RIGHT INVERSES OF LÉVY PROCESSES: THE EXCURSION MEA- SURE IN THE GENERAL CASE

Elect. Comm. in Probab. 15 (21), 572 584 ELECTRONIC COMMUNICATIONS in PROBABILITY RIGHT INVERSES OF LÉVY PROCESSES: THE EXCURSION MEA- SURE IN THE GENERAL CASE MLADEN SAVOV 1 Department of Statistics, University of Oford, 1 South Parks Road, Oford OX1 3TG, UK email: savov@stats.o.ac.uk MATTHIAS WINKEL Department of Statistics, University of Oford, 1 South Parks Road, Oford OX1 3TG, UK email: winkel@stats.o.ac.uk Submitted March 1, 21, accepted in final form December 6, 21 AMS 2 Subject classification: 6G51 Keywords: Lévy process, right inverse, subordinator, fluctuation theory, ecursion Abstract This article is about right inverses of Lévy processes as first introduced by Evans in the symmetric case and later studied systematically by the present authors and their co-authors. Here we add to the eisting fluctuation theory an eplicit description of the ecursion measure away from the (minimal) right inverse. This description unifies known formulas in the case of a positive Gaussian coefficient and in the bounded variation case. While these known formulas relate to ecursions away from a point starting negative continuously, and ecursions started by a jump, the present description is in terms of ecursions away from the supremum continued up to a return time. In the unbounded variation case with zero Gaussian coefficient previously ecluded, ecursions start negative continuously, but the ecursion measures away from the right inverse and away from a point are mutually singular. We also provide a new construction and a new formula for the Laplace eponent of the minimal right inverse. 1 Introduction Evans [5] defined a (full) right inverse of a Lévy process X = (X t, t ) to be any increasing process K = (K, ) such that X K = for all. A partial right inverse [8] is any increasing process K = (K, < ξ K ) such that X K = for all < ξ K for some (random) ξ K >. The eistence of partial right inverses is a local path property that has been completely characterised [4, 5, 8] in terms of the Lévy-Khintchine triplet (a, σ 2, Π) of the Lévy process X, i.e. a, σ 2 and Π measure on with Π({}) = and (1 y2 )Π(d y) < such that e iλx t = e tψ(λ), where ψ(λ) = iaλ + 1 2 σ2 λ 2 + 1 e iλy + iλy1 { y 1} Π(d y). 1 SUPPORTED BY AN ESMÉE FAIRBAIRN JUNIOR RESEARCH FELLOWSHIP AT NEW COLLEGE, OXFORD 572

Right inverses of Lévy processes 573 We recall Evans construction of right inverses. Recursively define for each n times T (n) =, T (n) k+1 = inf t T (n) : X k t = (k + 1)2 n, k 1, and a process K (n) if the limit = T (n), k2 n < (k + 1)2 n, k. Then, a pathwise argument shows that, k K = inf sup y> n K (n) y (1) is finite for < ξ K and ξ K >, it is the minimal partial right inverse: for all right-continuous partial right inverses (U, < ξ ), we have ξ ξ K and U K for all < ξ. Where right inverses eist, the minimal right-continuous right inverse is a subordinator, and for partial right inverses (and ξ K maimal), a subordinator run up to an independent eponential time ξ K. In the sequel, we focus on this minimal right-continuous (partial) right inverse and denote its Laplace eponent by ρ(q) = ln e qk 1 ; ξ K > 1 = κ K + η K q + 1 e qt Λ K (d t). (2) Evans [5] showed further that the reflected process Z = X L is a strong Markov process, where L t = inf{ : K > t}, t < K ξk, L t = ξ K, t K ξk ; and L is a local time process of Z at zero. In analogy with the classical theory of ecursions away from the supremum (see below), there is an associated ecursion theory that studies the Poisson point process (e Z, < ξ) of ecursions away from the right inverse, where (, ) e Z (r) = Z K +r, r K = K K, e Z (r) =, r K. Specifically, we denote by n Z its intensity measure on the space (E, ) of ecursions E = {ω D : ω(s) = for all s ζ(ω) = inf{r > : ω(r) = }} equipped with the restriction sigma-algebra induced by the Borel sigma-algebra associated with Skorohod s topology on the space D = D([, ), ) of càdlàg paths ω: [, ). The entrance laws n Z r (d y) = nz ({ω E : ω(r) d y, ζ(ω) > r}) are characterised by their Fourier- Laplace transform e qr e iλ y n Z ρ(q) iλ r (d y)dr = q + ψ(λ) η K, (3) see [5, 8]. The sub-stochastic semi-group within these ecursions is the usual killed semi-group P t (y, dz) = y ({ω D : ω(t) dz, ζ(ω) > t}), with y = ((X t ζ(x ), t ) X = y) as canonical measure on E D of the distribution of X starting from y and frozen when hitting zero. More eplicit epressions for n Z are available from [8] in two cases: when σ 2 >, then n Z (dω) = 2 σ 2 nx (dω; ω(s) < for all < s < ɛ and some ɛ > ) (4) is proportional to the intensity measure n X of ecursions of X away from zero restricted to those starting negative; in the bounded variation case (BV), σ 2 = and (1 y )Π(d y) <, n Z (dω) = 1 y (dω)π(d y), b where necessarily b = a y1 { y 1} Π(d y) >. (5)

574 Electronic Communications in Probability Recall now from [4, 5, 8] that a Lévy process X possesses a partial right inverse if and only if σ 2 > or BV, b >, Π((, )) < or not BV, 1 2 Π(d ) ( 1 <. Π(, s)dsd y y)2 In the present paper, we describe n Z for a general Lévy process that possesses a partial right inverse. This seems to answer the final open question [4] related to the notion of right inverse of a Lévy process. However, this study can also be seen in the light of more general subordination [7] of the form X T = Y, where (T, Y ) is a bivariate Lévy process, increasing in the T-component. To formulate our main result, we recall some classical fluctuation theory [1, 2]. With any Lévy process X we associate the ascending ladder time and ladder height processes (τ, H), a bivariate subordinator such that H = X τ = X τ visits all suprema X t = sup{x s, s t}, t. If X t as t, then τ = (τ, < ξ) is a subordinator run up to an eponential time ξ. We write the Laplace eponent of (τ, H) in Lévy-Khintchine form as k(α, β) = ln e ατ 1 βh 1 ; ξ > 1 = κ + ηα + δβ + [, ) 2 1 e αs β y Λ(ds, d y). (6) It was shown in [8] that δ > whenever there eists a partial right inverse. In the sequel, we will always normalise the ascending ladder processes so that δ = 1. Also, when partial right inverses eist, then (T {z} < ) > for all z >, where T {z} = inf{t : X t = z}. In particular, the q-resolvent measure U q (dz) = e qt (X t dz)d t then admits a bounded density u q (z) that is continuous ecept possibly for a discontinuity at zero in the bounded variation case, see [6, Theorem 43.19]. Now R = X X is a strong Markov process with τ as its inverse local time; its ecursions, with added height H at freezing, e R (r) = R τ +r, r < τ = τ τ, e R (r) = H, r τ, form a Poisson point process whose intensity measure we denote by n R. For ω D, we write ζ + (ω) = inf{r > : ω(r) > }. For ω 1 D with ζ + (ω 1 ) < and ω 2 D, we concatenate ω = ω 1 ω 2, where ω(r) = ω 1 (r), r < ζ + (ω 1 ), ω(ζ + (ω 1 ) + r) = ω 2 (r), r. Theorem 1. Let X be a Lévy process that possesses a partial right inverse. Then the ecursion measures n Z away from the right inverse and n R away from the supremum are related as where the stochastic kernel n Z (dω) = (n R )(dω), (7) (ω 1, dω 2 ) = ω 1 (ζ + (ω 1 )) (dω 2), if ζ + (ω 1 ) <, (ω 1, dω 2 ) = δ otherwise, associates to a path ω 1 that passes positive a Lévy process path ω 2 that starts at the first positive height of ω 1 and is frozen when reaching zero, and where (n R )(dω) is the image measure of (ω 1, dω 2 )n R (dω 1 ) under concatenation (ω 1, ω 2 ) ω = ω 1 ω 2.

Right inverses of Lévy processes 575 For the case σ 2 = and X of unbounded variation, it was noted in [4] that a.e. ecursion under n X starts positive and ends negative, while a.e. ecursion under n R starts negative; by Theorem 1, the two measures n Z and n X are therefore singular, in contrast to (4) in the case σ 2 >. When X is of bounded variation, the discussion in [8, Section 5.3] easily yields the compatibility of (7) and (5); it can happen that a jump in the ladder height process H occurs without an ecursion away from the supremum, τ =, and so n Z can charge paths with ω() >, while in the unbounded variation case we have ω() = for n Z -a.e. ω E. Other descriptions of n Z follow: let n R t (dz) = nr ({ω E : ω(t) dz, ζ + (ω) ζ(ω) > t}). Corollary 2. In the setting of Theorem 1, the entrance laws of n Z are given by n Z t (dz) = nr t (dz) + P t s (y, dz)λ(ds, d y), (8) [,t] (, ) and the semi-group of n Z, or rather n Z ({ω E : (ω(t), t < ζ(ω)) }, is (P t (y, dz), t ). We also record an epression for the Laplace eponent ρ of the partial right inverse K. Theorem 3. Let X be a Lévy process which possesses a partial right inverse. Then ρ(q) = κ + ηq + 1 e qs uq ( y) u q Λ(ds, d y), with u q () = u q (+), (9) () [, ) [, ) where κ, η and Λ are as in (6), respectively, the killing rate and the drift coefficient of the ascending ladder time process and the Lévy measure of the bivariate ladder subordinator (τ, H). In particular, the characteristics (κ K, η K, Λ K ) of K in (2) are given by κ K = κ + Λ K (d t) = [, ) 2 (T { y} = )Λ(ds, d y), η K = η, [, ) [, ) (s + T { y} d t; T { y} < )Λ(ds, d y). (1) We stress that Λ({}, d y) is the zero measure unless X can jump into its new supremum from the position of its current supremum. The latter can happen only when the ascending ladder time has a positive drift η >, i.e. in particular only when X is of bounded variation. Let us note a simple consequence of Theorem 3 which can also be seen directly using more elementary arguments: (ξ K >, K t) > for some t > implies (ξ K > ) = 1. Corollary 4. A recurrent Lévy process has a partial right inverse iff it has a full right inverse. We proceed as follows. Sections 2 and 3 contain proofs of Theorem 3 and Corollary 4 eploiting recent developments [3] on joint laws of first passage variables, and, respectively, of Theorem 1 and Corollary 2. In an appendi, we introduce an alternative construction of right inverses and indicate an alternative proof of Theorem 3. 2 The Laplace eponent of K; proof of Theorem 3 Before we formulate and prove some auiliary results, we introduce some notation. Denote by (ds, d y) = (τ ds, H d y; ξ > )d the potential measure of the bivariate ascending

576 Electronic Communications in Probability ladder subordinator (τ, H), by T + = inf{t : X t (, )} the first passage time across level >, by X t = sup s t X s the supremum process, by G t = sup{s t : X s = X t or X s = X t } the time of the last visit to the supremum and by O = X T + the overshoot over level. Then on {(t, s, w, y) : t, s, w >, y }, we have (T + G T + d t, G T + ds, O dw, X T + d y; T + < ) = Λ(d t, du + y) ( d y, ds), (11) by a corollary of the quintuple law of Doney and Kyprianou [3]. Suppose that X possesses a partial right inverse. Recall construction (1). A crucial quantity there is the hitting time of levels, T {}. A key observation for our developments is that T {} = T + + T { O } a.s. on {T + < }, (12) where T { O } = inf{t : X t = O } for X = (X T + +t X T +, t ) independent of (T +, O ). For q >, let z n = 2 n 1 e qt {2 } n = 2 n 1 e q(t + 2 n + T { O2 n }), with the convention that e = and T + + T 2 n {O2 n } = on {T + = }. As was already eploited 2 n by Evans [5], construction (1) allows us to epress ρ(q) = ln lim e qk(n) 1 = ln lim 1 z 2 n n 2 n = lim z n, (13) because K (n) 1 is the sum of 2 n independent random variables with the same distribution as T {2 n }. To calculate this limit, we will use (12) and also decompose z n, as follows, setting z n = 2 n 1 e qt {2 n } 1{O2 n >,T + 2 n < } and z n = z n z n. (14) Lemma 5. Let X be a Lévy process of unbounded variation which possesses a partial right inverse. Then lim z n = 1 e qt uq ( h) u q Λ(d t, dh). (15) () (t,h) (, ) 2 Proof. According to [6, Theorems 43.3, 43.19 and 47.1], the resolvent density u q is bounded and continuous for all q > and (e qt {} ) = u q ()/u q () for all. We use (11) to obtain 1 e q(t + + T { O }) 1 {O >} = 1 e q(g T + +T + G T + + T { O }) 1 {O >} = 1 e q(s+t) uq ( w) u q Λ(d t, dw + y) (ds, d y) () (s, y) (, ) [,] (t,w) (, ) 2 = 1 e q(s+t) uq ( h + y) u q (ds, d y)λ(d t, dh). () (t,h) (, ) 2 (s, y) (, ) [, h] Therefore it will be sufficient to show that as n tends to infinity, we have the convergence 2 n 1 e q(t+s) uq ( h + y) u q (ds, 2 n d y)λ(d t, dh) () (t,h) (, ) 2 (s,y) (, ) [,2 n h] 1 e qt uq ( h) u q Λ(d t, dh). (16) () (t,h) (, ) 2

Right inverses of Lévy processes 577 First fi (t, h) (, ) 2 and consider the bounded and continuous function f : [, ) 2 [, ) given by f (s, y) = 1 e q(t+s) uq ( h + y) u q (17) () and the measures ϑ n,h (ds, d y) = 1 [,2 n h](y)2 n (ds, 2 n d y) on [, ) 2. Since H has unit drift, we have that (H t t for all t ) = 1 and so for all ɛ >, we have lim ϑ n,h (([, ɛ] [, ɛ]) c ) = lim 2 n = lim 2 n 2 n τ > ɛ, H 2 n d τ > ɛ, H 2 n d lim (τ 2 n > ɛ) =, whereas H t /t 1 a.s., as t, implies that (H 2 n (1 ɛ) 2 n ) 1 ɛ for n sufficiently large, and so 2 n 2 n (1 ɛ) 1 2 n (τ, H 2 n )d 2 n (H 2 n )d (1 ɛ) 2. This shows convergence ϑ n,h (ds, d y) = 1 [,2 n h](y)2 n (ds, 2 n d y) δ (,) weakly as n tends to infinity, where δ (,) is the Dirac measure in (, ); in particular [, ) 2 f (s, y)ϑ n,h (ds, d y) f (, ). (18) To deduce (16), and hence (15), from (18), we use the Dominated Convergence Theorem and for this purpose we show that f n (t, h) = 1 e q(t+s) uq ( h + y) u q (ds, 2 n d y) (19) () (s,y) (, ) [,2 n h] is bounded above by a Λ-integrable function. We shall prove the bound f n (t, h) (1 e qt ) + k(q, ) + 12 k(, ρ(q)) (1 h) + 1 uq ( h) u q, (2) () where we recall from (6) that k is the Laplace eponent of (τ, H). The integrand in (19) is f (s, y) = (1 e qt ) + e qt (1 e qs ) + e q(t+s) 1 uq ( h + y) u q () (1 e qt ) + (1 e qs ) + 1 uq ( h + y) u q, () three terms, where f (s, y) is defined in (17). First note that as before since H has unit drift Therefore 2 n ϑ n,h ([, ) 2 ) = 2 n P(H t 2 n )d t 1. (1 e qt )ϑ n,h (ds, d y) 1 e qt. (21) (s,y) [, ) 2

578 Electronic Communications in Probability For the second term we use (H u H t u t for all u t ) = 1 to write (1 e qs )ϑ n,h (ds, d y) [, ) 2 (1 e qs )2 n = 2 n 2 n s [, ) [,2 n ] [,2 n ] r [, ) 1 e qτ r 1{Hr [2 n h,2 n ]} dr τ r ds, H r [2 n h, 2 n ] dr 1 e qτ 2 n 1 {Hr [2 n h,2 n ]} dr 2 n (h 1) 1 e 2 n k(q,) (h 1)k(q, ). (22) For the third term we mimick the previous calculation to get 1 uq ( h + y) u q ϑ n,h (ds, d y) () [, ) 2 = 2 n 1 uq (( h + H r ) ) u q dr =: Υ(h). () r [,2 n h] We now eploit the fact [8, Corollary 2] that e ρ(q) u q ( ) is decreasing, and also 1 e 2 /2, to see, for h 2 n and similarly for h > 2 n, Υ(h) 2 n h h uq ( h) u q () = 2 n h uq ( h) u q () 1 uq ( h) u q () e ρ(q)h r dr 1 k(, ρ(q)) (1 e hk(,ρ(q)) ) + h 1 k(, ρ(q)), 2 Υ(h) 2 n 2 n uq ( h) u q 2 n k(, ρ(q)) 2n + 2. () 2 Together, this yields an upper bound for all h (, ) Υ(h) 1 uq ( h) u q + 1 k(, ρ(q))(h 1). (23) () 2 Thus (2) follows from (21), (22) and (23). In view of the fact that Λ(d t, dh) is a Lévy measure of a subordinator the RHS of (2) will be Λ(d t, dh)-integrable if 1 u q ( h)/u q () is Λ(d t, dh)- integrable. First using Fubini s Theorem in (16), followed by Fatou s Lemma, because of (18) and the simple inequality 1 uq ( h) u q () 1 uq ( h) e qt u q (),

Right inverses of Lévy processes 579 we get lim inf ẑn (t,h) (, ) 2 1 uq ( h) u q Λ(d t, dh). () On the other hand, (13) gives lim z n = ρ(q) <. Moreover, z n z n and we conclude that 1 u q ( h)/u q () is Λ(d t, dh)-integrable. Thus (2), together with the Dominated Convergence Theorem, implies (16) and then (15). Lemma 6. Assume that X is of unbounded variation and that X possesses a partial right inverse. Then lim z n = κ + (, ) (1 e qu )Λ(d t, {}). Proof. This proof is based on [1, Theorem VI.18], which yields (1 e qt + ) = k(q, )V q (), where V q () = (e qτ s ; Hs )ds. Also, V q () as, since V q is differentiable with v q (+) = 1, by dominated convergence, as H has unit drift coefficient. Then note that z n = 2 n 1 e qt {2 n } 1{O2 n =,T + = 2 n 1 e qt +2 n 1 {O2 n =,T +2 n < or T +2 n = } 2 n < or T + 2 n = } and so, we obtain the required formula from (6) noting η = in the unbounded variation case 1 1 e qt + 1 1 e qt + 1{O >,T + < } κ + (t,h) (, ) [, ) (1 e qt )Λ(d t, dh) (t,h) (, ) 2 (1 e qt )Λ(d t, dh). Although this is not necessary for our proof of Lemma 6, we would like to mention that we can calculate eplicitly z n or, as Andreas Kyprianou pointed out to us, we can etend (11): Proposition 7. In the relevant case δ = 1 of the setting of (11), cf. (6), we also have (T + d t, O = )d = (T + = G T + d t, O =, X T + = ; T + < )d = (d t, d ), Proof. Indeed, note that for H 1 T + = τ H 1 (, ) = = inf{s : H s > } we have O = iff H H 1 for a.e. a.s., so that as H has unit drift coefficient δ = 1, t (, ) e ατ H 1 e αt β (T + βh H 1 dh 1 d t, O = )d = e ατ H 1 βh H 1 1 { HH 1 =}d = and = e ατ s βh s ds = e αt β (d t, d ). (,t) (, ) 2 Proof of Theorem 3. Let X be a Lévy process that possesses a partial right inverse. If X is of unbounded variation, we have lim z n = ρ(q) by (13); and (14) together with Lemma 5 and Lemma 6 proves the claim.

58 Electronic Communications in Probability When X is of bounded variation we refer to [8, Section 5.3] which discusses in this case the Laplace eponent ρ(q) of the partial right inverse and how the right inverse relates to ladder processes. This establishes (9). The characteristics can now be read off by inverting the Laplace transform u q ( y)/u q () = (e qt { y} ; T{ y} < ) for y, where we recall u q () = u q (+), which entails (T {} = ) = 1. Let us briefly eplore the contet of the last part of this proof. For y =, note that (T {} = ) = 1 is a trivial consequence of the definition T {} = inf{t : X t = }. Indeed, this is the appropriate notion to use in the light of Theorem 1, where ecursions below the supremum ending at zero before passing positive do not get marked by a further return time T > {} = inf{t > : X t = }, which in the bounded variation case would have Laplace transform (e qt > {} ; T > {} < ) = u q ( )/u q (+) < 1, see e.g. [6, Theorem 43.21]. Proof of Corollary 4. Suppose that X has a partial right inverse. In terms of the characteristics (κ K, η K, Λ K ) of the minimal partial right inverse K, this is indeed a full right inverse if ρ(+) = κ K =. But if X is recurrent (and has a partial right inverse), then X does not drift to, so κ =, and (T { y} = ) = for all y, so indeed κ K = κ + [, ) 2 (T { y} = )Λ(ds, d y) =. Similarly, it is also straightforward to show that in the transient case X possesses a full right inverse if and only if X drifts to + and has no positive jumps in that Π((, )) =. 3 The ecursion measure away from K; proof of Theorem 1 Although Theorem 1 is more refined than Theorem 3, it is now a straightforward consequence. Proof of Theorem 1 and Corollary 2. The proof relies crucially on [8, Theorem 2]. First recall that the semigroup within ecursions away from the right inverse is (P t (y, dz), t ). With this, the ecursion measure n Z is uniquely determined by its entrance laws. Let us first check that the entrance laws given in (8) satisfy (3), which is (vi) of [8, Theorem 2]. A standard ecursion measure computation similar to the proof of that theorem together with the Wiener- Hopf factorization gives directly e qt e iλy n R k(q, ) t (d y)d t = e iλ (R γ(q) d ) q \{} k(q, ) = (e iλr γ(q) ) (R γ(q) = ) = k(q, ) k(q, ) k(q, iλ) q q η = η, (24) k(q, iλ) q + ψ(λ) where k(α, β) is the Laplace eponent of the bivariate descending ladder process, which is the ascending ladder process of X, and where γ(q) is an independent eponential random variable with rate parameter q; we recall R γ(q) d = X γ(q) = inf{x s, s γ(q)}; the Wiener-Hopf identity k(q, ) k(q, iλ) k(q, ) q = k(q, iλ) q + ψ(λ)

Right inverses of Lévy processes 581 can be found in [2, Formulas (4.3.4) and (4.3.7)]. Net we compute the joint transform of the remaining part of the RHS of (8). e qt e iλz P t s (y, dz)λ(ds, d y)d t = = 1 q [,t] [, ) [, ) (, ) [, ) (, ) e qs e q(t s) y (e iλx t s ; T {} > t s)d tλ(ds, d y) s e qs y (e iλx γ(q) ; T {} > γ(q))λ(ds, d y), where γ(q) is an independent eponential random variable with rate parameter q. Also recall so that y (T {} γ(q)) = (e qt { y} ) = uq ( y) u q (), where uq () = u q (+), qe iλy q + ψ(λ) = y(e iλx γ(q) ) = y (e iλx γ(q) ; T {} > γ(q)) + y (e iλx γ(q) ; T {} γ(q)) = y (e iλx γ(q) ; T {} > γ(q)) + y (T {} γ(q))(e iλx γ(q) ) = y (e iλx γ(q) ; T {} > γ(q)) + uq ( y) u q () q q + ψ(λ). Thus we have computed y (e iλx γ(q) ; T{} > γ(q)). Hence e qt e iλz P t s (y, dz)λ(ds, d y)d t [,t] [, ) 1 = q + ψ(λ) [, ) (, ) e qs e iλy uq ( y) u q Λ(ds, d y). () Net observe that by using (6) and adding the numerator of the first term in (24) we get k(q, iλ) + e qs e iλy uq ( y) u q Λ(ds, d y) () [, ) (, ) = κ + ηq iλ + 1 e qs+iλy + e qs+iλy e qs uq ( y) u q Λ(ds, d y) = ρ(q) iλ, () [, ) 2 where the last equality holds by Theorem 3. Together with (24), this proves (8) since (3) holds for the measure on the RHS of (8). To finish the proof of Theorem 1, we note that the RHS of (7) is Markovian with semi-group (P t (y, dz), t ) and check that the RHS of (7) has as entrance laws the RHS of (8). Specifically, (n R )({ω D : ω(t) dz; ζ(ω) > t}) = n R ({ω 1 D : ω 1 (t) dz; ζ + (ω 1 ) > t}) + {ω 1 D:ζ + (ω 1 ) t} = n R t (dz) + [,t] [, ) (ω 1 ; {ω 2 D : ω 2 (t ζ + (ω 1 )) dz; ζ(ω 2 ) > t ζ + (ω 1 )})n R (dω 1 ) P t s (y, dz)λ(ds, d y)

582 Electronic Communications in Probability since n R ({ω 1 D : ζ + (ω 1 ) ds, ω 1 (ζ + (ω 1 )) d y}) = Λ(ds, d y). A Alternative proof of Theorem 3 Suppose that X possesses right inverses. We introduce an alternative construction that we first use to sketch a heuristic proof of Theorem 3. Informally, for all n we approimate K by the ascending ladder time process τ, but when an ecursion away from the supremum with e R ( τ ) = H > 2 n appears, our approimation K(n) of K makes a jump whose size is the length of this ecursion plus the time needed by X to return to the starting height H = X τ = X τ of the ecursion, after which we iterate the procedure. Then X K (n) evolves like an ascending ladder height process H with jumps of sizes eceeding 2 n removed. Let us formalize this. Fi n. Consider the bivariate subordinator (τ, H). To begin an inductive definition, let S 1 (n) = inf : H > 2 n K S1 (n)(n) = inf K (n) = τ, < S 1 (n) t τ S1 (n) : X t = X τs1 (n), if S 1 (n) <. Given (K (n), S m (n)) and T m (n) = K Sm (n)(n) <, let X (m) t (n) = X Tm (n)+t X Tm (n), t. With (τ (m) (n), H (m) (n)) as the bivariate ladder subordinator of X (m) (n), define S m+1 (n) = S m (n) + inf : H (m) (n) > 2 n K Sm (n)+(n) = T m (n) + τ (m) (n), < S m+1 (n) S m (n) K Sm+1 (n)(n) = T m (n) + inf t τ (m) (m) (n) : X S m+1 (n) S m (n) t (n) = X (m) τ (m) (n) (n) S m+1 (n) Sm(n) Thus we have defined K (n) for all and n a.s. Now it must be epected that K = inf sup y> n K Ln (y)(n) = lim K (n), where L n (y) = inf{ : X K (n) y 2 n }. (25) To derive formula (1) of Theorem 3, consider the Poisson point process ( K (n), ) whose intensity measure we can calculate from (( τ, H ), ) using standard thinning (keep if H 2 n, modify if H > 2 n ), marking by independent T { H } if H > 2 n and mapping ( τ, H, T { H }) τ + T { H } = K (n) of Poisson point processes, as Λ K(n) (d t) = y [,2 n ] Λ(d t, d y) + (s, y) [, ) (2 n, ) s + T { y} d t Λ(ds, d y), which converges to the claimed epression, as n. We now make this approach rigorous, the main task being to rigorously establish a variant of (25). To simplify notation, let us assume in the sequel that X possesses a full right inverse. In the case where only partial right inverses eist we can follow the construction above until the first m for which H (m) (n) is killed before its first jump of size eceeding 2 n. If we denote the resulting process by (K (n), < ξ(n)), we can insert suitable restrictions to events such as { ξ(n) > } into the following arguments..

Right inverses of Lévy processes 583 Lemma 8. Let K (n) be as above. Then ((K (n), X K (n)), ) is a bivariate subordinator with drift coefficient (η, 1) and Lévy measure Λ n (d t, dz) = Λ(d t, dz [, 2 n ]) + (s + T { y} d t, dz)λ(ds, d y). (s,y) [, ) (2 n, ) Proof. Let H (n) = X K (n). Then ( K(n), H(n)) inherits the drift coefficient (η, 1) from (τ, H). By standard thinning properties of Poisson point processes, (( K (n), H (n)), < S 1 (n)) has the distribution of a Poisson point process with intensity measure Λ(d t, dz [, 2 n ]) run up to an independent eponential time S 1 (n) with parameter λ = Λ([, ) (2 n, )), and ( K S1 (n) d t, H S1 (n) dz) = λ 1 (s + T { y} d t, dz)λ(ds, d y). [, ) (2 n, ) By the strong Markov property of X at T m (n), m 1, the process (( K (n), H (n)), ) with points at S m (n), m 1, removed, is a Poisson point process with intensity measure Λ(d t, dz [, 2 n ]) independent of the removed points, which we collect in independent and identically distributed vectors (S m (n) S m 1 (n), K Sm (n)(n), H Sm (n)(n)), m 1. By standard superposition of Poisson point processes, the result follows. Lemma 9. With K (n) as above, we have K = inf sup y> n K Ln (y)(n), where L n (y) = inf{ : X K (n) y 2 n }. Proof. By construction, the process (X K (n), ) has no jumps of size eceeding 2 n, so that 2 n X K L n()(n). Note that we have K Ln ()(n) K. Let us define K by K = lim inf K Ln ()(n) = lim inf m n K Lm ()(m) sup K Ln ()(n) K, n where the limit in the middle member of this sequence of inequalities is an increasing limit of stopping times. Since X is right-continuous and quasi-left-continuous [1, Proposition I.7], we obtain 2 n X infm n K X L K m() = lim X infm n K = a.s. L m() Now, it is standard to argue that X K q = q holds a.s. simultaneously for all q [, ) and, since K is increasing and X right-continuous, inf y> K y = inf q (, ) K q K is a rightcontinuous right inverse. Since K is the minimal right-continuous right inverse, inf y> K y = K. Lemma 1. Let K (n) be as above. Then for all there is convergence along a subsequence (n k ) k of lim k K (n k ) = K a.s. Proof. Denote by eb Lebesgue measure on [, ). By Lemma 8, H(n) has drift coefficient 1, so H (n) and eb({ H z (n), z }) = a.s., and the distribution of H (n) is such that e β H (n) = ep β [,2 n ] (1 e β y )Λ H (d y) e β for all β.

584 Electronic Communications in Probability Therefore, H (n) in probability, and there is a subsequence (n k ) k along which convergence holds almost surely. Now let ɛ >. Then there is (random) N such that for all n k N, we have H (n k ) + ɛ. Therefore, lim sup K (n k ) inf K +ɛ = K. k ɛ> Since L n () a.s., the previous lemma implies the claimed convergence. Proof of Theorem 3. This proof is for the case where X has a full right inverse and we only prove (9). The general case can be adapted. By Lemma 1, we can approimate K = lim k K (n k ). The Laplace eponent of K (n k ) follows from Lemma 8 and this yields ρ(q) = ln e 1 qk = lim ln e q K 1 (n k ) k = ηq + lim k = ηq + + lim k [, ) [,2 n k ] [, ) (2 n k, ) [, ) {} (1 e qs )Λ(ds, d y) [, ) [, ) (1 e qs )Λ(ds, d y) + (1 e qt )(s + T { y} d t, dz)λ(ds, d y) [, ) (, ) 1 e qs e qt { y} Λ(ds, d y). References [1] Bertoin, J. (1996) Lévy Processes. Cambridge University Press. MR146564 [2] Doney, R. (27) Fluctuation theory for Lévy processes. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour. Lecture Notes in Mathematics 1897. Springer, Berlin. MR232889 [3] Doney, R.A. and Kyprianou, A.E. (26) Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16, No. 1, 91 16. MR229337 [4] Doney, R. and Savov, M. (21) Right inverses of Lévy processes. Ann. Prob. 38, No. 4, 139 14. MR2663631 [5] Evans, S. (2) Right inverses of non-symmetric Lévy processes and stationary stopped local times. Probab. Theory Related Fields. 118, 37 48. MR1785452 [6] Sato, K. (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press. MR173952 [7] Simon, T. (1999) Subordination in the wide sense for Lévy processes. Probab. Theory Related Fields. 115, 445 477. MR1728917 [8] Winkel, M. (22) Right inverses of non-symmetric Lévy processes. Ann. Prob. 3, No. 1, 382 415. MR1894112