On the excursion theory for linear diffusions

Transcription

1 On the excursion theory for linear diffusions Paavo Salminen, Pierre Vallois, Marc Yor To cite this version: Paavo Salminen, Pierre Vallois, Marc Yor. On the excursion theory for linear diffusions. Japanese Journal of Mathematics, Springer Verlag, 27, 2 1, pp <1.17/s y>. <hal-12183> HAL Id: hal Submitted on 22 Dec 26 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 On the excursion theory for linear diffusions Paavo Salminen Åbo Akademi University, Mathematical Department, Fänriksgatan 3 B, FIN-25 Åbo, Finland, phsalmin@abo.fi Pierre Vallois Université Henri Poincaré Département de Mathématique F-5456 Vandoeuvre les Nancy, France vallois@iecn.u-nancy.fr Marc Yor Université Pierre et Marie Curie, Laboratoire de Probabilités et Modèles aléatoires, 4, Place Jussieu, Case 188 F Paris Cedex 5, France December 7, 26 Abstract We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions in stationary state. We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein s representations that, e.g., the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss Ornstein-Uhlenbeck processes. Keywords: Brownian motion, last exit decomposition, local time, infinite divisibility, spectral representation, Ornstein-Uhlenbeck process AMS Classification: 6J65, 6J6. 1

3 1 Introduction and preliminaries 1.1 Throughout this paper, we shall assume that X is a linear regular recurrent diffusion taking values in R + with an instantaneously reflecting boundary. Let P x and E x denote, respectively, the probability measure and the expectation associated with X when started from x. We assume that X is defined in the canonical space C of continuous functions ω : R + R +. Let C t := σ{ωs : s t} denote the smallest σ-algebra making the co-ordinate mappings up to time t measurable and take C to be the smallest σ-algebra including all σ-algebras C t, t. The excursion space for excursions from to associated with X is a subset of C, denoted by E, and given by E := {ε C : ε =, ζε > such that εt > t, ζε and εt = t ζε}. The notation E t is used for the trace of C t on E. As indicated in the title of the paper our aim is to gather a number of fundamental results concerning the excursion theory for the diffusion X. In Section 2 the classical descriptions, the first one due to Itô and McKean and the second one due to Williams, are presented. In Section 3 the stationary excursions are discussed and, in particular, the description due to Bismut is reviewed. After this, in Section 4, we proceed by analyzing excursions straddling an exponential time. The paper is concluded with an example on Ornstein-Uhlenbeck processes. Our motivation for this work arose from different origins: First, we would like to contribute to Professor Itô s being awarded the 1st Gauss prize, by offering some discussion and illustration of K Itô s excursion theory, see [21], when specialized to linear diffusions. The present paper also illustrates Pitman and Yor s discussion see [42] in this volume of K. Itô s general theory of excursions for a Markov process. In the literature there seems to be lacking a detailed discussion on the excursion theory of linear diffusions. Information available has a very scattered character, see, e.g., Williams [52], Walsh [51], Pitman and Yor [38], [39], [4], [41], Rogers [46], Salminen [49]. The general theory of excursions has been developed in Itô [21], Meyer [35], Getoor 2

4 [13], Getoor and Sharpe [15], [14], [16], [17], Blumenthal [4]. Although the case with Brownian motion is well studied and understood, for textbook treatments see, e.g., Revuz and Yor [43] and Rogers and Williams [47], we find it important to highlight the main formulas for more general diffusions using the traditional Fellerian terminology and language. To generalize some recent results see Winkel [53] and Bertoin, Fujita, Roynette and Yor [2] on infinite divisibility of the distribution of the length of the excursion of a diffusion straddling an independent exponential time. The Ornstein-Uhlenbeck process is one of the most essential diffusions. To present in detail formulae for its excursions is important per se. One of the key tools hereby is the distribution of the first hitting time H y of the point y from which the excursions are observed. For y = this distribution can be derived via Doob s transform see Doob [8] which connects the Ornstein-Uhlenbeck process with standard Brownian motion see Sato [5], and Göing-Jaeschke and Yor [19]. For arbitary y the distribution is very complicated; for explicit expressions via series expansions, see Ricciardi and Sato [44], Linetsky [32] and Alili, Patie and Pedersen [1]. We will focus on excursions from to and relate our work to earlier papers by Hawkes and Truman [2], Pitman and Yor [4], and Salminen [48]. Due to the symmetry of the Ornstein-Uhlenbeck process around, it is sufficient for our purposes to consider only positive excursions - the treatment of negative ones is similar - and view the process with values in R + and being a reflecting boundary. 1.2 In this subsection we introduce the basic notation and facts concerning linear diffusions needed in the sequel. A main source of information remains Itô and McKean [22], see also Rogers and Williams [47], and Borodin and Salminen [6]. i Speed measure m associated with X is a measure on R + which satisfies for all < a < b < < ma, b <. For simplicity, it is assumed that m does not have atoms. An important fact is that X has a jointly continuous transition density pt; x, y 3

5 with respect to m, i.e., P x X t A = A pt; x, y mdy, where A is a Borel subset of R +. Moreover, p is symmetric in x and y, that is, pt; x, y = pt; y, x. The Green or the resolvent kernel of X is defined for λ > as R λ x, y = dt e λt pt; x, y. ii Scale function S is an increasing and continuous function which can be defined via the identity P x H a < H b = where H. denotes the first hitting time, i.e., Sb Sx, a < x < b, 1 Sb Sa H y := inf{t : X t = y}, y. We normalize by setting S =. Due to the recurrence assumption it holds S+ = +. Recall that {SX t H : t } is a continuous local P x -martingale for every x see, e.g., Rogers and Williams [47] p It is easily proved that SX = {SX t : t } is a recurrent diffusion taking values in R +. The scale function associated with SX is the identity mapping x x, x, and we say that SX is in natural scale. Clearly, also for SX the boundary point is instantaneously reflecting. Using the Skorokhod reflection equation it is seen that SX is a P x -submartingale cf. Meleard [34] Proposition 1.4 where the semimartingale decomposition is given in case there are two reflecting boundaries. iii The infinitesimal generator of X can be expressed as the generalized differential operator G = d d dm ds acting on functions f belonging to the appropriately defined domain DG of G see Itô and McKean [22], Freedman [12], Borodin and Salminen [6]. In particular, since is assumed to be reflecting then f DG implies that f + := lim x fx f Sx S =. 4

6 iv The distribution of the first hitting time of a point y > has a P x - density: P x H y dt = f xy t dt. This density can be connected with the derivative of a transition density of a killed diffusion obtained from X. To explain this, introduce the sample paths X y t := { X t, t < H y,, t H y, where is a point isolated from R + a cemetary point. Then y { X t : t } is a diffusion with the same scale and speed as X. Let ˆp denote the transition density of X y with respect to m. Then, e.g., for x > y ˆpt; x, z f xy t = lim z y Sz Sy. 2 For a fixed x and y, the mapping t f xy t is continuous, as follows from the eigen-differential expansions and discussion in Itô and McKean p. 153 and 217 see also Kent [24], [25]. Recall also the following formula for the Laplace transform of H y E x e αh y = R α x, y R α y, y, 3 which leads to mdx E x e αh y = 1 αr α y, y. v There exists a jointly continuous family of local times {L y t : t, y } such that X satisfies the occupation time formula t ds hx s = hyl y t mdy, where h is a nonnegative measurable function see, e.g., Rogers and Williams [47] 49.1 Theorem p Consequently, L y t = lim δ 1 my δ, y + δ 5 t 1 y δ,y+δ X s ds.

7 For a fixed y introduce the inverse of L y via τ y l := inf{s : L y s > l}. Then τ y = {τ y l : l } is an increasing Lévy process, in other words, a subordinator and its Lévy exponent is given by E y exp λτ y l = exp l/r λ y, y = exp l ν y dv1 e λv, 4 where ν y is the Lévy measure of τ y. The assumption that the speed measure does not have atoms implies that τ y does not have a drift. In case y = we write simply L, τ and ν. 1.3 Assuming that X is started from we define for t > G t := sup{s t : X s = } and D t := inf{s t : X s = }. 5 The last exit decomposition at a fixed time t states that for u < t < v P G t du, X t dy, D t dv = pu;, f y t u f y v t du dv mdy. 6 In fact, this trivariate distribution is only the skeleton of a more complete body of processes: {X u : u G t }, {X Gt+v : v t G t }, and {X t+v : v D t t} 7 the distributions of which we now characterize following Salminen [49]. For general approaches; see Getoor and Sharpe [15], [14], and Maisonneuve [33]. Let x, y R + and u > be given. Denote by X x,u,y, P x,u,y the diffusion bridge from x to y of length u constructed from X, i.e., the measure P x,u,y governing X x,u,y is the conditional measure associated with X started from x and conditioned to be at y at time u. The bridge X x,u,y is a strong non-time-homogeneous Markov process defined on the time axis [, u. For the first component in 7, we have conditionally on G t = u {X s : s < G t } d = {X,u, s : s < u} 8 For the second component in 7 consider the process X y as introduced in iv above with y =. We write simply X instead of X. For positive 6

8 x and y let X x,u,y denote the bridge from x to y of length u constructed, as above, from X. The measure P x,u,y governing X x,u,y can be extended by taking in the weak sense P,u,y := lim x Px,u,y. We let X,u,y denote the process associated with P,u,y. Then, conditionally on G t = u and X t = y, {X Gt+s : s < t G t } d = { X,t u,y s : s < t u}. 9 For the final part in 7, by the Markov property, we have conditionally on X t = y where X = y. {X t+s : s < D t t} d = { X s : s }, 1 2 Two descriptions of the Itô measure 2.1 Description due to Itô and McKean We discuss the description of the Itô measure n where the excursions are studied by conditioning with respect to their lifetimes. Let X be as in section 1.3 and ˆpt; x, y its transition density with respect to the speed measure, in other words, P x X t dy = P x X t dy; t < H = ˆpt; x, y mdy. The Lévy measure ν of τ is absolutely continuous with respect to the Lebesgue measure, and the density - which we also denote by ν - is given by νv := νdv/dv = lim x lim y ˆpv; x, y SxSy =: p v;,. 11 In Section 1.3 we have introduced the bridge X x,t,y and the measure P x,t,y associated with it. The family of probability measures { P x,t,y : x >, y > } is weakly convergent as y thus defining P x,t, for all x >. Intuitively, this is the process X conditioned to hit at time t. Moreover, letting now x we obtain a measure which we denote by P,t, which governs a non-time homogeneous Markov process X,t, starting from, staying positive on the time interval, t and ending at at time t. 7

9 Theorem 1. a. The law of the excursion life time ζ under the Itô excursion measure n is equal to the Lévy measure of the subordinator {τ l } l and is given by nζ dv = νdv = p v;, dv. 12 b. The Itô measure can be represented as the following integral ndε = nζ dv P,v, dε. 13 Moreover, the finite dimensional distributions of the excursion are characterized for < t 1 < t 2 < < t n and x i >, i = 1, 2,..., n by nε t1 dx 1, ε t2 dx 2,..., ε tn dx n = mdx 1 f x1 t 1 ˆpt 2 t 1 ; x 1, x 2 mdx ˆpt n t n 1 ; x n 1, x n mdx n. In particular, the excursion entrance law is given by and it holds nζ > t = nε t dx = mdx f x t, nε t dx = mdx f x t. 15 Combining the formulas 12 and 13 with the last exit decomposition 6 leads to a curious relation between the transition densities p and p. Proposition 2. The functions pt;, and p t;, satisfy the identity t du pu;, t u Proof. From 12 and 15 we may write t dv p v;, = nζ > t = dv p v;, = Consequently, identity 16 can be rewritten as t du pu;, mdx f x t. mdx f x t u = 1, 17 but, in view of the last exit decomposition 6, identity 17 states that the last exit from when starting from takes place with probability 1 before t, in other words, P G t t = 1, which, of course, is trivially true. 8

10 Remark 3. For another approach to 16 notice that it follows from 4 and 11 1 R λ, = dv p v;, 1 e λv. Hence, from the definition of the Green kernel, Consequently, 1 = 1 λ = = du e λu pu;, du e λu pu;, du from which 16 is easily deduced. dv p v;, 1 e λv. 18 dv e λv dv e λu+v pu;, 2.2 Description due to Williams v v ds p s;, ds p s;,, In the approach via the lengths of the excursions the focus is first on the time axis. In Williams description see Williams [52], and Rogers [45], [46] the starting point of the analysis is on the space axis since the basic conditioning is with respect to the maximum of an excursion. To formulate the result, let for ε E Mε := sup{ε t : < t < ζε}. The key element in Williams description is the diffusion X obtained by conditioning X not to hit. We use the notation P and E for the measure and the expectation associated with X. To define this process rigorously set for a bounded F t C t, t >, E xf t := lim a + E xf t ; t < H a H a < H E x F t ; t < H a < H = lim a + P x H a < H = lim a + E x F t SX t ; t < H a H, Sx where the Markov property and formula 1 for the scale function are applied. The monotone convergence theorem yields E xf t = 1 Sx P x F t SX t ; t < H, 9

11 in other words, the desired conditioning is realized as Doob s h-transform of X by taking h to be the scale function of X. It is easily deduced that the transition density and the speed measure associated with X are given by p t; x, y := ˆpt; x, y SySx, m dy := Sy 2 mdy. We remark that the boundary point is entrance-not-exit for X and, therefore, X can be started from after which it immediately enters, and never hits. Theorem 4. a. The law of the excursion maximum M under the Itô excursion measure n is given by nm a = 1 Sa. b. The Itô excursion measure n can be represented via ndε = nm da Q,a dε, where Q,a is the distribution of two independent X processes put back to back run from until they first hit level a. As an illustration, we give the following formula ζ n1 exp = ds V ε s nm da 1 E exp Ha If V, this quantity is the Lévy exponent of the subordinator { τl } ds V X s : l, that is, E { exp τl } α ds V X s 2 du V ω u. { ζ } = exp l n 1 exp α ds V ε s. 1

12 Comparing the descriptions of the Itô excursion measure in Theorem 1 in particular formula 14 and in Theorem 4 hints that the processes X and X have, in addition to conditioning relationship, also a time reversal relationship. This is due to Williams [52], who particularized to the case of diffusions the general time reversal result, obtained by Nagasawa [36]. See also [43] p. 313, and [6] p. 35. Proposition 5. Let for a given x > Λ x := sup{t : ωt = x} denote the last exit time from x. Then { X s : s < H } d = {X Λ x s : s < Λ x}, 19 where X = x and X =. 3 Stationary excursions; Bismut s description Consider the diffusion X with the time parameter t taking values in the whole of R. In the case mr + < the measure governing X can be normalized to be a probability measure. Indeed, in this case the distribution of X t is for every t R defined to be PX t dx = mdx/mr + =: mdx. Recall from 5 the definitions of G t and D t, and introduce also t := D t G t. Theorem 6. Assume that mr + <. Then the joint distribution of t G t and D t t is given by Pt G t du, D t t dv/dudv = Consequently, for t it holds mdy f y u f y v = νu + v/mr +. P t du/du = u νu/mr +. 2 Moreover, the law of the process {X Gt+v : v t } is given by ζε ndε/mr +, 21 where ndε is the Itô measure as introduced in Theorem 1 and 4 and ζ denotes the length of an excursion. 11

13 Proof. The density of t G t, D t t is derived using the time reversibility of the diffusion X, i.e., {X t : t R} d = {X t : t R}, and the conditional independence given X t. The fact that the density can be expressed via the density of the Lévy measure is stated and proved in Proposition 12 below, see formulas 3 and 31. To compute the distribution of t is elementary from the joint distribution of t G t and D t t. For these results, we refer also to Kozlova and Salminen [28]. The statement concerning the law of {X Gt+v : v t } has been proved in Pitman [37] see Theorem p. 29 point iii and the formulation for excursions on p. 293 and 294 all that remains for us to do is to find the right normalization constant, but this is fairly obvious, e.g., from the density of t. If mr + = the measure associated with X is still well-defined but only σ-finite. In this case, the distribution of X t is plainly taken to be m. From 21 it is seen that we are faced with a representation of the Itô measure via stationary excursions valid in both cases mr + < and mr + =. We focus now on this representation as displayed in 22 below, and present a proof of the representation using the diffusion theory this provides, of course, also a proof of 21. We remark that in Pitman [37] a more general case concerning homogeneous random sets is proved, and, hence, it seems worthwhile to give a direct proof in the diffusion case. Theorem 7. Let F be a measurable non-negative functional defined in the excursion space E. Then up to a normalization 1 nf ε = E F X Gt+s : s t. 22 t In particular, the process {X Gt+s : s t } conditionally on t = v is identical in law with the excursion bridge X,v, as introduced in Section 2.1. Proof. Without loss of generality, we take t =. From 2 we have 1 nfζ = fa νa da = E f. Therefore, it is enough cf. Theorem 1 to prove that nf ε ζ = u = E F X G +s : s = u

14 Define for s 1 < s 2 < < s n u and consider E A 1,n = u = A 1,n := {X G +s 1 dy 1,..., X G +s n dy n }, = u y= v= u y= v= E A 1,n, G dv, X dy = u E A 1,n = u, G = v, X = y P G dv, X dy = u. From the description of the process X, the conditional independence and the equality of the laws of the past and future given X, and using formula 2 we obtain P G dv, X dy = u = 1 u νu f y,v f y, u v mdydv. 24 Letting k be such that v + s k < t < v + s k+1, if any, we write applying again the conditional independence E A 1,n = u, G = v, X = y = E A 1,k A k+1,n = u, G = v, X = y = E A 1,k G = v, X = y E A k+1,n D = u v, X = y. Recall from Introduction section 1.2 iv the notation X for the diffusion X killed when it hits. As in section 1.3 of Introduction we may construct the bridge X y,v, starting from y having the length v and ending at. We let P y,v, denote the measure associated with X y,v,. With these new notations, and E A 1,k G = v, X = y = P y,v, ω v sk dy k,..., ω v s1 dy 1 = 1 f y v pv s k; y, y k mdy k ps k s k 1 ; y k, y k 1 mdy k 1... ps 2 s 1 ; y 2, y 1 mdy 1 f y1 s 1 E A k+1,n D = u v, X = y = P y,v, ωsk+1 v dy k+1,..., ω sn v dy n = 1 f y u v ps k+1 v; y, y k+1 mdy k+1 ps k+2 s k+1 ; y k+1, y k+2 mdy k+2... ps n s n 1 ; y n 1, y n mdy n f ynu s n. 13

15 Using now 24 and formulas above we have after some rearranging and applying the symmetry of the transition density p E A 1,n = u = 1 u νu mdy 1 f y1 s 1 ps 2 s 1 ; y 1, y 2 mdy 2... u dv Performimg the integration yields mdy pv s k ; y k, y ps k+1 v; y, y k+1... ps n s n 1 ; y n 1, y n mdy n f ynu s n. E A 1,n = u = 1 νu mdy 1 f y1 s 1 ps 2 s 1 ; y 1, y 2 mdy 2... ps n s n 1 ; y n 1, y n mdy n f ynu s n, and this means that 23 holds completing the proof. Remark 8. The formula 22 was derived for Brownian motion by Bismut [3]. The connection with the Palm measure and stationary processes is discussed in Pitman [37]. In fact, Bismut describes in the Brownian case the law of the process {X Gt+s : s t } in terms of two independent 3-dimensional Bessel processes started from and killed at the last exit time from an independent level distributed according to the Lebesgue measure see [3] and [43] for details. 4 On the excursion straddling an independent exponential time In the literature one can find several papers devoted to the properties of excursions straddling a fixed time t; first of all, Lévy s fundamental paper [31], which contains a lot about the zero set of Brownian motion, its inverse local time, excursions, and so on. See also Chung [7] starting from Lévy s paper [31], Durret and Iglehart [9], and Getoor and Sharpe [16], [17]. In fact, the last exit decomposition 6 lies in the heart of these studies see Getoor and Sharpe [15], [14]. However, it seems to us that excursions straddling an exponential time are not so much analyzed. Here we make some remarks on this subject. 14

16 Let T be an exponentially distributed random variable with parameter α >, independent of X, and define and G T := sup{s T : X s = }, D T := inf{s T : X s = }, T := D T G T. The Lévy exponent of the inverse local time at is denoted by Φ, in other words, E exp λτ l = exp l Φλ Recall the relation cf. 4 with y = 4.1 Last exit decomposition at T Φλ R λ, = We begin by discussing the last exit decomposition at the exponential time T. Theorem 9. i The processes {X u : u G T } and {X GT +v : v T } are independent. ii The law of {X u : u G T } may be described as follows: a L T := L GT is exponentially distributed with mean 1/Φα, b The process {X u : u G T } conditionally on L T = l is distributed as {X u : u τ l } under the probability exp α τ l + l Φα P. iii The law of the process {X GT +v : v T } is given by 1 1 e αζε ndε. 26 Φα where ndε is the Itô measure associated with the excursions away from for X and ζ denotes the length of an excursion. 15

17 Proof. Let F 1 and F 2 be two nonnegative functionals of continuous functions and consider E F 1 X u : u G T F 2 X GT +v : v T = α dt e αt E F 1 X u : u G t F 2 X Gt+v : v t τl = αe dt e αt F 1 X u : u τ l F 2 X τl +v : v τ l τ l l τ l = E dl e α τ l F 1 X u : u τ l ndε 1 e α ζε F 2 ε s : s ζε = ΦαE dl e α τ l F 1 X u : u τ l 1 e α ζε ndε F 2 ε s : s ζε. Φα where the third equality is based on the properties of the Poisson random measure associated with the excursions see Revuz and Yor [43] Master Formula p Remark 1. Notice that letting α in 26 and using lim α α Φα = lim α α R α, = 1/mR + 27 yield the probability law of the excursion straddling a fixed time in the stationary setting, cf. 21 in Theorem 6. As a corollary of Theorem 9 we have the following results which show that after conditioning the quantities do not depend on α, and in this context α is entirely contained in G T and T. The formulas should be compared with 8, 9, and 1. The distributions of G T and T are given, respectively, in 4 and 35 below. Corollary 11. For any nonnegative functionals F 1 and F 2 of continuous functions it holds E F 1 X u : u G T G T = g = E F 1 X u : u g X g = = E,g, F 1 ω u : u g 28 16

18 and E F 2 X GT +v : v T T = h = E F 2X v : v h X h =. = Ê,h,F 2 ω s : s h 29 Proof. The statement 28 can be obtained from the corresponding result for fixed time as presented in 8. Also 29 can be derived from the fixed time result but we prefer to present here a proof based on the Master Formula. For this consider for δ > E F2 X GT +v : v T 1 {h T <h+δ} 1 e α ζε = ndε F 2 ε s : s ζε 1 Φα {h ζε<h+δ}. Using the description 13 of the Itô excursion law we obtain E F2 X GT +v : v T 1 {h T <h+δ} h+δ 1 e α u = du νu Ê,u, F 2 ω s : s u. Φα h Applying the explicit form of the distribution of T letting δ leads to 29. as given in 35 and 4.2 On the distribution of G T, D T In this section the distributions of T G T, D T T and T := D T G T are studied in detail. Proposition 12. The joint distribution of T G T and D T T is given by P T G T du, D T T dv = dudv α R α, e αu mdy f y u f y v. 3 = α e αu νu + v Φα dudv. 31 In particular, νu + v = mdy f y u f y v

19 Proof. From 6, and, hence, P t G t du, D t t dv = dudv pt u;, 1 {u t,v } mdy f y u f y v, P T G T du, D T T dv = dudv α u dt e αt pt u;, mdy f y u f y v, from which 3 follows. To derive 31, we apply again the Master Formula see Revuz and Yor [43] p For this, let u, v ϕu, v be a nonnegative and Borel measurable function and define Qϕ := E ϕt G T, D T T. Letting τ l denote the inverse of the local time L at we have Qϕ = E ϕt τ l, τ l T 1 {τl <T <τ l } l = E ϕt τ l, z + τ l T 1 {τl <T <τ l +z}νz dzdl R 2 + since {l, τ l : l } is a Poisson point process with Lévy measure dl dν, and T is independent of {τ l : l }. Apply next that T is exponentially distributed to obtain Qϕ = E νzdzdl R 2 + τl +z τ l dt α e αt ϕt τ l, z + τ l t = αe νze αx+τl ϕx, z x 1 {x z} dxdzdl R 3 + where we have substituted x = t τ l. Furthermore, setting y = z x yields Qϕ = αe νy + xe αx+τl ϕx, y dxdydl R 3 + = αe e α τ l dl ϕx, y e α x νy + x dxdy, R 2 + and 31 follows now easily from 4. consequence of 3 and , The equality 32 is an immediate,

20 Corollary The densities for T G T, D T T, and T respectively, by are given, and P T G T du/du = P D T T dv/dv = α Φα e αu α Φα eαv v u νzdz, 33 e αz νzdz, 34 P T da/da = 1 e αa νa. 35 Φα 2. The joint density of T G T and T is P T G T du, T da/du da = 3. The density of T G T conditionally on T = a is P T G T du T = a/du = α Φα e αu νa, u a. 36 α 1 e αa e αu, u a. 37 Proposition 14. The joint Laplace transform of G T and D T is given by In particular, E exp γ 1 G T γ 2 D T = Φγ 2 + α Φγ Φγ 1 + γ 2 + α E e γ T Φγ + α Φγ =, 39 Φα and the random variables G T and T are independent. The density of G T is given by P G T du/du = Φα e αu pu;,. 4 Proof. The formula 4 for the density of G T is obtained from 6 by integrating. The independence of G T and T follows immediately from 38. To derive the joint Laplace transform of G T and D T, consider E exp γ 1 G T γ 2 D T v = dt α e αt e γ 1u γ 2 v P G t du, D t dv. u 19

21 Applying the last exit decomposition formula 6 yields E exp γ 1 G T γ 2 D T = = = = α du e γ 1u pu;, du e γ 1u pu;, u du e γ 1+γ 2 u pu;, dv e γ 2v v u dt α e αt mdy f y t u f y v t da e γ 2a+u a+u u dt α e αt pu;, mdy f y t u f y a + u t du e γ 1+γ 2 +αu pu;, = αr γ1 +γ 2 +α, To proceed, we have a da e γ 2a a mdy f y b f y a b mdy E y e γ 2+αH E y e γ 2 H = 1 R γ2 +α, R γ2, mdy db α e αb+u pu;, db e αb f y b f y a b da e γ 2a mdy E y e γ 2+αH E y e γ 2 H. mdy R γ2 +αy, R γ2 y,. 2

22 The integral term in this expression can be evaluated: mdy R α+γ2 y, R γ2 y, = = = = mdy dt e α+γ 2t dt e α+γ 2t dt e α+γ 2t pt; y, t ds e γ 2s pt + s;, du e γ 2u t pu;, du e γ 2u 1 e αu pu;,, α = 1 α R γ 2, R γ2 +α,. ds e γ 2s ps; y, where the Chapman-Kolmogorov equation and the symmetry of the transition density p is applied, and by 25 this completes the proof. Remark From Proposition 12 it is seen that the density of T can also be written in the form P T da/da = α Φα mdy a which taking into account 35 leads to the identity 1 e αa α νa = Let here α to obtain νa = mdy mdy a a db e αb f y b f y a b, db e αb f y b f y a b. db a f yb f y a b. 41 It is interesting to compare this expression with the following one obtained from 32 νa = mdy f y b f y a b. 42 The fact that the right hand sides of 41 and 42 do not depend on b can also be explained via the Chapman-Kolmogorov equation. 2. We may study distributions associated with G t, D t and t in the stationary case, i.e., if mr + <, by letting α, as observed in Remark 1. 21

23 From Proposition 12 and Corollary 13 we deduce the following results take t = : 1 P G du, D dv/dudv = νu + v. mr + P G du/du = PD du/du = P da/da = 1 mr + 1 mr + a νa. u νv dv, Moreover, letting Z T := T G T / T then Z T, T converges in distribution as α to U,, where U and are independent with U uniformly distributed on, 1 and is distributed as cf. Theorem Infinite divisibility In the paper by Bertoin et al. [2] it is proved that the distribution of T for a Bessel process with dimension d = 21 α, < α < 1, is infinitely divisible in fact, self-decomposable and the Lévy measure associated with this distribution is computed. In this section we show that the distribution of T is infinitely divisible in general, i.e., for all regular and recurrent diffusions. Moreover, we also prove that the distributions of T G T and D T T have this property. The key to these results is the Krein representation of the density of the Lévy measure ν see Knight [26], Kent [25], Küchler and Salminen [3], and in general on Krein s theory of strings Kotani and Watanabe [27], Dym and McKean [1] according to which νa = where the measure M has the properties e az Mdz, 43. Mdz zz + 1 < and Mdz z =. Theorem 16. The distributions of T G T, D T T and T are infinitely divisible. Proof. As seen from 33, 34, and 35, the intrinsic term in the densities of T G T, D T T and T is the density νa of the Lévy measure of 22

24 the inverse local time at. We consider first the distribution of T G T. Applying the Krein representation 43 in 33 yields with P T G T du/du = α R α, e αu da = α Φα e αu = α Φα = M α dz = u Mdz z Mdz z e uz e α+zu α + z e α+zu Mdz, α Φα Mdz e az Mdz zα + z. 44 The claim of the theorem follows now from the fact that mixtures of exponential distributions are infinitely divisible. see Bondesson [5]. For D T T we compute similarly from 34 via the Krein representation P D T T dv/dv = α Φα eαv = v z e zv Mdz. e αa νada. To analyze the distribution of T we use the Krein representation in 35 to obtain P T da/da = 1 e za e α+za Mdz. 45 Φα Notice that for a fa; z, α = zα + z α e za e α+za is a probability density as a function of a. In fact, letting T 1 and T 2 be two independent exponentially distributed random variables, with respective parameters z and α + z, then the sum T 1 + T 2 has the density fa; z, α. In other words, the distribution of T 1 + T 2 is a so called gamma convolution. Next we notice that letting Π z,α dx := zα + z α x 2 dx, 23 z < x < α + z

25 we may represent the distribution of T 1 + T 2 as a mixture of Gamma2- distributions as follows fa; z, α = Combining the representation 46 with 45 yields P T da/da = α Φα = x 2 a e xa Π z,α dx. 46 fa; z, α zα + z Mdz x 2 a e xa Πα dx, 47 where Π α is a probability measure on R + given for any Borel set A in R + by Π α A = M α dzπ z,α A. 48 The claim that the distribution of T is infinitely divisible follows now from 47 by evoking the result that mixtures of Gamma2-distributions are infinitely divisible see Kristiansen [29]. Remark Recall from Bondesson [5] that a probability distribution F on R + is called a generalized gamma convolution GGC if its Laplace transform can be written as e sa F da = exp µs + t log Udt, 49 t + s where µ and U is a measure on, satisfying Udt log t Udt < and t,1] 1, <. It is known see [5] Theorem p. 49 that if β is the total mass of U then the distribution F in 49 is a mixture of Gammaβ-distributions 2. The distribution of the length t of an excursion straddling a fixed time t for a stationary diffusion with stationary probability distribution is given in Theorem 6 2 as P t da = a νa mr + da. 24

26 Also in this case the distribution of t is a mixture of Gamma2-distributions and, hence, it is infinitely divisible. In fact, P t da/da = z 2 a e za Mdz. where the probability measure M is given in terms of the Krein measure M via Mdz = Mdz/mR + z 2. 5 Case study: Ornstein-Uhlenbeck processes In this section we give some explicit formulas for excursions from to associated with Ornstein-Uhlenbeck processes. It is possible to obtain such formulas due to the symmetry of the Ornstein-Uhlenbeck process around. Analogous results for excursions from an arbitrary point x to x are less tractable. 5.1 Basics Let U denote the Ornstein-Uhlenbeck diffusion with parameter γ >, i.e., U is the solution of the SDE du t = db t γu t dt with U = u, and most of the time, but not always, we take u =. Recall that the speed measure and the scale function of U can be taken to be mdx := 2 e γx2 dx and Sx := x e γy2 dy, respectively. Moreover, see [6] p. 137, the Green kernel of Ornstein-Uhlenbeck process with respect to the speed measure is given for x y by R λ x, y = Γλ/γ γx 2 2 γπ exp D 2 λ/γ x 2γ γy 2 exp D 2 λ/γ y 2γ, where D denotes the parabolic cylinder function. In particular, since D λ/γ = 1 π 2 λ/2γ Γ λ + γ/2γ, 25

27 we have, after some manipulations, π Γλ/γ 2 R λ, = 2 2 λ/2γ Γ λ + γ/2γ. γ Consequently, using the formula we obtain Γx = 2 x 1 π Γx + 1/2 Γx/2 R λ, = 1 Φλ = Γλ/2γ 4 Γλ + γ/2γ. 5 We remind also that U can be represented as the deterministic time change Doob s transformation of Brownian motion via U t = e γt u + β at, where β is a standard Brownian motion and a t := e 2γt 1/2γ see Doob [8]. 5.2 Killed Ornstein-Uhlenbeck processes We consider now the Ornstein-Uhlenbeck process killed at the first hitting time of, and denote this process by Û. Let Y be the diffusion on R + satisfying the SDE 1 dy t = db t + γy t dt, Y = y >. Y t Recall that Y may be described as the radial part of the three-dimensional Ornstein-Uhlenbeck process. In [6] p. 138 the basic properties of such processes are presented. In particular, we record that is an entrance-notexit boundary point and the process is positively recurrent its stationary distribution being the Maxwell distribution, i.e., the distribution with the density proportional to the speed measure of Y, that is, m Y dx := 2x 2 e γx2 dx, x >. We remark that there is a misprint in [6] p. 139; the stationary distribution in the general case is not a χ 2 -distribution. The transition density of Y with respect to its speed measure m Y is γ e p Y 3γt/2 t; x, y = exp γe γt x 2 + y 2 γxy sinh 2π sinhγt xy 2 sinhγt sinhγt 26

28 and can be computed from the transition density of a Bessel process using Doob s transform for an approach via inverting the Laplace transform see Giorno et al. [18]. In Salminen [48] it is proved that P x Ût dy = P x U t dy, t < H = e γt p Y t; x, y ϕ γy ϕ γ x my dy, 51 where ϕ γ x = 1/x is the unique up to multiplicative constants decreasing positive solution of the ODE associated with Y killed at rate γ: u x + x γx u x = γux. From 51 we obtain Proposition 18. The transition density with respect to its speed measure m of the Ornstein-Uhlenbeck killed at the first hitting time of is given by γ e γt/2 ˆpt; x, y = exp γe γt x 2 + y 2 γxy sinh. 52 2πsinhγt 2sinhγt sinhγt Combining the expression of the transition density in 52 with formula 2 yields the distribution of H see also Sato [5] and Going-Jaeschke and Yor [19]. Proposition 19. The density of the first hitting time of for the Ornstein- Uhlenbeck process {U t } is given by γ 3/2 x e γt/2 f x t = exp γe γt x πsinhγt 3/2 2sinhγt 5.3 Lévy measure of inverse local time and densities of T, T G T and D T T The density of the Lévy measure of the inverse local time at is obtained by applying formula 11 see also Hawkes and Truman [2]. Moreover, using 5 in formula 4 leads to an explicit expression for the Bernstein function associated with the inverse local time at. Proposition 2. The density of the Lévy measure of the inverse local time at is νt = γ 3/2 e γt/2 2πsinhγt 3/2 = 2γ 3/2 e 2γt. 54 2π e 2γt 1 3/2 27

29 Let {τ l : l } be the inverse local time at. Then E exp λτ l = exp l 4 Γ λ + γ/2γ Γλ/2γ Next we display the distributions of T, T G T, and D T T. Recall that these distributions are infinitely divisible and the densities are expressable via the density of the Lévy measure, as stated in Corollary 13 formulae 33 and 34, and in Theorem 16. To simply the notation, we take γ = 1. Proposition 21. With Φα as in 5, the distributions of T, T G T and D T T are given, respectively, by and P T da/da = 1 e αa Φα P T G T da/da = α e αa Φα P D T T da/da = α eαa Φα 5.4 The Krein measure a. 2 π e 2a e 2a 1 3/2, 55 2 π e 2a 1 1/2, 56 du e αu 2 π e 2u e 2u 1 3/2. 57 As seen in Section 4.3, the Krein representation plays a central rôle in the proof of infinite divisibility of the distributions of T G T, D T T, and T. Therefore, it seems motivated to compute the measure M cf. 43 in this representation for Ornstein-Uhlenbeck processes. To start with, we give the spectral representation of the transition density of ˆp of the Ornstein-Uhlenbeck process killed at the first hitting time of. Instead of computing from scratch, we exploit the spectral representation for p Y with γ = 1 as presented in Karlin and Taylor [23] p. 333: p Y t; x, y = n= w 1 n,1/2 e 2nt L 1/2 n x 2 L 1/2 n y 2, 58 where {L n 1/2 : n =, 1, 2,... } is the famly of Laguerre polynomials with parameter 1/2 normalized via 2 π L n 1/2 x 2 m Y dx = 2 28 n =: w n n,1/2. 59

30 Notice that we consider the symmetric density with respect to the speed measure m Y. From 51 and 58 the spectral representation of ˆp is now obtained immediately and is given by ˆpt; x, y = n= w 1 n,1/2 e 2n+1t xl 1/2 n x 2 yl 1/2 n y 2. 6 The normalization 59 coincides with the normalization in Erdelyi et al. [11] see formula 2 p. 188 where the notation for the norm is h n. Therefore, from [11] formula 13 p. 189 we have L 1/2 n = n n 61 and, consequently cf. 53, we obtain the spectral representation for the density of the first hitting time of f x t = w 1 n,1/2 e 2n+1t xl n 1/2 x 2 L n 1/2. n= = 2 π n= e 2n+1t xl 1/2 n x To find the spectral representation for the density of the Lévy measure we apply formula 54 which yields νt = 2 n e 2n+1t. 63 π n In view of 43, we have n= Proposition 22. The measure M in the Krein representation of ν for the Ornstein-Uhlenbeck process is given by Mdz = 2 n δ π n {2n+1} dz, where δ is the Dirac measure. Notice that n= νt = 2 π e t 1 e 2t 3/2, and, hence, 63 is obtained also from the MacLaurin expansion of x 1 x 3/2 evaluated at x = e 2t. Acknowledgement. We thank Lennart Bondesson for co-operation concerning gamma convolutions. 29

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