∈ (0, 1) such that condition E0 (exp(θ H1 )) = 1 holds. 2. There is at most one self-similar recurrent extension that leaves 0 continuously. Proposition 4 Assume h-(5) and x > 0. There exists a unique recurrent 1-self-similar x ˜ positive extension Z t of Z t†,x that leaves 0 continuously. t≥0
t≥0
We will show later that this extension corresponds in fact to Z tx t≥0 (see Sect. 2.3). Proof We check Cramer’s condition. We apply Ito’s formula to the semi-martingale ln(Z t†,x ) for t < U ∗ (recall (7)), ln(Z t†,x ) = ln(x) +
t
t 0
+
†,x Z u−
0
= ln(x) −
dZ u†,x β1 du Z u†,x
u≤t
+
u≤t
Consider the jump process Jt =
†,x †,x + Z u†,x ) − ln(Z u− )− ln(Z u−
ln 1 +
Z u†,x †,x Z u−
Z u†,x
†,x Z u−
.
(28)
†,x β2 (Z u− ,eu ) Z u†,x = u≤t ln 1+ . †,x u≤t ln 1+ †,x
Z u−
Z u−
Its compensator can be easily computed, and we have
∞ (1 − β1 )β2 β2 (h, e) a ν(h, da) = ln 1 + ln 1 + , dnβ1 (e) = h h (1 + β2 )h C0→0 0
and hence Jt = Jt −
(1−β1 )β2 (1+β2 )
t
du 0 Z u†,x
is a compensated jump process.
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J Theor Probab
Using (28), we can write ln(Z t†,x ) = ln(x) + θ
t
0
du Z u†,x
+ Jt ,
(29)
β2 −β1 (1+2β2 ) 1 )β2 < 0. with θ = (1−β (1+β2 ) − β1 = 1+β2 From (28), we have that
Htln x := ln(Z ξ†,x x ) = ln(x) − β1 t + Jξtx , t
where Jξtx =
ln 1 +
Z ξ†,x x
t ≥0
(30)
u Z ξ†,x x− u
is a subordinator possessing only pure jumps. The generator A† of the killed process Z †,x has domain D A† consisting of functions laying in the resolvent set u≤t
D A† = f ∈ C 0 ([0, ∞)) : ∃g ∈ C 0 ([0, ∞)) s.t. f (x) +∞ e−t g(Z t†,x )dt , ∀x ≥ 0 = Ex 0
and if f ∈ D A† , then there exists g ∈ C 0 ([0, ∞)) such that f is solution of the resolvent equation f (x)−A† f (x) = f (x)+β1 f (x)−
∞
[ f (x +a)− f (x)]ν(x, da) = g(x), ∀x ≥ 0.
0
(31) In particular, f ∈ C 1 ([0, ∞)). Letting x tend to 0 in the previous equation we obtain that there exists a constant δ = 0 such that f (0) + δ f (0) = g(0). The representation of f is given by f (x) = E
x
+∞
e 0
−t
g(Z t†,x )dt
=E
U
x
e 0
−t
g(Z t†,x )dt
+g(0)E
x
∞
U
−t
e dt
which is seen to tend to g(0) as x 0 thanks to (16). So that by continuity of f at 0, we deduce δ f (0) = 0. Consequently, D A† = f ∈ C 1 ([0, ∞)) : f (0) = 0 .
123
J Theor Probab
Hence, A† f (h) = −β1 f (h) +
∞
0
= −β1 f (h) +
∞
[ f (h + a) − f (h)]ν(h, da) [ f (h + a) − f (h)]
0
(32)
a −γ κ 1 + da. h2 h
(33)
for h > 0 and f an element of C 1 [0, ∞) bounded on [0, ∞) satisfying f (0) = 0. From the theory of time changes (see, e.g., Lamperti [12] p. 217), we may compute the generator B of (Ht0 )t≥0 , and we easily find that
Bg(h) = −β1 g (h) +
= −β1 g (h) +
∞
0 ∞
[g(h + a) − g(h)]κe(γ −1)h e−(a+h)(γ −1) da [g(h + a) − g(h)]κe−(γ −1)a da
(34)
0
for h > 0 and g an element of C 1 (0, ∞) bounded on [0, ∞). Consequently, the Lévy-Khintchine Formula implies ! ! E exp −λH10 = eλβ1 E exp −λJξ 1 1 ∞ 1 − e−λy e−(γ −1)y dy = exp λβ1 − κ 0 1 1 − . = exp λβ1 − κ γ −1 λ+γ −1 So that if ξ := λ = 0 or λ=
1 2β1
−
1 2β2
# " (as in [9]), we have that E exp −λH10 = 1 if either
1−β1 κ 1 + β2 β2 − β1 − 2β1 β2 − (γ −1) = − = = ξ − 1, β1 (γ −1) 2β1 2β2 2β1 β2
and Cramer’s condition is satisfied ! E exp (1 − ξ )Ht0 = 1 Observe that for β1 , β2 > 0, (1 − ξ ) > 0 ⇔ β1 > assumption h-(5).
∀t > 0. β2 1+2β2 ,
(35)
which is guaranteed by
2.3 Identification of the Difference Process as the Unique Positive Recurrent Extension of the Killed Difference Process that Leaves Zero Continuously Remember Definition 1 ; the aim of this section is to prove the following crucial result.
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J Theor Probab
Theorem 4 Assume h-(5) and x > 0. The process Z tx t≥0 is the positive Markov 1-self-similar recurrent extension of Z t†,x that leaves 0 continuously. t≥0
The key of the proof relies on the preliminary results stated below. 2.3.1 Existence and Uniqueness for Solutions of the Skew Brownian Equation Driven by Bessel Bridges Before getting started, let us introduce some notations. Notations : In the next computations, we use the notation (ρ(a)t )t≥0 for the 3(3) dimensional Bessel process starting from a, and Pa for its law. When a = 0, we simply write (ρt )t≥0 . P B stands for the law of the standard Brownian motion. The notation X x,β (ω) stands for some solution of the skew Brownian motion equation driven by (ωt )t≥0 ∈ a + W (where W is the Wiener space), namely a solution of x,β
Xt
(ω) = x + ωt + β L 0t X x,β (ω) , t ≥ 0
t 1 I[−ε,ε] X x,β (ω) ds. where L 0t X x,β (ω) = lim ε0+ 2ε 0 The key of our proof relies on the following result (3)
Lemma 3 Assume −1 < β < 1. Under P0 , there is strong existence and uniqueness for X 0,β (ρ) solution of the skew Brownian motion equation. (3) Moreover, if −1 < β < 1, L 0t X 0,β (ρ) = 0 for any t > 0, P0 -a.s., and 0,β X t (ρ) = ρt for any t > 0, P(3) 0 -a.s. (3)
Proof Remember that L 0t (ρ) = 0, P0 -a.s. so that ρ is itself a solution of the skew Brownian equation driven by itself. It remains to prove that it is the only solution, (3) namely that necessarily X x,β (ρ) = ρ under P0 . Let h > 0 and a > 0 be positive parameters. Then, by Girsanov’s Theorem, the process X −h,β (ρ(a)) is solution of −h,β Xt (ρ(a)) = −h + ρ(a)t + β L 0t X −h,β (ρ(a)) , t ≥ 0 (36) where (ρ(a)t )t≥0 is a standard Brownian motion starting from a > 0 under the ρ(a)
t∧T (ρ(a)) Pa |Ft . Because of the probability measure Pa defined by Pa(3) |Ft = a strong existence and uniqueness of the skew Brownian motion equation under Pa , this absolute continuity relation ensures that there is strong existence and uniqueness (3) for the solution of (36) under Pa . Let us now look at X −h,β (ρ), a possible solution of −h,β (3) Xt (ρ) = −h + ρt + β L 0t X −h,β (ρ) , t ≥ 0 under P0 . (37)
123
0
J Theor Probab
Using the Markov property of ρ, we see from the previous that Eq. (37) possesses a (3) unique solution (under P0 ) on [T h/2 (ρ), +∞[. But the skew equation guarantees that −h,β (ρ) cannot have increased its local time on the interval [0, T h/2 (ρ)] the process X −h,β and X t (ρ) = ρt for any t ∈ [0, T h/2 (ρ)]. We thus have a solution of (37) on the whole time interval [0, ∞) and this solution is likely seen to be unique. 1 Let T (ρ) = inf{t > 0 : ρt = 1}, we are going to give a lower bound on (3) 0 P0 L T 1 (ρ) X −h,β (ρ) ≤ ch where c > 0 is some fixed constant satisfying 0 < 1 c < |β| . Note that since the local time L 0 X −h,β (ρ) does not increase on [0, T h/2 (ρ)], the Markov property for the 3-dimensional Bessel process ρ applied at T h/2 (ρ) gives that
0 −h,β P(3) L 0T 1 (ρ) X −h,β (ρ) ≤ ch = P(3) (ρ(h/2)) ≤ ch . 0 h/2 L T 1 (ρ(h/2)) X (38) (3) B (see [18] Chap XI, Exercice 1.22 The absolute continuity between Ph/2 and Ph/2 p.450) for events in FT 1 (ρ(h/2)) = FT 1 (ρ(h/2))∧T 0 (ρ(h/2)) ensures that (3) Ph/2 L 0T 1 (ρ(h/2)) X −h,β (ρ(h/2)) ≤ ch BT 1 (B)∧T 0 (B) B . = Eh/2 I L 0 ( X −h,β (B))≤ch × h/2 T 1 (B)
(39)
B (Note that this absolute continuity relation is ensured by the fact that Eh/2 1 T (B) ∧ T 0 (B) < +∞ and the random stopping theorem for martingales applies). B IL 0 Let us evaluate Eh/2
T 1 (B))
B Eh/2 IL 0
( X −h,β (B))≤ch × BT 1 (B)∧T 0 (B) . We have
T 1 (B))
( X −h,β (B))≤ch × BT 1 (B)∧T 0 (B)
B IL 0 = Eh/2
T 1 (B))
( X −h,β (B))≤ch × IT 1 (B)≤T 0 (B)
B {there exists an excursion (necessarily positive) es of process X −h,β that reaches 1 − h + βs while s ≤ ch} = Eh/2 ! ∩ {no negative excursion es of process X −h,β reaches − h + βs while s ≤ ch} ! B {there exists an excursion (necessarily positive) es of process X −h,β that reaches 1 − h + βs while s ≤ ch} = Ph/2 ! B {no (necessarily negative) excursion es of process X −h,β reaches − h + βs while s ≤ ch} , × Ph/2
where the last equality comes from the independence of the processes of positive and 1 ). negative excursions and because c ∈ (0, |β| We compute " B Ph/2 {there exists an excursion (necessarily positive) es of process X −h,β that reaches 1 − h + βs while s ≤ ch} βch −δ(β) =1− 1+ 1+h
!
123
J Theor Probab
and B Ph/2 {no negative excursion es of process X −h,β reaches − h + βs while s ≤ ch}
= (1 − cβ)
1−β 2β
!
.
Finally, from (38) and (39), we deduce (3) P0
L 0T 1 (ρ)
2 X −h,β (ρ) ≤ ch = h
βch 1− 1+ 1+h
−δ(β)
(1 − cβ)
1−β 2β
. (40)
In particular, we see that there exists p > 0 satisfying (3)
P0
L 0T 1 (ρ) X −h,β (ρ) ≤ ch ≥ p > 0
(41)
uniformly for any h > 0 sufficiently small. We are now ready to show uniqueness for solutions X 0,β (ρ) for β ∈ (−1, 1) (case h = 0). The difficult case is when β ∈ (−1, 0), so we now assume that β ∈ (−1, 0). Let us denote 0−,β x,β (ρ) = sup X t (ρ) ; L 0t X 0−,β (ρ) = sup L 0t X x,β (ρ) . Xt x<0,x∈Q
x<0,x∈Q
Suppose that we have proven that ⎛ (3) P0 ⎝
&
{L 0t
⎞ X 0−,β (ρ) = 0, ∀t ∈ [0, s]}⎠ = 1.
(42)
s∈[0,T 1 (ρ))
Then, remember that there exists simultaneous solutions of (37) when h ∈ Q+ . The comparison theorem for solutions of the skew Brownian motion equation ensures −h,β that a.s. for any t ≥ 0, the family of r.v. {X t h ∈ Q+ } is a.s. increasing w.r.t. −h, so that h L 0t X −h,β (ρ) ≥ + L 0t X 0,β (ρ) β for any fixed h > 0. Letting h ∈ Q+ tend to 0 in the previous inequality gives L 0t X 0−,β (ρ) ≥ L 0t X 0,β (ρ) . Consequently, if (42) is proved, then there exists a (possibly random) s > 0 such that 0,β L 0t X 0,β (ρ) = 0, ∀t ∈ [0, s]P(3) 0 -a.s. But if this is the case, then X s (ρ) = ρs > 0, and since ρ does not hit 0 after time s > 0, X 0,β (ρ) will never hit 0 either after s > 0
123
J Theor Probab
because it satisfies the skew Brownian equation driven by ρ. This proves that X 0,β (ρ) cannot increase its local time, and the result of the lemma follows directly. It remains to prove (42). From (41), for any ε > 0 ⎛
&
⎝ P(3) 0
h∈(0,ε),h∈Q+
L 0T 1 (ρ)
⎞ X −h,β (ρ) ≤ ch ⎠ ≥ p
and Fatou’s lemma implies ⎛
)
(3) P0 ⎝
&
ε∈(0,1) h∈(0,ε),h∈Q+ (3)
≥ lim sup P0 h0
So that (3) P0
lim sup h0
L 0T 1 (ρ)
X
−h,β
L 0T 1 (ρ)
⎞ X −h,β (ρ) ≤ ch ⎠
L 0T 1 (ρ) X −h,β (ρ) ≤ ch ≥ p.
(3) (ρ) ≤ ch = P0 L 0T 1 (ρ) X 0−,β (ρ) = 0 ≥ p.
(43) In particular, since s → L 0s X 0−,β (ρ) is a.s. increasing and T 1 (ρ) > 0 a.s., we have ∪s∈[0,T 1 (ρ)) {L 0t X 0−,β (ρ) = 0, ∀t ∈ [0, s]} (3) L 0T 1 (ρ) X 0−,β (ρ) = 0 ≥ p. ≥ P0
(3)
P0
Note that the event ∪s∈[0,T 1 (ρ)) {L 0t X 0−,β (ρ) = 0, ∀t ∈ [0, s]} belongs to the germ σ -field J0+ = ∩s>0 σ (ρt , 0 < t < s) . So, applying Blumenthal’s zero-one law and since p > 0, we necessarily conclude that (3) P0 ∪s∈[0,T 1 (ρ)) {L 0t X 0−,β (ρ) = 0, ∀t ∈ [0, s]} = 1, which is exactly (42), and the result is proved.
(3),r
Let P0,0 the law of the Bessel bridge of dimension 3 over [0, r ]. From the previous lemma, we readily deduce the following corollary. (3),r
Corollary 3 Assume −1 < β < 1. Under P0,0 , there is strong existence and uniqueness for X 0,β (ω) solution of the skew Brownian motion equation.
123
J Theor Probab
0,β (3),r Moreover, L 0t X 0,β (ω) = 0 for any 0 ≤ t ≤ r , P0,0 -a.s., and X t (ω) = ωt for (3),r
any 0 ≤ t ≤ r , P0,0 -a.s. . Proof We give the main idea of the proof, leaving the details to the reader. (3),r Using the result of Lemma 3 and the absolute continuity relationship between P0,0 (3)
(3),r
and P0 (see, e.g., [18] Exercice 3.11 p.468), we get that under P0,0 , there is existence and uniqueness for X 0,β (ω) solution of the skew Brownian motion equation on any time interval 0 ≤ t < r − ε (0 < ε < r ). Moreover, we have L 0t X 0,β (ω) = 0 for any 0 ≤ t < r − ε (0 < ε < r ). 0,β (3),r But if this is the case, X r −ε (ω) = ωr −ε > 0, and since under P0,0 the trajectory ω does not hit 0 before time r > 0 a.s., X 0,β (ω) will never hit 0 either before r (because (3),r it satisfies the skew Brownian equation driven by ω). This proves that P0,0 -a.s. X 0,β cannot increase its local time before the end time r > 0 and the result of the corollary follows.
2.3.2 Conclusion: Proof of Theorem 4 Proof We will now use the master formula (25) Ex
∞
Vs .F ◦ θs = n(F).Ex 0
s∈G
Vs d0s (Z )
for any predictable positive process V and bounded G ∗ measurable functional F. Let us T > 0 satisfying Ex 0T (Z ) > 0 (remember the normaliza ∞choose 0 −s 0 tion E 0 e s (Z )ds = 1 so that it is easily seen that such T exists) and set F = Ilimu0 Z u >0 = I Z 0 >0 = Ilimu0 X x0 >0 and Vs ≡ I[0,T ] (s). τu
x = 0. Fact 1 : first note that ∀t ∈ G, Z t− Indeed, by definition of G, if t lies in G, there exists a sequence of times (u n ) such that u n ↑ t with Z uxn = 0. Since Z has a.s. rcll trajectories, we have 0 = limu n ↑t Z uxn = x . Z t− 0 = τ 0 . Fact 2 : second, note that if t ∈ G and Z tx > 0, then necessarily τt− t x,β 0 , then so Indeed, by the preceding (fact 1), Z τx0 = X τ 0 2 = 0. But if τt0 = τt− x,β2
would we have Z tx = X τ 0 t
t−
t−
= 0, violating our hypothesis Z tx > 0.
Let us denote Z = {s ∈ R+ : X s 1 = 0} the zero set of X 0,β1 and Zg the left-hand points of the maximal intervals in Z c .
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0,β
J Theor Probab
From fact 1 and fact 2, we deduce 0 ≤ Ex
Vs .F ◦ θs = Ex
0 =τ 0 s∈G,τs− s
s∈G
≤ Ex
Vs .I X x,β2 =0 .F ◦ θs
Vs .I X x,β2
0 =τ 0 } {s|τs− s
≤ Ex
0 τs−
0 +R(e ) τs− s
>0,X
x,β2 0 =0 τs−
Vs .H ◦ θs
(44)
s∈Zg
with H = I L 0 (X 0,β2 (e))>0 . Now applying the master formula for the skew Brownian R(e) motion (with parameter β1 ) and the result of Corollary 3 gives Ex
Vs .H ◦ θs = nβ1 L 0R(e) (X 0,β2 (e)) > 0 Ex L 0T (X 0,β1 ) = 0
s∈Zg
where we used nβ+1 L 0R(e) (X 0,β2 (e)) > 0 =
∞
0
1 (3),r P0,0 L r0 (X 0,β2 (ω)) > 0 dr = 0 √ 2 2πr 3
thanks to the result of Corollary 3. Of course, the same holds for nβ−1 L 0R(e) (X 0,β2 (e)) > 0 . Coming back to (44), this implies 0 = Ex
F ◦ θs = n(F).Ex 0T (Z )
(45)
s∈G∩[0,T ]
and n(F) = n(Z 0 > 0) = 0. The theorem is thus proved.
2.3.3 The Difference Process as a Solution of a S.D.E with Jumps Driven by the Poisson Excursion Process of the Underlying S.B.M Corollary 4 The result of Theorem 4 ensures finally that β2
u
x =0 I Z u−
τu τu−
Ieu (s)<0 d L 0s X x = 0
(46)
almost surely. From (17), (46) and the result of Corollary 2 eq-(24), we finally arrive at the following description of the difference process after time U , which is an extension of Theorem 1.
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J Theor Probab
Theorem 5 Assume h-(5) and x > 0. Then, (Z tx )t≥0 is solution of the following stochastic differential equation with 0,β jumps, driven by the excursion process (eu )u>0 of (X t 1 )t≥0 : Z tx = x − β1 t +
x β2 (Z u− , eu ),
t ≥ 0.
(47)
0
For h > 0, the law of e → (h, e) under nβ1 is described by (8). In particular, we have that K ≡ 0 in (17). From this last equation (47), we see that the description of Z using the usual theory of reflected jumping processes fails to describe the trajectories at the reflecting boundary 0. 2.4 The Law of the Difference Process Starting at Zero Let us define the family of probability measures (Px ; x > 0) on the Skorokhod space D by (48) Px (A) = P (t → Z tx ) ∈ A . As a consequence of the previous study, we know that the process (Z x , Px ; x > 0) is a self-similar Markov process. The Markov property implies that the process (Z Ux +t , t ≥ 0) is independent of the process (Z Ux +t , t ≤ U ) and its law does not depend on x. Moreover, the scaling property implies that lim x→0 U = 0 a.s.; hence, this shows that the family of measures (Px ; x > 0) converges weakly, as x goes to 0, toward the law of the process (Z Ux +t , t ≥ 0). Let us mention that in this case, Rivero [17] gives a construction of an entrance law for the process (Z Ux +t , t ≥ 0) in terms of exponential functionals of the underlying Levy process. We will denote by Z 0 , P0 a process whose law P0 is that of the process (Z Ux +t , t ≥ 0). The process Z 0 , P0 will a “difference process starting from zero.” By construction, the processes 0 be0called Z , P and (Z x , Px ) share a common infinitesimal generator. 2.4.1 An Itô-Dynkin Formula for Flat Functions Near Zero The aim of this paragraph is to prove the following proposition. Proposition 5 Assume that f : R+ → R is a C 1 function such that there exists a neighborhood V ⊂ [0, 1) of {0} and some δ > 0 such that | f (y)| ≤ C y δ for any y ∈ V . Then, we have the following Itô-Dynkin formula for f : t E f Z sx ds E f (Z tx ) − f (x) = −β1 0 t ∞ x x ! x x >0 ν + 1 Z s− ds E f Z s− + a − f Z s− ˜ Z s− , da . 0
0
Before proving Proposition 5, we need the following Lemma :
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(49)
J Theor Probab
Lemma 4 We have that
P Z tx = 0 = 0,
(50)
for any t ≥ 0, for any x > 0. Proof From (23) and taking expectations E that the equality (50) is satisfied for almost every t > 0 outside a negligible Nx set. By the absurd suppose there exists t0 > 0 and x > 0 with P(Z tx0 = 0) > δ (i.e., t0 ∈ Nx ). There exists a > 1, t1 = at0 >> t0 with P Z tx1 = 0 = 0. Then, by the comparison principle for time changed skew Brownian motions, we have x x/a 0 = P Z tx1 = 0 = P Z at = 0 = P Z t0 = 0 ≥ P Z tx0 = 0 > δ 0
yielding the contradiction. Proof (of Proposition 5) 0 = 0, Let ε > 0 be fixed. Set τε0 = τ2ε k : Z sx = ε) for any integer k ≥ 0 ; τεk+1 = inf(s ≥ τ2ε k τ2ε = inf(s ≥ τεk : Z sx ≥ 2ε) for any integer k ≥ 1.
For convenience, we introduce the notation [ f ] (y, a) = f (a + y) − f (y) for y ≥ 0, a ≥ 0. Using the Markovian nature of Z tx and using repeatedly Itô’s formula on the k , τ k+1 ] (allowed by the fact that on these intervals, the generator of [τ2ε ε intervals x Z t t≥0 is completely known), for x > 2ε, we may write f (Z tx ) − f (x) ∞ ! f (Z xk+1 ) − f (Z τxk ) + f (Z τxk ) − f (Z τxk ) 1τ k ≤t = 1 Z tx ≥2ε
τε
k=0
2ε
− f (Z xk+1 ) − f (Z tx ) 1τ k ≤t≤τεk+1 τε 2ε +1 Z tx <2ε f (Z tx ) − f (x) ∞ f f M k+1 − M k − β1 = 1 Z tx ≥2ε +
k=0
k ,τ k+1 ] (τ2ε ε
−1 Z tx ≥2ε
τε
τ2ε
∞
ds
∞ k=0
[ 0
M
f τεk+1
f]
x ,a Z s−
− Mt
f
ε
2ε
τεk+1 k τ2ε
2ε
f Z sx ds
x ν˜ Z s− , da 1τ k ≤t 2ε
τεk+1
− β1 t
f Z sx ds
123
J Theor Probab
τεk+1
+
∞
ds
f]
[
t
0 ∞
+1 Z tx ≥2ε
x ,a Z s−
2ε
f (Z τxk ) − f (Z τxk ) 1τ k ≤t + 1 Z tx <2ε f (Z tx ) − f (x) . ε
2ε
k=0
x ν˜ Z s− , da 1τ k ≤t≤τεk+1
2ε
So that, by adding and subtracting the missing bounded variation terms on the intervals k ] in order to complete the integrals, we have (τεk , τ2ε f (Z tx ) − f (x) ∞ f f = 1 Z tx ≥2ε M k+1 − M k − β1
t
+ −
ds 0
−β1
k τ2ε
τεk
k=0
+
f]
[
0 ∞
k ] (τεk ,τ2ε
−1 Z tx ≥2ε +1 Z tx ≥2ε
τ2ε
τε
k=0 ∞
x ,a Z s−
f]
[ 0
∞ k=0 ∞
M
0
1
x >0 Z s−
f τεk+1
x ,a Z s−
− Mt
f
x Z s− , da
x >0 ν 1 Z s− ˜
x Z s− , da
1τ k ≤t 2ε
1τ k ≤t≤τεk+1 2ε
f (Z τxk ) − f (Z τxk ) 1τ k ≤t + 1 Z tx <2ε f (Z tx ) − f (x) . ε
2ε
k=0
ν˜
f Z sx ds
f Z sx ds
∞
ds
t
2ε
We may complete the martingale increments too, and we may write (with a slight abuse of notation) f (Z tx ) − f (x) f,ε f,ε = 1 Z tx ≥2ε Mt − M0 − β1 +
∞
ds 0
∞
∞
−β1
+1 Z tx ≥2ε
x x >0 ν 1 Z s− ˜ Z s− , da
! f (Z τxk ) − f (Z τxk − ) 1τ k ≤t
k τ2ε
τεk
∞ k=0
x ,a Z s−
2ε
k=0
k=0
123
f]
[
0
+1 Z tx ≥2ε −
0
t
f Z sx ds
t
f
Z sx
2ε
2ε
ds +
k ] (τεk ,τ2ε
ds
∞
[ 0
f]
x Z s− ,a
x >0 ν 1 Z s− ˜
f (Z τxk − ) − f (Z τxk ) 1τ k ≤t + 1 Z tx <2ε f (Z tx ) − f (x) 2ε
ε
2ε
x Z s− , da
1τ k ≤t 2ε
J Theor Probab
f,ε k ]. is a martingale which is constant on the intervals of type (τεk , τ2ε where Mt So that f,ε f,ε | f (Z tx ) − f (x) − Mt − M0 t ∞ t x x x >0 ν f Z sx ds + ds , a 1 Z s− ˜ Z s− , da | −β1 [ f ] Z s− 0
0
≤ 1 Z tx ≥2ε
∞ k=0
+
∞
| f (Z τxk ) − f (Z τxk − )|1τ k ≤t 2ε
2ε
2ε
2ε
1τ k ≤t
k=0
0
k −] (τεk ,τ2ε
ds sup | f (z)|
∞ 0
z∈[0,2ε)
x x >0 ν a1 Z s− ˜ Z s− , da 1τ k ≤t 2ε
+ |β1 |t sup | f (z)|+2ε1 Z tx ≥2ε sup | f (z)| z∈[0,2ε)
z∈[0,2ε)
∞
1τ k ≤t +1 Z tx <2ε | f (Z tx )− f (x)|.
k=0
2ε
We now take expectations; since Z x leaves 0 continuously, we have that necessarily k −. Using Z τ k > 0; thus, we may scale the law of the jumps at the jumping times τ2ε 2ε− this fact and scaling gives f,ε f,ε E| f (Z tx ) − f (x) − Mt − M0 t ∞ t x x x
x >0 ν −β1 f Z s ds + ds , a 1 Z s− ˜ Z s− , da | [ f ] Z s− 0 0 0 ∞ x x ≤ E 1 Z tx ≥2ε | f (Z τ k − Jk ) − f (Z τ k − )|1[Z xk Jk ]≥2ε 1τ k ≤t +
∞
2ε
τ2ε −
2ε
E 1τ k ≤t
k=0
2ε
k=0
k −] (τεk ,τ2ε
ds sup | f (z)| 0
z∈[0,2ε)
∞
x >0 ν a1 Z s− ˜
+ |β1 |t sup | f (z)| + 2εE 1 Z tx ≥2ε sup | f (z)|
z∈[0,2ε)
+ E 1 Z tx <2ε | f (Z tx ) − f (x)|
2ε
z∈[0,2ε)
∞
x Z s− , da
1τ k ≤t 2ε
1τ k ≤t
k=0
where (Jk ) is a sequence of independent r.v. with density Thus,
1+β2 2β2
2ε
(1 + a)−γ .
f,ε − M0 t t ∞ x x x >0 ν f Z sx ds + ds , a 1 Z s− ˜ Z s− , da | −β1 [ f ] Z s− 0 0 0 ∞
≤ (2ε) sup | f (z)|E 1 Z tx ≥2ε |Jk − 1|1[Z xk Jk ]≥2ε 1τ k ≤t
E| f (Z tx ) − f (x) − Mt
z∈[0,2ε)
f,ε
k=0
τ2ε −
2ε
123
J Theor Probab
+
∞ k=0
E 1τ k ≤t 2ε
k −] (τεk ,τ2ε
a −γ x 1 1 1 + k Z s− >0 τ ≤t x )2 x 2ε (Z s− Z s− 0 ∞ sup | f (z)| 1τ k ≤t + E 1 Z tx <2ε | f (Z tx ) − f (x)| .
ds sup | f (z)| z∈[0,2ε)
+ |β1 |t sup | f (z)| + 2εE 1 Z tx ≥2ε z∈[0,2ε)
∞
da κ
z∈[0,2ε)
k=0
a
2ε
∞ −γ A change in variable ensures that 0 κ ha2 1 + ah 1h>0 da < +∞ and does not depend on h. Moreover, from the fact that (Z tx ) is a process having only positive jumps ∞ β1 t a.s. and that its slope is −β1 , we deduce that for any ε > 0, 1 k ≤ k=0 τ2ε ≤t ε This implies that f,ε f,ε E| f (Z tx ) − f (x) − Mt − M0 t ∞ t x x x >0 ν −β1 f Z sx ds + ds , a 1 Z s− ˜ Z s− , da | [ f ] Z s− 0
0
0
sup | f (z)| + C sup | f (z)|
≤Ct
z∈[0,2ε)
z∈[0,2ε)
sup | f (z)|+C sup | f (z)|
≤Ct
z∈[0,2ε)
≤Ct
z∈[0,2ε)
sup | f (z)| + CP Z tx < 2ε
∞ k E 1τ k ≤t (τ2ε − τεk ) + CP Z tx < 2ε
k=1 ∞
2ε
k+1 k E 1τ k ≤t (τ2ε − τ2ε ) +CP Z tx < 2ε
k=0
2ε
z∈[0,2ε)
where C is some constant depending on f and x and possibly changing from line to line. The last term is easily seen to tend to zero as ε 0 because of our assumptions on f,ε f,ε the function f . Since Mt − M0 is of zero expectation, we deduce the Itô-Dynkin formula (49).
2.4.2 Computation of the Law of Z 10 This section is devoted to the proof of the following result: Theorem 6 Assume h-(5). For all t > 0, the law of Z t0 has density p Z (t, 0, y) = c1
where c1 is defined by γ + ξ − 1 .
c1−1
1 y
tβ1 y
:=
∞
γ −1 tβ1 1−γ −ξ 1+ I y>0 , y
(51)
∗
z γ −2 (1 + z)1−γ −ξ dz = (γ − 1) (ξ )/
0
The main ingredient in the proof is that because of the self-similarity of Z , the generator A acts as a multiplier for Mellin’s transform. Let us recall that for f :
123
J Theor Probab
[0, ∞) → R, one defines Mellin’s transform of f as
∞
M [ f ] (ξ ) =
x ξ −1 f (x)dx,
0
for all ξ ∈ C such that the latter integral is well defined. It is clear that if f is bounded and with exponential decay near ∞, then ξ → M [ f ] (ξ ) is well defined and holomorphic on the half plane {ξ ∈ C | Re(ξ ) > 0}. For such functions f , we recall the four following properties which are easily derived from the definition of Mellin’s transform: M [x → f (x(1 + y))] (ξ ) = (1 + y)−ξ M [ f ] (ξ ), for Re(ξ ) > 0,
(52)
M [x → f (x)/x] (ξ ) = M [ f ] (ξ − 1), for Re(ξ ) > 1, (53) " # 1 M f (ξ ) = (1−ξ )M [ f ] (ξ −1), if f ∈ C (0, ∞) and Re(ξ ) > 1, (54) # "
1 M x → x f (x) (ξ ) = −ξ M [ f ] (ξ ), if f ∈ C (0, ∞) and Re(ξ ) > 0. (55) Proof Choose ξ with Re(ξ ) ∈ (2, 23 + to f (y) := y ξ −1 , we deduce,
E
ξ −1 Z tx
1 2β2 ).
Applying the Itô-Dynkin formula (49)
t ξ −2 Z sx − β1 (ξ − 1)E I Z sx >0 ds 0 t ∞ x ξ −1 x ξ −1 x ν˜ Z s− , da , Z s− + a +E ds − Z s−
=x
ξ −1
0
0
where the moments above are finite (see Remark 2). Performing the change in variable a˜ = Zax gives s
t x ξ −2 ξ −1 Zs = x ξ −1 − β1 (ξ − 1)E I Z sx >0 ds E Z tx 0 t ∞ κ ξ −1 +E ˜ ξ −1 − 1 ds Z sx I Z sx >0 ˜ −γ d a˜ (1 + a) (1 + a) Z sx 0 0 t x ξ −2 I Z sx >0 ds Zs = x ξ −1 − β1 (ξ − 1)E 0 t ∞ ξ −2 +E ˜ ξ −1 − 1 κ (1 + a) ds Z sx I Z sx >0 ˜ −γ d a. ˜ (1 + a) 0
0
123
J Theor Probab
ξ −2 x ξ −2 Since ξ > 2, Z tx = Zt I Z tx >0 and t ξ −1 ξ −2 E Z tx = x ξ −1 − β1 (ξ − 1) E Z sx ds 0 t ξ −2 κ κ − + E Z sx ds. γ −ξ γ −1 0 For any λ > 0, we set wλ,x (ξ ) :=
∞ 0
parts, we have
x ξ −1 wλ,x (ξ − 1) wλ,x (ξ ) = + λ λ
ξ −1 e−λt E Z tx dt. From an integration by
κ κ − − β1 (ξ − 1) . γ −ξ γ −1
λwλ,x (ξ ) = x ξ −1 + wλ,x (ξ − 1) κ κ − − β1 (ξ − 1) . γ −ξ γ −1 Set u λ,x (dy) :=
∞
e−λt p Z (t, x, dy)dt.
0
We have for x = 0, M[u λ,0 ](ξ ) = =
∞
0 ∞ 0
ξ −1 e−λt E Z t0 dt ξ −1 e−λt E Z 10 t ξ −1 dt
ξ −1 = (ξ )E Z 10 . Fubini’s theorem implies that " # # " 1 λM u λ,x (ξ ) = (ξ − 1)M y → I0
So that λ (γ − ξ ) (γ − 1) M [u λ ] (ξ ) = − (ξ − 1)β1 (γ − ξ ) (γ − 1) M [u λ ] (ξ − 1) + κ (ξ − 1) M [u λ ] (ξ − 1),
123
J Theor Probab
and λγ (γ − 1) M [u λ ] (ξ ) − λ (γ − 1) ξ M [u λ ] (ξ ) = β1 γ (γ − 1) (1 − ξ )M [u λ ] (ξ − 1) + β1 (γ − 1) ξ(ξ − 1)M [u λ ] (ξ − 1) + κ (ξ − 1) M [u λ ] (ξ − 1). Using the properties of Mellin’s transform gives # " λγ (γ − 1) M [u λ ] (ξ ) + λ (γ − 1) M y → yu λ (y) (ξ ) ! " # " #
= β1 γ (γ − 1) M u λ (ξ ) + β1 (γ − 1) M y → yu λ (y) (ξ ) − κM u λ (ξ ). Dividing by γ − 1 and using the fact that κ/(γ − 1) = (1 − β1 )/2, # " λγ M [u λ ] (ξ ) + λM y → yu λ (y) (ξ ) ! " #
= (β1 γ − (1 − β1 )/2) M u λ (ξ ) + β1 M y → yu λ (y) (ξ ) ! " #
= β1 (2 − ξ )M u λ (ξ ) + β1 M y → yu λ (y) (ξ ) with ξ :=
1 2β1
−
1 2β2 .
Inverting Mellin’s transform gives (for y > 0) :
β1 yu λ (y) − λy + β1 (ξ − 2) u λ (y) − λγ u λ (y) = 0. Let us set λ υλ (y) := uλ β1
β1 y λ
.
Then, υλ is solution of the Kummer’s equation (for y > 0) :
yυλ (y) + (2 − ξ ) − y υλ (y) − γ υλ (y) = 0. From the theory of solutions of Kummer’s equation, if M and U are the confluent geometric functions, we may deduce that there exist two constants b1 (λ) and b2 (λ) such that for y > 0 : υλ (y) = b1 (λ)U γ , 2 − ξ , y + b2 (λ)M γ , 2 − ξ , y . Here, M and U denote the confluent hypergeometric functions (the functions M and U are defined by formulas 13.1.2 and 13.1.3 in chapter 13 of [1], and see also formula 13.2.6 of [1] for an integral representation of U ). So that u λ (y) = b1 (λ)
β1 λ
∞
e
− λyt β1 γ −1
t
(1 + t)1−γ −ξ dt + b2 (λ)M γ , 2 − ξ , y .
0
123
J Theor Probab
M (γ , 2 − ξ , y) tends to ∞ with y (see [1] 13.1.4. p.504), so that necessarily b2 (λ) = 0. Moreover, the homogeneity of u λ ensures that b1 (λ) ∝ λ. Identification of the Laplace’s transform ends the proof.
Remark 4 One may want to see whether the above formula satisfies (at least formally) the Fokker–Planck forward equation, which in our case may be written as ⎧ ⎨ ∂ p Z (t, 0, y) = A∗ p Z (t, 0, y) ∂t ⎩ p Z (t, 0, y)dy −−→ δ0 (dy).
(t > 0, y > 0),
(56)
t0
Here, A∗ denotes the formal adjoint of A. It is defined for any ϕ ∈ C 1 ((0, ∞)) such that A∗ ϕ < +∞ by
a −γ 1+ y 0 −γ a κ 1+ −ϕ(y − a) 10
−γ γ −2 [ϕ(y) − ϕ(y − a)]y (y − a) 10
∗
A ϕ(y) = β1 ϕ (y) −
∞
ϕ(y)
κ y2
0
β1 Let us denote the function (t, y) → y c1−1 p Z (t, 0, y). We have
tβ1 y
γ −1
1−γ −ξ tβ1 +1 = y
−(γ +ξ ) tβ1 tβ1 (γ − 1) − ξ +1 ; y y 2 −(γ +ξ ) tβ1 tβ1 γ −1 tβ1 ∂ β1 ξ −1 β1 (t, y) = +1 −γ ; ∂y y y y y ∂ (t, y) = ∂t
β1 y
2
tβ1 y
γ −2
and
y
κ 0
(t, y − a)y −γ (y − a)γ −2 da = κ
× =−
123
y 0
tβ1 y−a
β1 y−a
1−γ −ξ +1 y −γ (y − a)γ −2 da
β1 κ 2 − γ − ξ y2
tβ1 y
γ −2
2−γ −ξ tβ1 +1 . y
tβ1 y−a
γ −1
J Theor Probab
Hence, −γ −ξ 1 tβ1 γ −2 tβ1 ∂ − A∗ (t, y) = 2 −1 ∂t y y y tβ1 tβ1 tβ1 − β12 ξ −1 × β12 (γ − 1) − ξ −γ y y y 2 tβ1 tβ1 tβ1 κβ1 κβ1 . +1 + +1 + γ −1 y y 2 − γ − ξ y
One may easily check at once that κβ1 κβ1 κβ1 + β12 1 − ξ + = β12 (γ − 1) + γ − 1 2 − γ − ξ 2 − γ − ξ κβ1 2κβ1 = −β12 ξ + β12 γ + = 0. + γ − 1 2 − γ − ξ So that "∂
∂t
# − A∗ p Z (t, 0, y) = 0.
Remark 5 (A remark about the law of Z starting from x > 0) Set u λ,x (y) :=
∞
e−λt p Z (t, x, y)dt. We may retake the previous computations
0
without letting x tend to 0. Then, Fubini’s theorem implies that " " # # 1 λM u λ,x (ξ ) = (ξ − 1)M y → I0
(γ − 1) 1 − e−2iπ xθ − 2iπ θ xe−2iπ xθ ζ0 + β1 (1 − ξ )u λ,x (0 )θ −γ (2iπβ1 θ − λ)ξ +γ −2 dθ 1−γ × ζ 1−γ (2iπβ1 ζ − λ)ξ −(1−γ ) − ζ0 (2iπβ1 ζ0 − λ)ξ −(1−γ ) . ζ
where uˆ λ,x stands for the Fourier transform of u λ,x . Unfortunately, this formula does not seem easily invertible.
123
J Theor Probab
2.4.3 Mean Longtime Behavior From this expression of p Z (t, 0, y)dy, we deduce the following results. Proposition 6 Assume h-(5). 1 Set π(y) = cξ1 y 1−ξ defined for y > 0. Then, for any fixed y > 0, β1
|t ξ p Z (t, 0, y) − π(y)| −−−−→ 0. t→+∞
Proposition 7 Assume h-(5).
∞ For any real Borel function f defined on [0, ∞) such that 0
1 t 1−ξ
t 0
E f Z s0 ds −−−−→ t→+∞
1 1 − ξ
+∞
f (y) y 1−ξ
< ∞,
f (y)π(dy).
0
Proof As already noticed, under hypothesis h-(5), we have that 0 < 1 − ξ ∗ < 1. It is
t not hard to see that 0 p Z (s, 0, y)ds = O(t 1−ξ ) and is strictly increasing as t tends to ∞. Moreover, from the Fubini–Tonelli theorem and a simple change of variable, we have that
1 t 1−ξ
t 0
E f Z s0 ds =
+∞
t→+∞
with X
+∞
t 1−ξ
−ξ c1 β1 y = f (y) 1−ξ y tβ1 0 +∞ −−−−→ f (y)π(dy)
t
1
0 0 1−ξ
f (y) p Z (s, 0, y)dyds
tβ1 y
u
γ −1
(1 + u)
1−γ −ξ
du dy
0
0
1 1 −ξ dy = lim c1 c2 β1 , c := 2 1−ξ 1−ξ X →+∞ X y 1−ξ u γ −1 (1 + u)1−γ −ξ du and where the last line comes from Lebesgue’s domπ(dy)
0
ination theorem.
:=
We may check that π(dy) thus defined is in fact an invariant measure for (Z t )t≥0 meaning that A∗ π(dy) = 0.
3 Applications In this section, we present two applications that may be derived from the expression of the law of Z x given in Theorem 6.
123
J Theor Probab
3.1 Excursions of the Difference Process From the result of Theorem 2, we have that Ex (1−exp(−U )) 1 lim = x→0+ 1 x 1−ξ b 1−ξ , 1−β 2β1 where b(a, b) =
1 0
u a−1 (1 − u)b−1 du =
∞
(1−e−u )β1 (β1 u)ξ
−2
du ∈ (0, ∞)
0
(57)
(a) (b) (a+b) .
x (Z ) has been shown to be the positive self-similar recurrent extension of Since Z x,† that leaves 0 continuously, the theory of self-similar recurrent extensions asserts from (57) that the excursion measure n is self-similar with index 1 − ξ . In turn, the self-similarity of index 1 − ξ for the excursion measure n of the process Z tx t≥0 implies that there exists a constant c (namely, c = n (R(e) > 1)) such that
n (R(e) > t) = c t −(1−ξ ) ,
t > 0.
∞ (see iii) Lemma 2 in [16]). In particular, n 1 − e−λR(e) = λ 0 e−λθ n (R(e) > θ ) dθ
∞ ∝ λ 0 e−λθ θ ξ −1 dθ (from the previous) and we get that n 1 − e−λR(e) ∝ λ1−ξ ,
λ > 0.
(58)
In particular, this implies that the Levy measure ν(dt) of the subordinator ςt0 , P0 takes the form dt ν(dt) = c(1 − ξ ) 2−ξ t 3.1.1 Entrance Law Let us denote Q u f (x) = Ex
f (Z u†,x )Iu
(59)
We have Q u (x, A) = Q au (ax, a A). Let (nt (dy))t>0 := n Iet ∈dy It≤R(e) t>0 denote the family of entrance laws satisfying nt Q s = nt+s (t, s > 0) and related to n. The family of entrance measures (nt (dy))t>0 may then be described in terms of the underlying Levy process thanks to the result of [17] (See Theorem 2. formula (3)), which in our case and with our notations reads 1 nt ( f ) = 1−ξ E t (1 − ξ )E J −ξ
∞ t 0 −ξ f , with J := J e−Hs ds J 0 (60)
123
J Theor Probab
and where E denotes the Lamperti transform of the canonical Lévy process under the h-transform probability measure (with respect to h : y → e(1−ξ )y ) of the law of (Ht )t≥0 . We shall prove the following result, Theorem 7 Assume h-(5). The family of entrance laws (nt (dy))t>0 related to the description of n(de) is given by (61) nt (dy) ∝ t γ +ξ −2 (y + β1 t)−γ dy. Proof From the theory of self-similar recurrent extensions (master formula applied
R(e) −λu firstly for Vs = e−λs and F = 0 e f (eu )du and secondly to for f ≡ 1), for any positive Borel function, we have u [ f ] (λ) :=
∞
e−λt
0
∞
f (y) p Z (t, 0, y) dy dt =
0
∞ 0
e−λt n f (et ) It≤R(e) dt . n 1 − e−λR(e)
(62) A notable consequence is that using (58) 1
λ1−ξ u [ f ] (λ) =
λ(ξ −1)+1
λu λ,0 ( f ) ∝
∞
e−λt n f (et ) It≤R(e) dt
0
:= L [nt ( f )] (λ). This decomposition allows to invert the Laplace transforms in order to guarantee the integrability of the integrands, and for any t > 0, ∞ t ξ −1 ∂ f (y) p (s, 0, y)dy ds ∝ n f (et ) It≤R(e) . − s) (t Z ∂s 0 0 From the previous computation, for any positive function f ∈ C 1 ([0, ∞)) such that f ∈ L 1 (π ), we get from Fokker–Planck’s equation that ∞ t nt ( f ) ∝ (t − s)ξ −1 A∗ p Z (s, 0, y) ds f (y)dy 0 0 t ∞ A∗ ∝ (t − s)ξ −1 p Z (s, 0, y) ds f (y)dy 0 0 ∞ t ∝ (t − s)ξ −1 p Z (s, 0, y) ds A f (y)dy 0 0 ∞ t ∝ (t − s)ξ −1 s γ −1 (y + sβ1 )1−γ −ξ ds A f (y)π(dy). 0
0
So that ! 1 γ −1 1−γ −ξ (λ) L s (. + sβ ) , A f 1 π λξ ! 1 ∝ ξ L s γ −1 (. + sβ1 )1−γ −ξ (λ), A f π . λ
L [ns ( f )] (λ) ∝
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J Theor Probab
Note that using the change in variable θ = sβ1 /y in the integral gives,
1 λξ
∞
0
e−λs s γ −1 (y + sβ1 )1−γ −ξ ds
y 1−ξ = γ ξ β1 λ = ∝
y 1−ξ
γ β1 y 1−ξ
λ
∞
−λ βyθ
θ γ −1 (1 + θ )1−γ −ξ dθ
−λ βyθ
(λθ )γ −1 (λ + λθ )1−γ −ξ dθ
e
0 ∞
e
1
1
0 ∞
e−uy u γ −1 (λ + uβ1 )1−γ −ξ du.
0
Hence, ! 1 γ −1 1−γ −ξ (λ), A f π L s (. + sβ ) 1 λξ ! 1 ∝ y → L s γ −1 (λ + sβ1 )1−γ −ξ (y), A f λ 1 ∞ γ −1 ∝ s (λ + sβ1 )1−γ −ξ y → A∗ e−ys , f ds. λ 0
L [ns ( f )] (λ) ∝
Note that the inverse Laplace transform of λ → (λ + sβ1 )1−γ −ξ is t → 1 γ +ξ −2 e−tsβ1 . Inverting Laplace’s transforms, we finally find that (γ +ξ −1) t nt ( f ) ∝
∞
s
γ −1
A e−ys , f ∗
t
e
0
−u sβ1 γ +ξ −2
u
du ds,
0
and
t
nt (dy) ∝
u
∝
γ +ξ −2
0 t
u γ +ξ
A∗
∞
s
γ −1 −(y+uβ1 )s
e
ds
du dy
0 −2
A∗ (y + uβ1 )−γ dudy.
0
Let us compute A∗ y → (y + uβ1 )−γ . We get that κ y −1 (y + uβ1 )−γ A∗ (y + uβ1 )−γ = −β1 γ (y + uβ1 )−γ −1 − γ −1 y −γ + κy (θ + uβ1 )−γ θ γ −2 dθ 0
κ y −1 (y + uβ1 )−γ = −β1 γ (y + uβ1 )−γ −1 − γ −1 y uβ1 −γ −2 −γ ) θ dθ + κy (1 + θ 0
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J Theor Probab
= −β1 γ (y + uβ1 )−γ −1 κ κ − y −1 (y + uβ1 )−γ + y −1 (uβ1 )−1 (y + uβ1 )1−γ γ −1 γ −1 κ y κ − β1 γ = (y + uβ1 )−γ −1 + γ − 1 uβ1 γ −1 which in turn implies that nt (dy) ∝
t
u
γ +ξ −3
(y + uβ1 )
−γ −1
0
κ 2 (y + uβ1 ) − β1 γ u dudy. γ −1
Note that γ + ξ − 3 > −1 ⇔ −3/2 + 1/(2β1 ) > −1 ⇔ 1/(2β1 ) > 1/2 guaranteed by the assumption 0 < β1 < 1, and so the above integral is definite. κ 1 − 1 = 1−3β Moreover, setting δ := (γ −1)β 2β1 . We see that 1 nt (dy) ∝ β1
t
u γ +ξ
−3−δ
0
∂ δ+1 u (y + β1 u)−γ dudy. ∂u
But γ + ξ − 3 = −3/2 + 1/(2β1 ) = δ. So γ + ξ − 3 − δ = 0 and finally nt (dy) ∝ t
1−β1 2β1
(y + β1 t)−γ dy.
Remark 6 Let us check that
∞
nt (dy)dt ∝ π(dy)
0
as announced by the theory (see for example Proposition 3(i) in [16]) : indeed, note 1−β1 1 that our assumption h implies that 1−β 2β1 > −1 and γ − 2β1 = 2 − ξ > 1, so that the integral below is definite, and we have that
∞ 1−β ∞ 1−β y −γ 1 1 + β1 nt (dy)dt ∝ t 2β1 (y + β1 t)−γ dt ∝ t 2β1 t −γ dt t 0 0 ∞ 0 2−ξ −γ −γ ∞ 1 y 2−ξ y y 1 y +β1 +β1 dt ∝ π(dy) dt ∝ π(dy) y t t y t t 0 ∞ 0 ∝ θ −ξ (θ + β1 )−γ dt π(dy). ∞
0
3.1.2 Last Exit Decomposition Before Time t = 1 and Azema’s Projection Let us introduce
gt := sup s ≤ t : Z s0 = 0
and the subfiltration (Fˇ t ) := (σ {Hgt : H ranges through (Ft ) optional processes }).
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J Theor Probab
From (62), we have (see Theorem 7.2 in [19]) p Z (1, 0, y) ∝ ∝
1
0 1
(1 − s)−ξ s γ +ξ sξ
−1
(1 − s)−ξ
0
−2
(y + β1 s)−γ ds
−γ 1 y + β1 ds. s s
In particular, we readily deduce the following result : Proposition 8 Assume U is a r.v. with a scaled type of Beta law B (ξ , 1 − ξ ), U ∼ c4 s ξ −1 (1 − s)−ξ Is∈(0,1) and M1 is a r.v. independent of U with law M1 ∼ c5 (z + −γ β1 ) Iz∈(0,∞) . We have the identity in law L
Z 10 ∼ M1 × U. Moreover, since Z t0 t≥0 is auto-similar with index 1, we have Proposition 9
L g1 = sup t ≤ 1 : Z t0 = 0 ∼ 1 − U.
Proof Introduce du := inf s ≥ u : Z s0 = 0 . From Markov’s property applied to 0 Z t t≥0 , we have for u ≤ 1 P0 (g1 ≤ u) = P0 (du ≥ 1) =
∞
p Z (u, 0, y)P y U ∗ ≥ 1 − u dy
0
∞ β tβ ξ −2 uβ1 1−γ −ξ β1 uβ1 γ −1 1 1 1+ ∝ y y y y y 0 1−u ξ +γ −3 y 1− It≥ y dtdy β1 β1 t ∞ β1 t γ −1 ξ −2−(ξ +γ −3) ∝u t (y +uβ1 )1−γ −ξ (β1 t − y)ξ +γ −3 dydt 1−u 0 ∞ γ −1 t u ∝ (y + u)1−γ −ξ (t − y)ξ +γ −3 dydt t 1−u 0 ∞ γ −1 t u 1 ∂ y + u 2−γ −ξ ∝ dydt − u + t 0 ∂y t−y 1−u t −ξ u u/(1−u) θ 1 1 −ξ 1 dτ ∝ τ dθ ∝ θ 1+τ 1−θ 1 + 1−θ (1 − θ )2 0 0 u θ −ξ (1 − θ )ξ −1 dθ. ∝
∞
0
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J Theor Probab
For any fixed t > 0, set At := t − gt . If F is a positive measurable function of the state space of excursions, set q (s, F) := n (R(e) > s)−1
F (e) n (de) . R>s
Proposition 10 (Azema’s projection) For any fixed t > 0, ! E0 F egt |Fˇ t = q (At , F)
a.s.
(63)
Proof We follow [18] Chap XII Proposition 3.3 pp 489-490 with the use of the master formula (26). We know that a.s. t is not a zero of Z , hence 0 < gt < t and q (At , F) is defined. For s ∈ G ∩ [0, t], we have s = gt if and only if s + R ◦ θs > t (where (θs ) denotes the usual shift operator on ). Consequently, if H is a positive (Ft )-predictable process, we have applying (26) : ⎡ ⎤ " # E0 Hgt F egt = E0 ⎣ Hs F(es )I R◦θs >t−s ⎦ = E0 s∈G ∞
0
=E
0
⎡ = E0 ⎣
0
Hs q (t − s, F) n (R(e) > t ⎤
∞
Hs n (FI R>t−s ) d0s (Z )
− s) d0s (Z )
" # Hs q (t − s, F) I R◦θs >t−s ⎦ = E0 Hgt q (At , F)
s∈G
Corollary 5 For any function f such that f (Z t ) is integrable, ! E f (Z t ) |Fˇ t = q At , f egt (At ) ∝ (At )1−ξ n At ( f ) ∝ (t − gt )γ −1 ∞ f (y) (y + β1 (t − gt ))−γ dy. (64) 0
Let us now follow the lines of the construction of Azema martingales as in [18] Chap.XII Exercice 4.16 p.505. We have that for any (Ft )-martingale Mt , its projection onto (Fˇ t ) is a (Fˇ t )-martingale. As an application, for f (y) = y, and Mt = Z t + β1 t +
κ κ t − γ −1 γ −2
! and using formula (64) for the computation of E Z t |Fˇ t , we find that there exists a constant c such that
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J Theor Probab
m t := β1 +
κ κ − t + c (t − gt ) γ −1 γ −2
is an (Fˇ t ) martingale. The constant c is determined by E (m t ) = 0. 1−β1
Remark 7 (Integral representation of p Z (1, 0, dy)) Choose nˆ t (dy) = t 2β1 (y + β1 t)−γ dy ∝ nt (dy) as the renormalization of the entrance law nt (dy). Then, nˆ t Q s (dy) = n t+s (dy) = nˆ t+s (dy) (s, t > 0) and
1
p Z (1, 0, dy) = c0
(65)
du(1 − u)−ξ nˆ u (dy)
(66)
0
with c0 =
γ −1 1 1−γ B (ξ ,1−ξ ) . β1
3.2 Markovian Dependance on β ∈ (0, 1) ∩ Q 3.2.1 Markov Property β 0 X 0,β , For β = 0 let L 0t := 1× L 0t X 0,0 = L 0t (B) and for β ∈ (0, 1) let L := β L t t β where L 0t X 0,β is the local time process corresponding to X t with parameter β and starting from x = 0. Unique strong solutions exist for all rational β ∈ [0, 1) simultaneously. β Fix some rational β2 ∈ (0, 1) ∈ Q and let β1 < β2 < β3 . The process X t 2 is a skew Brownian motion. Let E + = {(s, es )+ }s∈G β2 be the Poisson point process + β2 β2 of positive excursions of X t . Here, G + is the set of all s ≥ 0 such that for 0,β 0,β 0,β some 0 < gs < ds , we have s = L 0gs X 0,β2 , X gs 2 = 0 = X ds 2 , X v 2 > 0 for 2 for u ∈ [0, ds − gs ). v ∈ (gs , ds ) and es+ (u) = X gs +u We define in analogous way the Poisson point process E − = {(s, es )− }s∈G β2 of − β negative excursions of X t 2 . The processes E + and E − are independent.
0,β
β
β ,β2
Let Ls 3 = Ls 3
β
β
:= inf{L t 3 : L t 2 > s} = L
β3
β
β
2 − τs/β
. Define Ls 1 and L0s in a
2
similar way. β2 β1 We have the inequalities L t ≥ L t for all t ≥ 0 a.s. On the intervals where β β β X t 2 is strictly negative, the value of the processes L t 2 , L t 1 , and L 0t does not β
change. In turn, this implies that Ls 1 , L0s are measurable with respect to the filtraβ tion F + = σ {(s, es )+ }s∈G β2 ,s≤u . The same reasoning ensures that Ls 3 is adapted to + β σ {(s, es )− }s∈G β2 ,s≤u . The random time T = inf s : L0s ≥ 1 = L τ 02 is a stopping +
1
β
time relative to F + . By independence of E + and E − , the random elements LT1 and
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J Theor Probab β
β
eT+ are independent of LT3 given the value of T = L τ 02 . But this can be restated as the β
β
β
β
1
β
β
independence of L τ 01 and L τ 03 given L τ 02 , since LT3 = L τ 03 and L τ 01 is a function of 1 1 1 1 1 β
β
LT1 and eT+ . This proves the Markov property for the process β ∈ [0, 1) ∩ Q → L τ 0 . 1 β
3.2.2 The Description of the Process β ∈ (0, 1) ∩ Q → X τ 0 1
β
Conditionally on L τ 0 = a, we have that 1
β+ε
β
− Xτ0 = X
Xτ0 1
1
β+ε β τa
−X
β
β β τa
.
β
β
In particular, conditionally on L τ 0 = a 1
β+ε
Xτ0 1
In particular for β + ε > β >
β+ε
= Za
β
β+ε 1+2(β+ε) β+ε
Xτ0 1
> 0 i.e., for β+ε
= Za
β
Hence, we find that for any ε ∈ Q such that L
β+ε β X τ 0 |X τ 0 1 1
=a
+ a. 2β 2 1−β
>ε>0
+ a.
2β 2 1−β
> ε > 0,
1+(β+ε) − 1+β 2β 2(β+ε) a a 1+ I y>a dy y−a y −a 1+(β+ε) − 1+β 2β 2(β+ε) y a 1 = c1 (β, β + ε) I y>a dy y−a y−a y−a := qβ,β+ε (a, y)dy.
1 = c1 (β, β +ε) y −a
β
The process β ∈ (0, 1) ∩ Q → X τ 0 is an a.s. increasing process. 1
Let us check the Chapman–Kolmogorov equations. For notational convenience, we set δ(r ) := 1+r 2r (r > 0). Then, for any y > a, ε ∈ Q such that 2(β+ε)2 1−(β+ε)
>θ >0:
123
2β 2 1−β
> ε > 0, and θ ∈ Q such that
J Theor Probab
qβ,β+ε (a, z)qβ+ε,β+ε+θ (z, y)dz ∝
δ(β+ε) −δ(β) δ(β+ε+θ) −δ(β+ε) z y a z 1 dz z −a z −a y −z y −z y −z −δ(β) δ(β+ε+θ) z a y − z δ(β+ε) z 1 1 dz, z−a y z−a z−a y−z y−z
y
1 z −a
a
∝
y
a
and performing the change in variable ζ =
y−z z−a ,
qβ,β+ε (a, z)qβ+ε,β+ε+θ (z, y)dz −δ(β) δ(β+ε) ∞ y + aζ δ(β+ε+θ) 1 1 a δ(β+ε) y + aζ ζ dζ ∝ y y−a 0 y−a ζ (y − a) ζ 1−δ(β) δ(β+ε+θ) δ(β+ε) 1 1 a ∝ y y−a y−a ∞ × ζ δ(β+ε)−1−δ(β+ε+θ) (y + aζ )−δ(β)+δ(β+ε+θ) dζ, 0
so that qβ,β+ε (a, z)qβ+ε,β+ε+θ (z, y)dz δ(β+ε) −δ(β+ε)+1+δ(β+ε+θ) 1−δ(β) a a 1 y y y−a δ(β+ε+θ) ∞ δ(β+ε)−1−δ(β+ε+θ) a a −δ(β)+δ(β+ε+θ) 1 1+ ζ ζ × dζ. y−a y y 0
∝ y −δ(β)+δ(β+ε+θ)
Finally, qβ,β+ε (a, z)qβ+ε,β+ε+θ (z, y)dz δ(β+ε+θ) 1−δ(β) δ(β+ε+θ) a 1 1 I y>a ∝y y y−a y−a −δ(β) δ(β+ε+θ) a y 1 ∝ I y>a y−a y−a y−a ∝ qβ,β+ε+θ (a, y). −δ(β)+δ(β+ε+θ)
Let us now compute the infinitesimal generators Gβ β∈(0,1)∩Q of the inhomogeβ
neous Markov process β ∈ (0, 1) ∩ Q → X τ 0 . Using constant) :
1
1 (z)
∼0 zeρz (ρ is the Euler
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J Theor Probab
1 ε0+ ε
Gβ g(a) = lim
∞
(g(a + z) − g(a))
0
1 1 2β
−
1 2(β+ε)
1 a δ(β+ε) z + a −δ(β) × dz z z z 1+β ∞ 2β a 1 1 = dz. (g(a + z) − g(a)) 2 2β z a + z 0
Note that the ladder integral is not well definite in general if g is only assumed to be bounded in some neighborhood of a, so it is not clear to answer what is the domain of Gβ . If we assume that g is bounded and is uniformly a Hölder function in the sense that there exists η > 0 and δ > 0 such that sup sup
a∈R+
z∈[0,δ[
g(a + z) − g(a) < ∞, zη
then Gβ g is well defined of all β ∈ (0, 1) ∩ Q.
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