RECURRENT EXTENSIONS OF POSITIVE SELF SIMILAR MARKOV PROCESSES AND CRAMER S CONDITION II

Transcription

1 ECUENT EXTENSIONS OF POSITIVE SELF SIMILA MAKOV POCESSES AND CAME S CONDITION II Víctor ivero Comunicación Técnica No I-6-/ (PE/CIMAT

2 ecurrent extensions of positive self similar Markov processes and Cramér's condition II Víctor IVEO This version: May 3, 26 Abstract We prove that a positive self-similar Markov process (X, IP that hits in a nite time admits a self-similar recurrent extension that leaves continuously if and only if the underlying Lévy process satises Cramér's condition. Key words: Self-similar Markov processes, Lamperti's transformation, excursion theory, Lévy processes, exponential functionals of Lévy processes. MSC: (6G8,6G5,6J25 Introduction and main result Let IP = (IP x, x be a family of probability measures on Skorohod's space D +, the space of càdlàg paths dened on [, [ with values in I +. The space D + is endowed with the Skohorod topology and its Borel σ-eld. We will denote by X the canonical process of the coordinates and (G t, t will be the natural ltration generated by X. Assume that under IP the canonical process X is a positive self-similar Markov process (pssmp, that is to say that (X, IP is a [, [-valued strong Markov process and that it has the scaling property: there exists an α > such that for every c >, ({cx tc /α, t }, IP x Law = ({X t, t }, IP cx x. We will assume furthermore that (X, IP is a pssmp that hits in a IP-a.s. nite time T = inf{t > : X t = }, and dies. So IP is the law of the degenerated path equal to. According to Lamperti's transformation [6] the family of laws IP can be obtained as the image law of the exponential of a { }-valued Lévy process ξ with law P, time changed by the inverse of the additive functional t t exp{ξ s /α}ds, t. ( CIMAT A.C. Calle Jalisco s/n, Col. Valenciana, C.P Guanajuato, Guanajuato, Mexico. rivero@cimat.mx

3 As usual, any function f : is extended to { } by tacking f( =. Thus, the state { } will be taken as a cemetery state for ξ and denote by ζ its lifetime, viz ζ := inf{t > : ξ t = }, and by {F t, t }, the ltration of ξ. A consequence of Lamperti's transformation is that the law of T under IP x is equal to that x /α I under P, where I denotes the exponential functional associated to ξ I := ζ exp{ξ s /α}ds. Lamperti proved the following characterization for pssmp that hit in a nite time: either (X, IP hits by a jump and in a nite time IP x (T <, X T >, X T +t =, t =, x >, which happens if and only if P(ζ < = ; or (X, IP hits continuously and in a nite time IP x (T <, X T =, X T +t =, t =, x >, and this is equivalent to P(ζ =, lim t ξ t = =. eciprocally, the image law of the exponential of any { } valued Lévy processes time changed by the inverse of the functional dened in ( is the law of a pssmp that dies at its rst hitting time of. For more details, see [6] or [8]. The main purpose of this note is continue our study initiated in [8] on the existence and characterization of positive valued self-similar Markov processes, X, that behave like (X, IP before its rst hitting time of and for which the state is a regular and recurrent state. A such process X will be called a recurrent extension of (X, IP. We refer to [8],[2] and the references therein for an introduction to this problem and for background on excursion theory for positive self-similar Markov processes. We say that a σ-nite measure n on (D +, G having innite mass is an excursion measure compatible with (X, IP if the following are satised: (i n is carried by {ω D + T (ω > and X t (ω =, t T }; (ii for every bounded G measurable H and each t > and Λ G t where ι t denotes the shift operator. (iii n( e T < ; n(h ι t, Λ {t < T } = n(ie Xt (H, Λ {t < T }, Moreover, we will say that n is self-similar if it has the scaling property: there exists a < γ <, s.t. for all a >, the measure H a n, which is the image of n under the mapping H a : D + D +, dened by H a (ω(t = aω(a /α t, t, 2

4 is such that H a n = a γ/α n. The parameter γ will be called the index of self-similarity of n. See Section 2 in [8] for equivalent denitions of self-similar excursion measure. The entrance law associated to n is the family of nite measures (n t, t >, dened by n(x t dy, t < T = n t (dy, t >. It is known that there exists a one to one correspondence between recurrent extensions of (X, IP and self-similar excursion measures compatible with (X, IP, see e.g. [2] and [8]. So determining the existence of the recurrent extensions of (X, IP is equivalent to doing so it for self-similar excursion measures. We recall that the index of self-similarity of a self-similar excursion measure coincides with that of the stable subordinator which is the inverse of the local time at of the associated recurrent extension of (X, IP. We say that a positive self-similar Markov process for which is a regular and recurrent state leaves continuously (respectively, by a jump whenever its excursion measure, n, is carried by the paths that leave continuously (respectively, that leave by a jump n (X + > = ; (respectively, n(x + = =. Vuolle-Apiala [2] proved, under some hypotheses, that any positive self-similar Markov process for which is a regular and recurrent state either leaves continuously or by jumps. In fact his result still holds true in the general setting as it is proved in the following Lemma. Lemma. Let n be a self-similar excursion measure compatible with (X, IP, and with index of self-similarity γ ], [. Then either n(x + > = or n(x + = =. Proof. Assume that the claim of the Lemma does not hold. Let n c = c (c n {X+ =} and n j = c (j n {X+ >}, be the restrictions of n to the set of paths {X + = }, and {X + > }, respectively, and c (c and c (j are normalizing constants such that n c ( e T = = n j ( e T. The measures n c and n j are self-similar excursion measures compatible with (X, IP, and with the same self-similarity index γ. According to Lemma 3 in [8] the potential measure of n c and that of n j is given by the same purely excessive measure ( T ( T n c {Xt dy}dt = C α,γ y ( α γ/α dy = n j {Xt dy}dt, y >, (2 where C α,γ ], [ is a constant. So by Theorem 5.25 in [3], on the uniqueness of purely excessive measures, the entrance laws associated to n c and n j are equal. So, by Theorem 4.7 in [7] the measures n c and n j are equal. Which lead to a contradiction to the fact that the supports of the measures n c and n j are disjoint. 3

5 If n β is a self-similar excursion measure with index γ = βα ], [ and that is carried by the paths that leave by a jump then the self-similarity implies that n β has the form n β = c α,β IP ηβ, where < c α,β < is a normalizing constant and the starting measure or jumping-in measure η β is given by η β (dx = βx β dx, x >. The choice of the constant c α,β depends on the normalization of the local time at of the recurrent extension of (X, IP. In the work [8] we provided necessary and sucient conditions on the underlying Lévy process for the existence of recurrent extensions of (X, IP that leave by a jump. For sake of completeness we include an improved version of that result. Theorem. Let (X, IP be an α-self-similar Markov process that hits the cemetery point in a nite time a.s. and (ξ, P the Lévy process associated to it via Lamperti's transformation. For < β < /α, the following are equivalent (i E(e βξ, < ζ <, (ii E(I αβ <, (iii There exists a recurrent extension of (X, IP, say X (β, that leaves by a jump and its associated excursion measure n β is such that where c α,β is a constant. n β (X + dx = c α,β βx β dx, x >, In this case, the process X (β is the unique recurrent extension of (X, IP that leaves by a jump distributed as c α,β η β. The equivalence between (ii and (iii in Theorem is the content of Proposition in [8] and the equivalence between (i and (ii is a consequence of Lemma 2 below. Thus only the existence of recurrent extensions that leave continuously remains to be established. In this vein, we proved in [8] that under the hypotheses: (H2a (ξ, P is not-arithmetic, i.e. its state space is not a subgroup of r Z, for any r. (H2b Cramér's condition is satised, that is to say that there exists a θ > s.t. E(e θξ, < ζ =, (H2c for θ as in the hypothesis (H2b, E(ξ + e θξ, < ζ <. and provided < αθ <, there exists a recurrent extension of (X, IP that leaves continuously. In a previous work, Vuolle-Apiala [2] provided a sucient condition on the resolvent of (X, IP for the existence of recurrent extensions of (X, IP that leave continuously. Actually, in [8] we proved that in the case where the underlying Lévy process is not-arithmetic the conditions of Vuolle-Apiala are equivalent to the conditions (H2b-c above. So, it is natural to ask if the conditions of Vuolle-Apiala and ours are also necessary for the existence of recurrent extensions of (X, IP that leave continuously? The following counterexample answers this question negatively. 4

6 Counterexample. Let σ be a subordinator with law P such that its law is not arithmetic and has some exponential moments of positive order, i.e. E := {λ >, < E(e λσ < }. Assume that the upper bound of E, say q, belongs to E ], [ and that the function m(x := E ( {σ >x}e qσ, x >, is regularly varying at innity with index β, for some β ]/2, [. Let (ξ, P be the Lévy process with nite lifetime ζ, obtained by killing σ at an independent exponential time of parameter κ = log (E(e qσ. By construction, it follows that Cramér's condition is satised and by Karamata's Theorem the function m (x := x E(e qξ, < ζ =, E( {ξ >u}e qξ, < ζdu, x, is regularly varying at innity with index β. As a consequence, the integral E(ξ e qξ, < ζ is not nite. We will denote by P, the Girsanov type transformation of P via the martingale (e qξs, s, viz. P is the unique measure s.t. P = e qξt P, on F t, t. Let (X, IP be the -pssmp associated to (ξ, IP via Lamperti's transformation and let V λ denote its λ-resolvent, λ >. We claim that the following assertions are satised: (P For any λ >, lim x + m (log(/x V λf(x = x q Γ(βΓ( β { f(y E (exp λy for every f :], [, continuous and with compact support; } e ξs ds y q dy, (P2 the limit exists and C q ], [; lim x + m (log(/x IE x( e T := C x q q, (P3 there exists a recurrent extension of (X, IP that leaves continuously. That the properties (P-3 are satised in the framework of Counterexample will be proved in Section 3. 5

7 In the previous counterexample we have constructed a pssmp that does not satisfy the hypotheses of Vuolle-Apiala [2] nor all the hypotheses in [8], and that nonetheless admits a recurrent extension that leaves continuously. This allowed us to realize that only Cramér's condition is relevant for the existence of recurrent extensions of pssmp. That is the content of the main theorem of this paper. To state the result we need further notation. First, observe that if Cramér's condition is satised with index θ, then the process M := (e θξt, t is a martingale under P. In this case, we will denote by P the Girsanov type transform of P via the martingale M, as we did in the Counterexample. Under the law P the process ξ is a -valued Lévy process with innite lifetime and that drift to. We will denote by J, the exponential functional J := exp{ ξ s /α}ds. A straight consequence of Theorem in [5] and the fact that (ξ, P is a Lévy process that drift to, is that J <, P -a.s. More details on the construction of the probability measure P and its properties can be found in Section (2.3 in [8]. This being said we have all the elements to state our main result. Theorem 2. Let (X, IP be an α-self-similar Markov process that hits its cemetery state in a nite time IP-a.s. and (ξ, P be the Lévy process associated to (X, IP via Lamperti's transformation. The following are equivalent: (i There exists a < θ < /α, such that E(e θξ, < ζ =. (ii There exists a recurrent extension of (X, IP that leaves continuously and such that its associated excursion measure from, say n, is such that n( e T =. In this case, the recurrent extension in (ii is unique and the entrance law associated to the excursion measure n is given by, for any f positive and measurable ( ( t α n(f(x t, t < T = t αθ Γ( αθ E (J αθ E f J αθ, t >, (3 J α with θ as in the condition (i. Observe that the condition (ii in Theorem 2 implies that the inverse of the local time at for the recurrent extension of (X, IP is a stable subordinator of parameter αθ for some < θ < /α. It is implicit in the Theorem 2 that this is the unique θ > that fullls the condition (i, and viceversa. Moreover, the expression of the entrance law associated to n should be compared with the entrance law of Bertoin and Caballero [2] and Bertoin and Yor [4] for positive self-similar Markov processes that drift to. Besides, we can ask whether the recurrent extension in Theorem 2 is of the type obtained in Theorem 2 in [8]. 6

8 Corollary. Assume that there exists a recurrent extension of (X, IP that leaves continuously and let ĨP and n denote its law and excursion measure at, respectively. For θ as in Theorem 2 the integrability condition is satised if and only if Furthermore, the latter holds if and only if E(ξ + e θξ, < ζ <, (4 n(x θ, < T <. (5 ĨE x (X θ t <, x, t. (6 During the elaboration of this work we learned that in [2] P. Fitzsimmons essentially proved the equivalence between (i and (ii in Theorem 2. He proved that Cramér's condition and a moment condition for the exponential functional I are necessary and sucient for the existence of a recurrent extension of (X, IP that leaves continuously. Actually, the moment condition of Fitzsimmons is a consequence of Cramér's condition, as it is proved in Lemma 2 below. Besides, Fitzsimmons' arguments and ours are completely dierent. He used arguments based on the theory of Kusnetzov measures and time change of processes with random birth and death, while our proof uses some general results on the excursions of pssmp obtained in our previous work [8] and some facts from the uctuation theory of Lévy processes. The rest of this note is organized as follows: Section 2 is mainly devoted to the proof of Theorem 2 and in Section 3 we establish the facts claimed in Counterexample. 2 Proofs To tackle our task we need some notation. ψ : [, θ] { } dened by The Laplace exponent of (ξ, P is the function E(e λξ, < ζ := e ψ(λ, λ. Holder's inequality implies that ψ is a strictly convex function on the set E := {λ : ψ(λ < }. So, if Cramer's condition is satised then the equation ψ(λ =, λ >, has a unique root that we will denote hereafter by θ. Observe that [, θ] E, and that ψ is derivable from the right at and from the left at θ and E(ξ, < ζ = ψ +( [, [, E(ξ e θξ, < ζ = ψ (θ ], ]. Our rst purpose is to prove that (i and (ii in Theorem are equivalent and that in Theorem 2, (i implies (ii. To reach our end we will need the following Lemma. Lemma 2. Let (ξ, P be a Lévy process and let β > be such that E(e βξ, < ζ, 7

9 and assume that β < /α. Then Furthermore, E ( I αβ <. E(e βξ, < ζ <, if and only if E ( I αβ <. Proof of Lemma 2. For t >, let Q t denote the random variable Q t := t exp{ξ u /α} {u<ζ} du. The main argument of the proof uses that E(Q αβ t <, for all t >. Indeed, the strict convexity of the mapping λ E(e λξ, < ζ implies that for any p >, E ( e (β/pξt, t < ζ = e tψ(β/p <, t >. Thus, [ { E(Q αβ t t αβ E sup e βξ u } ] {u<ζ} <u t [( { = t αβ E sup e (β/pξ u } p] {u<ζ} <u t [( { t αβ E sup e (β/pξ u } p] e uψ(β/p {u<ζ} <u t ( p p [ {e t αβ (β/pξ E t } e tψ(β/p p ] {t<ζ} p ( p p t αβ e tpψ(β/p, p owing that the process e (β/cξu uψ(β/c, u, is a positive martingale and Doob's L p inequality. We now prove the rst claim in Lemma 2. On the one hand, using the well known inequality x αβ y αβ x y αβ, x, y, we get that [ ( αβ ( E exp{ξ s /α} {s<ζ} ds On the other hand, we have a.s. ( αβ ( exp{ξ s /α} {s<ζ} ds = αβ = αβ t t t ] αβ ( exp{ξ s /α} {s<ζ} ds E t αβ exp{ξ s /α} {s<ζ} ds ( αβ exp{ξ u /α} {u<ζ} exp{ξ s /α} {s<ζ} ds du u ( αβ exp{βξ u } {u<ζ} exp{ ξ r /α} {r< ζ} e dr du, Q αβ t <. 8

10 where ξ r = ξ r+u ξ u, r, and ζ = ζ u. Thus, by taking expectations, using Fubini's Theorem and the independence of the increments of ξ we get the identity ( ( αβ ( αβ E exp{ξ s /α} {s<ζ} ds exp{ξ s /α} {s<ζ} ds = αβ t E ( = αβ E(I αβ t ( αβ exp{βξ u } {u<ζ} exp{ ξ r /α} {r< ζ} e dr du t E ( exp{βξ u } {u<ζ} du. The rst claim in Lemma 2 follows. To prove the second assertion, we assume rst that E ( e βξ, < ζ <. Thus, by making t tend to innity and integrating in the latter equation we get the identity E ( I αβ = αβ ψ(β E ( I αβ. (7 This relation is well known, see e.g. [5] and [7]. Which together with the rst assertion of the Lemma implies that E(I αβ <. We now prove the reciprocal. If E(I αβ <, then we have that ( ( ζ αβ > E exp{ξ s /α}ds ( ( ζ αβ > E exp{ξ s /α}ds {<ζ} = E ( e βξ E ( ( = E ( e βξ {<ζ} E ( ( ζ αβ exp{(ξ +s ξ /α} {+s<ζ} ds {<ζ} αβ exp{ξ s /α}ds, owing to the fact that ξ is a Lévy process. So, we get that in this case E ( e βξ, < ζ <. Theorem 2, (i implies (ii. The proof of this result is based on Theorem 3 in [8], but to use that result we rst need to establish some weak-duality relations. By assumption (i and Lemma 2 we have that E(I αθ <. Moreover, let (ξ, P := ( ξ, P denote the dual of (ξ, P. Then (ξ, P drift to, because (ξ, P drift to, and as a consequence I <, P a.s. Furthermore, (ξ, P satises the hypotheses of Lemma 2 with β = θ due to the identity Ê ( e θξ = E ( e θξ = E ( e θξ e θξ, < ζ. Thus we can also ensure that Ê ( I αθ <. Now, let ÎP be the law of the α-pssmp associated to (ξ, P via Lamperti's transformation. Then (X, ÎP is an α-pssmp that hits continuously 9 (8

11 and in a nite time ÎP -a.s. and according to Lemma 2 in [4], (X, IP and (X, ÎP are in weak duality with respect to the measure α x /α dx, x >, and given that the law IP is the h-transform of the law IP via the invariant function h(x = x θ for the semigroup of (X, IP (see Proposition 5 in [8] it then follows that (X, IP and (X, ÎP are in weak duality w.r.t. the measure α x /α θ dx, x >. Furthermore, we have that for any λ >, α dxx /α θ IE x (e λt <, Indeed, for λ > ( α dxx /α θ IE x e λt = α = E α dxx /α θ ÎE x(e λt <. (9 (α ( dxx /α θ E e λx/α I dxx /α θ e λx/α I = λ αθ E(I αθ Γ( αθ <. The same calculation applies to verify the niteness of the second integral in equation (9. This being said, Theorem 3 in [8] ensures that there exists a unique recurrent extension of (X, IP, such that the λ-resolvent of its excursion measure, say n, is given by ( T n e λt f(x t dt = f(xx /α θ ÎE x(e λt dx, ( αγ( αθ Ê (I αθ for λ, and any function f, positive and measurable on [, [. An easy calculation proves that the λ-resolvent of n satises the self-similarity property in Lemma 2 in [8] and therefore the excursion measure n is self-similar. In particular, n( e T =, and the potential of n is given by ( T n f(x t dt = f(xx /α θ dx. αγ( αθ Ê (I αθ Which compared with the result in Lemma 3 in [8] implies that Ê ( I αθ = E ( I αθ. ( Actually, Theorem 3 cited above establishes also that there exists a recurrent extension of (X, ÎP with excursion measure n such that ( T n e λt f(x t dt = αγ( αθ E(I αθ f(xx /α θ IE x (e λt dx. Moreover, the recurrent extensions of (X, IP and (X, ÎP associated to n and n, respectively, still are in weak duality. To verify that n is carried by the paths that leave continuously, we claim that the image under time reversal at time T of n is n. This follows from the fact that n and n both have the same potential and an application of a result for time reversal of Kusnetzov measures established in Dellacherie et al. [] Section XIX.23. Thus, using the

12 Markov property and that (X, ÎP is a pssmp that hits continuously and in a nite time ÎP -a.s., given that the underlying Lévy process (ξ, P drifts to, we get that n is carried by the paths that hit continuously and therefore = n(x T > = n(x + >. We will next prove that in Theorem 2, (ii implies (i. The proof is long so we will give it in two main steps: with the rst we will prove that if it is possible to construct recurrent extensions of (X, IP that leave by a jump then the condition in (i in Theorem 2 is satised; and with the second step, we will prove that if (ii holds then it is indeed possible to construct recurrent extensions of (X, IP that leaves by a jump. 2. Step The Step of the proof that in Theorem 2, (ii implies (i relies on the following Proposition. Proposition. Assume that there exists a recurrent extension, X, of (X, IP that leaves continuously and let αϑ ], [ be the index of self-similarity of the subordinator inverse of the local time at of X. We have that for any < β < ϑ there exists a recurrent extension X (β of (X, IP with associated excursion measure n β = c α,β IP ηβ, where η β is the jumping-in measure dened in the Introduction and c α,β is a normalizing constant. The proof of this Proposition will be given in Subsection 2.2. oughly, the process X (β will be obtained by erasing randomly the debut of all the excursions out from of X. Formally, this will be done using a time change associated to a uctuating additive functional. If we take for granted the existence of X (β, for all β ], ϑ[, the rest of the proof is rather elementary as we shall next explain. Owing to the Theorem this has as a consequence that for any < β < ϑ the Lévy process ξ has positive exponential moments of order β, E(e βξ, < ζ <, β ], ϑ[. So we have that lim E ( e βξ, < ζ = E ( e ϑξ, < ζ. β ϑ Nevertheless, it does not happens that E ( e ϑξ, < ζ <. Because, if this were indeed the case Theorem would imply that (X, IP admits a recurrent extension that leaves by a jump and with jumping-in measure proportional to η ϑ. Thus, that the measure m = 2 n + 2 c α,ϑ IP ηϑ, is a self-similar excursion measure compatible with (X, IP, and with index of self-similarity αϑ; as before c α,ϑ is a normalizing constant. Therefore, there exists a recurrent extension of (X, IP with excursion measure m and that may leave by a jump and continuously, which leads to a contradiction to the fact that any recurrent extension of (X, IP either leaves by a jump or continuously. Therefore, Cramér's condition is satised.

13 2.2 Step 2. Construction of the auxiliary processes The main purpose of this section is to prove the Proposition. Our rst concern will be ensure that there exists a measure ĨP on Skorohod's space D of càdlàg real valued paths dened on [,, such that under ĨP the canonical process Y is a strong Markov self-similar process that has the following properties: (Y, ĨP is a real valued αself-similar Markov process that leaves continuously, is a recurrent and regular state for (Y, ĨP, (Y, ĨP is symmetric, (Y, ĨP killed at its rst hitting time of ], ] has the same law as (X, IP, the measure of the excursions from of (Y, ĨP, say W, is supported by where {ω D : T (ω >, ω(t =, t T (ω} ( D + D D + = { ω D : ω(t >, t ], T (ω[ }, D = { ω D : ω(t <, t ], T (ω[ }, the restriction of W to the set of positive càdlàg paths D + is equal to n. This implies that there exists a local time L = (L t, t at for (Y, ĨP and that the inverse of L, say L, is a stable subordinator of parameter αϑ. The spaces D +, D are endowed with the σ-algebras G +, G, generated by the coordinate maps, respectively. Proof. We rst introduce some notations. Let IP be the law on (D, G which is the image of ( X, IP. Under IP the canonical process of coordinates X is a self-similar Markov process taking values in ], ], that dies at its rst hitting time of. We will denote by ( P t +, t ( and P t, t the semigroups of (X, IP and (X, IP, respectively. We dene a sub-markovian semigroup on \{} as follows, for any f : + measurable P t f(x = P + t f(x {x>} + P t f(x {x<} + f( {x=}, t, x. The semigroup {Pt, t } is Fellerian on \{} and satises the hypothesis in [7] Chapter 5, since the semigroups (P t +, t and (Pt, t have those properties on ], [ and ], [, respectively. Thus, there exists a unique strong Markov process on whose law will be denoted by IP, having (Pt, t as semi-group and for which is a cemetery state. Observe that when started at some x > the process (Y, IP has the same law as (X, IP. Besides, let ň be the image of n under the measurable mapping Φ : D + D dened by Φ(ω = ω. The measure ň is an excursion measure compatible with the semigroup (Pt, t. Then there exists a unique 2

14 measure W on D + D endowed with the σ-algebra G = G + G, such that W D + = n and W D = ň, and is an excursion measure compatible with the semigroup (Pt, t. The version of the Itô extension Theorem of Blumenthal [7] Chapter 5, implies that there exists a unique recurrent extension of (Y, IP, with law, say ĨP, and with excursion measure W. By construction the process (Y, ĨP has the required properties. Next, we introduce other ingredients that will be useful to prove the Proposition. For q >, let A +, A,q be the additive functionals of Y dened by A + t = t {Ys>}ds, A,q t = t q /αϑ {Ys<}ds, t, and we introduce the time change τ (q, which is the generalized inverse of the uctuating additive functional A + A,q, that is τ (q (t = inf{s > : A + s A,q s > t}, t, inf =. Now, let Y (+,q be the process Y time changed by τ (q, { Y (+,q Y t = τ (q (t if τ (q (t < if τ (q (t = ; where is a cemetery or absorbing state. consequence of the following Lemma. The proof of Proposition is a straightforward Lemma 3. Under ĨP the process Y (+,q, is a positive α-self-similar Markov process for which is a regular and recurrent state and that leaves by a jump according to the jumping-in measure c α,ρϑ η ρϑ, with η ρϑ (dx = ρϑx ρϑ dx, x >, where ρ is given by ρ = 2 + ( ( q παϑ παϑ arctan + q tan ], [, 2 and < c α,ρϑ = n(x ϑρ, < T <, is a constant. That the process Y (+,q is a pssmp follows from standard arguments. We should prove that the measure n +,q of the excursions from of Y (+,q, is such that n +,q (Y dy = c α,ρϑ η ρϑ (dy. This will be a consequence of the following auxiliary Lemma. Lemma 4. (i The processes Z +, Z,q dened by Z + (Z + t = A +, t ; Z,q (Z,q L t t = A,q, t are independent stable subordinators of parameter αϑ, and their respective Lévy measures are given by π + (dx = cx αϑ dx, and π,q (dx = qcx αϑ dx, on ], [, and c ], [ is a constant. 3 L t

15 (ii The process Z = Z + Z,q is a stable process with parameter αϑ and positivity parameter ρ = P(Z >, with ρ as dened in Lemma 3 (iii The upward and downward ladder height processes H and Ĥ associated to Z are stable subordinators of parameter αϑρ and αϑ( ρ, respectively. The proof of Lemmas 3 and 4 uses arguments from the uctuation theory of Lévy processes, we refer to [] for background on this topic. Proof. That the processes Z +, Z,q are independent subordinators is a standard result in the theory of excursions of Markov processes and follows from the fact that they are dened in terms of the positive and negative excursions of (Y, ĨP, respectively. The self-similarity property for (Z +, Z,q follows from that of (Y, ĨP. The jump measure π + of Z + is given by while that of Z,q is π + (dt = n(t dt, π,q (dt = qn(t dt. And given that the jumps of the stable subordinator L are given by the measure n(t dt, it follows that n(t dt = ct αϑ dt, t >, for some constant < c <. The proof of the assertion in (ii is straightforward. It is well known in the uctuation theory of Lévy processes that the upward and downward ladder height subordinators associated to a stable Lévy process have the form claimed in (iii. Proof of Lemma 3. By construction, the closure of set of times at which the process Y (+,q visits is the regenerative set which is the closure of the image of the supremum of the stable Lévy process Z. So is a regular and recurrent state for Y (+,q. The length of any excursion out of for Y (+,q is distributed as a jump of Z to reach a new supremum or equivalently as a jump of the upward ladder height process H associated to Z. Let N denotes the measure of the excursions from of Z Z, the process Z reected at its current supremum, viz. (Z Z t := sup { Z s } Z t, t ; s t and let denote the lifetime of the generic excursion from of Z Z. Let Π Z be the Lévy measure of Z and V be the renewal measure of the downward ladder height subordinator Ĥ, that is ( V (dy = E { H b ds s dy}. It is known in the uctuation theory for Lévy processes that under N the joint law of Z and Z Z is given by N(Z dx, (Z Z dy = V (dxπ Z (dy {<x<y}. Moreover, the Lévy measure of H, say po(dx, is such that po]x, [= V (dsπ Z (du ]x, [ (u s, x >. { s u} 4

16 cf. [2]. In our framework, (Z Z denotes the length of the generic positive excursion from for (Y, ĨP and Z is the length of the portion of the generic positive excursion from of (Y, ĨP that is not observed while observing a generic excursion from of Y (+,q. Furthermore, (Z Z Z is the length of the generic excursion from for Y (+,q, so n +,q (T dt = po(dt. Let n( T = denote a version of the regular conditional law of the generic excursion under n given the lifetime T. Similarly, the notation n +,q ( T = will be used for the analogous conditional law under n +,q. These laws can be constructed using the method in [9]. Finally, it follows from the verbal description above, that for any positive and measurable function f : + n +,q (f(y = = = = = t ], [ t ], [ t ], [ t ], [ n +,q (T dun +,q (f(y T = u N(Z dt, (Z Z dsn(f(y t T = s s ]t, [ V (dt Π Z (dsn(f(y t T = s s ]t, [ V (dt n(t dsn(f(y t T = s s ]t, [ V (dtn(f(y t, t < T. Given that the downward ladder height subordinator Ĥ is a stable process with index αϑ( ρ, it follows that n +,q (f(y = αϑ( ρ k dtt αϑ( ρ n(f(y t, t < T t ], [ = αϑ( ρ k dtt αϑ( ρ t αϑ n(f(t α Y, < T t ], [ = αϑ( ρ k n(y dx, < T dtt αϑρ f(t α x = ϑ( ρ kn = c α,ρϑ η ρ f, ], [ t ], [ ( Y ϑρ, < T u ], [ duu ϑρ f(u where k is a constant that depends on the normalization of the local time at zero for the reected process Z Z; which without loss of generality can be and is supposed to be k =. emark. The previous proof is inspired by the work of ogers [9]. We nally have all the elements to prove the Proposition. Proof of Proposition. By construction the process Y (+,q is a recurrent extension of (X, IP that leaves by a jump according to the jumping-in measure c α,ϑρ η ϑρ. Thus, for any β < ϑ the 5

17 process X (β in Proposition is the process Y (+,q with q > such that ρϑ = β, recall that ρ and q are related by the formula in Lemma 3. Which nishes the proof of the Proposition because if q ranges in ], [, then ρ ranges in ], [. We have so nished the proof of the equivalence between the assertions (i and (ii in Theorem 2. Observe that the θ in the proof of the implication (i = (ii is equal to the ϑ in the implication (ii = (i. We next prove the uniqueness and characterization of the entrance law associated to the excursion measure claimed in Theorem Uniqueness and characterization Assume that there exist two recurrent extensions of (X, IP that satisfy the conditions in (ii in Theorem 2 and let n and n, be its associated excursions measures. Then there exist θ and θ 2 such that Cramér's condition is satised. The strict convexity of the mapping λ IE(e λξ, < ζ implies that θ 2 = θ = θ. As a consequence, the potential of both excursion measures is given by equation (2 with γ replaced by αθ. Therefore, arguing as in the proof of Lemma we show that n = n. Which nishes the proof of the unicity. The characterization of the entrance law follows from our proof of the fact that (i implies (ii in Theorem 2. On the one hand, by construction the resolvent of the excursion measure n is given by equation (. On the other hand, ( T n e λt f(x t dt = ( = n = e λt t αθ n(f(t α X, < T dt duα u /α θ f(ux θ /α e λu/α X /α {<T } ( duα u /α θ f(un X θ /α e λu/α X /α {<T }, where we used thee times Fubini's Theorem combined with the scaling property of n and a change of variables. Comparing the results in equations ( and (2 we get the identity ( { n X θ /α exp λu /α X /α } {<T } = for all λ and a.e. u >. As a consequence, n(x θ /α {<T } <. ÎE ( Γ( αθ Ê (I αθ u e λt, By the dominated convergence theorem, the latter identity holds for all λ and all u >. ecall that by Lamperti's transformation, T under ÎP u has the same law as u /α I under P. So by the uniqueness of Laplace transforms it follows that ( ( n X θ /α f X /α {<T } = 6 Ê (f (I, Γ( αθ Ê (I αθ (2

18 The claim in Theorem 2 follows from this identity using the scaling property of n, and that the law of I under P is equal to that of J under P. We have nished the proof of Theorem 2 and we next prove the Corollary. 2.4 Proof of Corollary Owing that the left derivative of ψ at θ exists, it follows from Proposition 3. in [8] that E ( J = Ê ( I = Ê (ξ = E(ξ e θξ, < ζ, and the leftmost quantity is nite if and only if E(ξ + e θξ, < ζ <. So, that (4 and (5 are equivalent is an easy consequence of the representation of the entrance law obtained in Theorem 2. We will next prove that (5 is equivalent to (6. Let V λ denote the λ-resolvent of (X, IP and U λ be the λ-resolvent of the unique recurrent extension of (X, IP that leaves continuously. The invariance of h(x = x θ, x > implies that n(x θ, < T = n(x θ t, t < T, t >. Using a well known decomposition formula and Fubini's theorem we get λu λ h(x = λv λ h(x + IE x (e λt λu λ h( ( T λn e λt h(x = h(x + IE x (e λt t dt n( e λt = h(x + IE x (e λt λ αθ λe λt n (h(x t, t < T = h(x + IE x (e λt λ αθ n (h(x, < T. Thus λu λ h(x < for all x if and only if n(x θ, < T <. From where we get that if (5 holds then ĨE x ( X θ t < for all x >, and a.e. t >. The self-similarity implies that in this case the latter holds for all x >, and all t >. Which nishes the proof of Corollary. 3 Proof of Counterexample A key tool in the establishment of (P and (P2 is the following version of Erickson's renewal theorem []. Lemma 5 (Erickson's renewal theorem []. Let G be a non-arithmetic probability distribution function on + such that G is a regularly varying function at innity with index γ ]/2, ], U the renewal measure associated to G, and m(x := x ( G(u du, x. Then (i for any directly iemann integrable function g : + +, t lim m(t g(t yu(dy = t 7 Γ(γΓ( γ g(ydy;

19 (ii For any directly iemann integrable function g : +, lim m(t g(y tu(dy = t Γ(γΓ( γ g(ydy. The statement in (i in Lemma 5 is the content of Erickson's renewal theorem 3 and so only (ii requires a proof, which is postponed to the end of this Section. Next, we proceed to prove the claims in Counterexample. To that end, observe that the law P is that of a subordinator with innite lifetime such that the tail probability P (ξ > x is a regularly varying function with index β ]/2, [. Let U be the renewal measure of the subordinator with law P, that is to say U (dy = P (ξ t dydt, y. According to Bertoin and Doney [3], the measure U is the renewal measure associated to the probability distribution function given by F ( = P (ξ e, where e is a standard exponential r.v. independent of ξ under P. Let IP be the law of the -pssmp associated to (ξ, P via Lamperti's transformation. The measure IP is such that IP = X q t IP on G t, t. It follows that the resolvents of (X, IP and (X, IP are related by λ f(x = V λfh q (x, x ], [, (3 h q (x V with h q (x := x q, x >. Moreover, we have that for any function f : + +, such that the application y f(e y e y, is directly iemann integrable lim x + m (log(/xv f(x = Γ( γγ(γ f(ydy. (4 Indeed, by applying Lamperti's representation and (ii in Lemma 5 we get [ ] m (log(/xv f(x = m (log(/x E f(xe ξt xe ξt dt = m (log(/x f(e y log(/x e y log(/x U (dy f(e y e y dy. x + Γ( γγ(γ And by making a change of variables in the rightmost quantity we obtain (4. Moreover, repeating the arguments in [4], we prove that for every f :], [, continuous and with compact support and λ >, lim x + m (log(/xv λ f(x = { }] f(y E [exp λy e ξs ds dy. (5 Γ( γγ(γ Therefore, the claim in (P is a straightforward consequence of (3 and (5. 8

20 Besides, in [8] Lemma 4, we proved that in general the exponential functional I satises the equation in law ( I Law = Q + MĨ, with (Q, M := exp{ξ s /α} {s<ζ} ds, e α ξ {<ζ}, and I Law = Ĩ, and the pair (Q, M is independent of Ĩ. Moreover, under the hypotheses (H2 in Lemma 4 in [8] we obtained, as a consequence of Goldie's Theorems 2.3 and 4. in [5], an estimate of the tail probability of I. A perusal of the proofs provided by Goldie to those theorems allows us to ensure that the arguments can be extended, using Erickson's renewal theorem instead of the classical renewal theorem, to prove the following Lemma. Lemma 6. Under the hypothesis of Counterexample we have that lim t m (log(tt q IP (T > t (( q ( = Γ( γγ(γ E exp{ξ s } {s<ζ} ds = Γ( γγ(γ q E(Iq ], [. q exp{ξ s } {s<ζ} ds Therefore, Lemma 6 and Karamata's Tauberian Theorem imply that the property (P2 is satised. emark 2. The expression of the value of the limit in Lemma 6 is a consequence of the proof of Lemma 2. Finally, to prove that the condition (P3 is satised, we argue as in [2] page to ensure that there exists a family of nite measures on ], [, say (n λ, λ >, such that V λ f(x n λ f = lim, x + IE x ( e T for any f, continuous and with compact support on ], [, and for λ >. Moreover, the family (n λ, λ > satises the resolvent type equation, for λ, µ > n λ V µ f = n µf n λ f, λ µ for any f continuous and with bounded support on ], [. Thus, Theorem 6.9 in [4] and Theorem 4.7 in [7] imply that there exist a unique excursion measure n such that its λ-potential is equal to n λ, ( T n e λt {Xt dy}dt = n λ (dy. for any λ >. In fact, all the results of Vuolle-Apiala [2] are still valid if we replace the power function that gives the normalization in his hypotheses (Aa and (Ab, by a regularly varying function. Therefore, Theorem.2 therein ensures that n(x + > =. According to Blumenthal's [6] theorem, associated to this excursion measure n there exists a unique recurrent extension of (X, IP that leaves continuously. Which nishes the proof of Counterexample. Now, we just have to prove that (ii in Lemma 5 holds. 9

21 Proof of Lemma 5. The claim in (i is Theorem 3 of Erickson [] and that (ii holds is a consequence of the latter. We next prove the result for step functions and the general case follows by a standard argument. Let (a k, k Z be a sequence of positive real numbers such that k Z a k <, and h > a constant. A consequence of Theorem of Erickson [] is that for any k N, m(t + kh {[kh,(k+h[} (y tu(dy C γ {[,h[} (ydy = C γ {[kh,(k+h[} (ydy, t with C γ = (Γ(γΓ( γ, and uniformly in k. Thus, given that m is an increasing function, we get that m(t a k {[kh,(k+h[} (y tu(dy k N a k m(t + kh {[kh,(k+h[} (y tu(dy. k N Therefore, lim sup m(t t k N a k {[kh,(k+h[} (y tu(dy C γ a k C γ k N k N Owing that m is regularly varying with positive index the following limit lim t m(t m(t + kh =, {[kh,(k+h[} (ydy a k {[kh,(k+h[} (ydy. holds uniformly in k N. A standard application of Fatou's Theorem and an easy manipulation gives that lim inf m(t a k {[kh,(k+h[} (y tu(dy C γ a k {[kh,(k+h[} (ydy. t k N Let g be the step function dened by k N g(t = k Z a k [kh,(k+h[ (t, t. It follows from the arguments above that lim m(t g(y t {y t } U(dy = C γ g(y {y } dy. t Moreover, the assertion in (i implies that lim m(t g(y t {y t<} U(dy = lim m(t t t From where the result follows. t g( (t yu(dy = C γ g( ydy. Acknowledgement. esearch supported by a esearch Award from the Council of Science and Technology of the state of Guanajuato, Mexico (CONCYTEG. 2

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