On maximal inequalities for stable stochastic integrals

Transcription

1 On maximal inequalities for stable stochastic integrals Aldéric Joulin Laboratoire de Mathématiques et Applications Université de La Rochelle Avenue Michel Crépeau 174 La Rochelle Cedex 1 France May 9, 6 Abstract Sharp maximal inequalities in large and small range are derived for stable stochastic integrals. In order to control the tail of a stable process, we introduce a truncation level in the support of its Lévy measure: we show that the contribution of the compound Poisson stochastic integral is negligible as the truncation level is large, so that the study is reduced to establish maximal inequalities for the martingale part with a suitable choice of truncation level. The main problem addressed in this paper is to give upper bounds which remain bounded as the parameter of stability of the underlying stable process goes to. Applications to estimates of first passage times of symmetric stable processes above positive continuous curves complete this work. Key words: Stable processes, stable stochastic integrals, maximal inequalities, first passage times. Mathematics Subject Classification. 6E15, 6G5, 6H5. 1 Introduction Given a filtered probability space Ω = Ω, F, F t t, P, consider on Ω a càdlàg real stable process Z = Z t t ajoulin@univ-lr.fr of index α, without Gaussian component 1

2 and let H = H t t be a sufficiently integrable predictable càdlàg process. The purpose of this paper is to give maximal inequalities for stable stochastic integrals H Z = H sdz s t. We show that their decay in the bilateral case is s P sup H τ dz τ x K αx α H α L Ω [,t], x x α, p > α, 1.1 st whereas in the unilateral case, if Z is symmetric and α 1,, it is s α/α 1 x P sup H τ dz τ x L α exp M α, x x α. st H L Ω,L α [,t] 1. Here L α, M α, x α and x α stand for positive numbers depending explicitly on α, whereas K is a positive constant independent of α. It is known since the early 8 s that stable stochastic integrals inherit regularly varying tails from the underlying stable process. For example, in order to prove the central limit theorem for stable stochastic integrals in the Skorohod space, Giné and Marcus established in [1] the maximal inequality sup x α P sup H s dz s x x> t1 D α α H α L α Ω [,t], 1.3 where D is a universal constant independent of α. However, as α tends to, the upper bound in their maximal inequality 1.3 goes to infinity. On the other hand, the extremal behavior of stochastic integrals driven by multivariate Lévy processes with regularly varying tails have been studied recently in [14] by Hult and Lindskog, and by Applebaum, see [3]. In particular, if Z is symmetric and H is square-integrable and satisfies further the uniform integrability condition E [ sup t [,1] H t ] < + for some p >, then Example 3. in [14] yields the extremal behavior s lim x + xα P sup H τ dz τ x = C α H α L Ω [,t], t [, 1], 1.4 α st where C α depends on α and remains bounded as α, ]. Therefore, as α gets close to, the maximal inequality 1.3 of Giné and Marcus does not recover the non-explosive asymptotic estimate 1.4. Our approach to establish maximal inequalities for stable stochastic integrals is based

3 on stochastic calculus for jump processes and allows us to avoid the limiting explosion of the upper bound described above. Following Pruitt in [19] for Lévy processes and more recently Houdré and Marchal in [13] in the specific case of stable random vectors, the method relies on the use of the Lévy-Itô decomposition of Z with a truncation level R in the support of its Lévy measure, in order to control the jump size of the martingale part: Z is split into the sum of a square-integrable martingale with infinitely many jumps bounded by R on each compact time interval, and a compound Poisson process which represents the large jumps of Z, plus a drift part. Constructing then the stable stochastic integral H Z with respect to the above semimartingale decomposition, we show that the contribution of the compound Poisson stochastic integral in both bilateral and unilateral cases is negligible as the truncation level is large, reducing the study to the proof of maximal inequalities for the martingale part of H Z. Using stochastic calculus for Poisson random measures, sharp estimates follow by choosing suitably the truncation level R. Let us describe the content of the paper. In Section, some notation and basic properties of stable processes are introduced. Then we apply a truncation method somewhat similar to that of Pruitt to derive maximal inequalities for stable stochastic integrals, and compare them with the corresponding results of Giné and Marcus, and Hult and Lindskog, see [1] and [14]. In particular, Proposition.4 slightly improves the estimate in [1, Theorem 3.5] when the index of stability α of the underlying stable process lies in 1, and under some integrability conditions. The main contribution of this paper is contained in Section 3, Theorem 3., where large range inequalities are given in the bilateral case 1.1, freeing us from the explosion of the upper bound as α goes to. Section 4 is devoted to small range tail estimates in the unilateral case 1.. As a result, we recover the classical maximal Gaussian inequality via Theorem 4. and a limiting procedure in the Skorohod space. Finally, we apply in Section 5 the results of Section and 3 to estimate first passage times of a symmetric stable process above several positive continuous curves. The method relies on an extension to the stable case of the results of [1, 18] established for Brownian motions. 3

4 Notation and preliminaries Let Ω = Ω, F, F t t, P be a filtered probability space and let Z be a càdlàg real stable process on Ω of index α, without Gaussian component. For the sake of briefness, by a stable process we will implicitly mean an F t t -adapted real càdlàg stable process in the remainder of this paper. Recall that its characteristic function is defined by ϕ Zt u = exp t iub + + where ν stands for the stable Lévy measure on R: νdy = c 1 {y<} + c + 1 {y>} e iuy 1 iuy 1 { y 1} νdy,.1 dy y α+1, c, c +, c + c + >.. As a Lévy process, Z is a semimartingale whose Lévy-Itô decomposition is given by Z t = bt + y µ σdy, ds + y µdy, ds, t,.3 y 1 where µ is a Poisson random measure on R [, + with intensity σdy, dt = νdy dt and b is the drift. In particular, if α < 1, then Z is a finite variation process whereas when α 1, we have a.s. y >1 Z s = +, t >, where Z s denotes the jump size of Z at time s >. st Z is said to be strictly stable if we have the self-similarity property Z kt t d = k 1 α Zt t, where k > and the equality d = is in the sense of finite dimensional distributions. If moreover c := c + = c, then Z is symmetric and its characteristic function.1 is computed to be ϕ Zt u = e tρα u α,.4 where ρ α := πγ α/ α α Γ1 + α/ c. 4

5 .1 The truncation method In order to control the jump size of the martingale part of the stable stochastic integral, let us introduce the truncation method of the stable Lévy measure.. For some truncation level R > 1, let Z R+ and Z R be the independent Lévy processes defined by Z R t := y R y µ σdy, ds, Z R+ t := y >R y µdy, ds, t. The first one has a compactly supported Lévy measure and is a square-integrable martingale with infinitely many jumps bounded by R on each compact time interval, whereas the second one is a compound Poisson process. The Lévy-Itô decomposition.3 rewrites as Z t = b R t + Z R t where b R := b + y νdy is a drift depending on R. 1< y R Given a predictable càdlàg process H, let H p,t := H L p Ω [,t] = + Z R+ t, t,.5 1 E [ H s p ] ds p, t, p >, and define P p resp. B p as the space of predictable càdlàg process H such that for all t, H p,t < + resp. H L Ω,L p [,t] < +. In particular, H is said integrable if H P 1 and square-integrable if H P. Following [, Chapter 4], we construct the stable stochastic integral of a squareintegrable predictable process H as the sum of L -type and Lebesgue-Stieltjes stochastic integrals: letting X R t := H s dz R s, X R+ t := H s dz s R+, A R t := b R H s ds, t, the first integral X R = H Z R is a square-integrable martingale, whereas the integrals X R+ = H Z R+ and A R are constructed in the Lebesgue-Stieltjes sense, and we define the stable stochastic integral as X t := H s dz s = A R t + X R t + X R+ t, t..6 5

6 We denote respectively by a b and a b the maximum and the minimum between two real numbers a and b. We finish by making two remarks on the maximal inequalities of type 1.1 or 1. we will establish in the remainder of this paper: Remark.1 The truncation level R is related to the deviation level x and to some L p -norm of the process H, and is chosen each time equal to its optimal value. Remark. Although they can be computed, the constants appearing in the upper bounds are not given explicitly in general, since their numerical value is not of crucial importance in our study.. A first maximal inequality In order to study the rates of growth of Lévy processes, Pruitt established in [19] some maximal inequalities whose proofs are based on a truncation method for general Lévy measures, with a particular choice of truncation level. Inspired by this work, we derive in this part a first maximal inequality for stable stochastic integrals by using the semimartingale decomposition.6. Fix t and x > H,t. Using the above notation, we have by.6: P sup st X s x P sup st P sup st A R s + sup st A R s x X R s + P + sup st X R+ s sup X s R x st x + P sup X s R+ >. st.7 First, we investigate the absolutely continuous part A R. By Chebychev s inequality, P sup A R s x t P H τ dτ x st b R [ 4b R t ] E H x τ dτ. 6

7 Using the elementary inequality a + b a + b, a, b R, and then Cauchy- Schwarz inequality, b R = b + 1< y R y νdy b + y νdy 1< y R b + ν {y R : 1 < y R} y νdy 1< y R b + ν {y R : y > 1} y νdy = b + c + c + α α R α. y R By Cauchy-Schwarz inequality again and since x > H,t, we have P sup A R s x 8t b + c + c + st x α α R α H,t < 8tb H α,t x α + 8tc + c + R α H,t α αx..8 Now, we show that the contribution of the compound Poisson stochastic integral X R+ is negligible as the truncation level R is sufficiently large. Recall that the integral X R+, and so its supremum process sup st X s R+ t, has piecewise constant sample paths and its distribution at any time has an atom at. Now, denote by T R 1 the first jump time of the Poisson process µ {y R : y > R} [, t] t on the set {y R : y > R}. If a.s. T R 1 occurs after time t, then the compound Poisson stochastic integral X R+ and so its supremum process is identically on the interval [, t]. Thus we have P sup X s R+ > st = 1 P sup X s R+ = st 1 P T1 R > t = 1 exp tν {y R : y > R} tν {y R : y > R} = c + c + t αr α,.9 7

8 where we used in the second equality above that T R 1 parameter ν {y R : y > R}, see e.g. [, Theorem 1.3]. is exponentially distributed with Recall now that X R is a square-integrable martingale involving the small jumps of Z. By Doob s inequality together with the isometry formula for Poisson stochastic integrals, P sup X s R x st [ 4 x E = 4 [ x E = 4 x y R y R y R y νdy H τ y µ σdy, dτ ] Hτ y νdydτ E [ H τ ] dτ, ] that is to say P sup X s R x 4c + c + H,t R α..1 st αx Finally, using.7 and choosing the truncation level R = x H,t > 1 in.8,.9 and.1 show that there exists K := Kb, c, c +, t >, independent of α, such that P sup st s Let us comment the estimate.11. H τ dz τ x K H α,t α αx, x > H,t..11 α If Z is symmetric with Lévy measure νdy = c y α 1 dy, c >, and H satisfies further the uniform integrability condition E [ sup t1 H t ] < +, p >, then Example 3. in [14] entails the asymptotic estimate s lim x + xα P sup H τ dz τ x = K α,c α st H τ α dτ, t [, 1],.1 where K α,c := { c π1 αγ α/ α+1 Γ αγ1+α/ cosπα/ if α 1, c if α = 1, 8

9 which remains bounded as α [, ]. It shows that.11 is sharp for α, and also as α converges to, but goes to infinity as α tends to, in contrast to.1. On the other hand, assuming H P α, then a combination of Theorem 3.5 and Example 3.7 in [1] implies the maximal inequality sup x α P sup D H s dz s x x> α α t1 1 E [ H t α ] dt,.13 where D is a universal constant independent of α. Therefore, the speed of explosion in.11 is better than in.13 since it is linear in α and not quadratic, but it is worse in terms of L p -norm of H, since the L -norm is involved instead of the optimal L α -norm. Before avoiding in Section 3 the explosion of its upper bound as α gets close to, let us now improve.11 in terms of L p -norm of H..3 A maximal inequality in optimal L α -norm First, we quote [4, Proposition.1], up to a minor modification related to the integrability property of H: Lemma.3 Consider a stable stochastic integral X := H Z, where Z is a symmetric stable process of index α 1, with generator L, and H is square-integrable. Let f be a C R-function with bounded first and second derivatives. Then the process M f given by is a martingale. M f t := fx t fx H s α L fx s ds, t, Now, we improve the upper bound in.11 in terms of L p -norm of H. Actually, the estimate in Proposition.4 below recovers via a different proof the inequality.13 of Giné and Marcus, and slightly improves it as α tends to, since the speed of the explosion of the upper bound is not quadratic but only linear in α: Proposition.4 Let Z be a symmetric stable process of index α 1, and Lévy measure νdz = c z α 1 dz, c >, and let H be square-integrable. Then there exists K α,c >, finite as α tends to, such that x α P sup H s dz s x sup x> t1 9 K α,c α 1 E [ H t α ] dt.

10 Proof. The present proof is an adaptation to the case of stable stochastic integrals of that of Bass in [5, Proposition 3.1]. Denote by L the infinitesimal generator of Z. Let f be a non-negative C R-function such that f =, fy = 1 if y 1 and whose first and second derivatives are bounded above in absolute value respectively by c 1 > and c >. Let x >, f x y := fy/x and let τ x := inf{t : X t x} be the first exit time of the stable stochastic integral X = H Z of the centered ball of radius x. If the process exits the ball before time 1, then f x X 1 τx = 1 and by Lemma.3 and a conditioning argument, P sup X t x t1 = P τ x 1 E [f x X 1 τx ] [ 1 τx ] = E H t α L f x X t dt Therefore, By the symmetry of ν, 1 P sup X t x t1 L f x L R L f x y = R z R E [ H t α L f x X t ] dt. 1 f x y + z f x y zf xy νdz c z x νdz + c 1 z x νdz z >R = c cr α αx + 4c 1cR 1 α α 1x. E [ H t α ] dt..14 If we choose the truncation level R = x, then denoting K α,c := c c+4c 1 c α/α 1, the calculus above implies the bound L f x L R K α,c αx α. Finally, plugging this into.14, the proof is complete. 1

11 3 Large range estimates for α close to The purpose of the present part is to control the upper bound in.11, freeing us from its explosion as α tends to. The price to pay is to require stronger integrability conditions on the process H and to reduce the range interval of the deviation level x. First, we recall Bihari s inequality, which is a Gronwall-type inequality. See e.g. [11, Chapter 1] for a proof of such an inequality. Lemma 3.1 Let T be a positive time horizon and let ρ, ψ and g be positive measurable functions such that ρ is monotone-increasing, s ψsρ gs is integrable on [, T ] and gs K T + s where K T. Then the Bihari inequality holds, where φx := x gt φ 1 φk T + dy ρy. We can now state the main result of this paper: ψτ ρgτ dτ, s [, T ], 3.1 T ψsds Theorem 3. Let Z be a stable process of index α 1, and Lévy measure ν given by.. Let p > α, ɛ > and let H P. Then for all t, there exists K := Kb, c, c +, t, p, ɛ >, independent of α, such that for all x α > H α,t max 1, p 4 ɛ αα + p 1 c + c + t, we have the maximal inequality P sup Proof. st s H τ dz τ x K H α,t. 3. x α We proceed as in the proof of inequality.11 and investigate first the absolutely continuous part A R analogously to.8. Fix t and x > H,t. By the elementary inequality a + b q q 1 a q + b q, a, b R, q 1, applied with q = α + p, together with Hölder s inequality, we get P A R s x sup st 11

12 1 t 1 H,t x = 1 t 1 H,t x 1 t 1 H α,t x α b + ν {y R : y > 1} 1 b + c + c + R p p α 1 b + c + c + R p H p,t p α 1 x p where we used in the last inequality x > H,t. y R y νdy, 3.3 Now, let us control the martingale part X R = H Z R. By Doob s and Burkholder s inequalities for martingales with jumps, see e.g. pp in [1], we have P sup X s R x [ st x E ] H s dz s R [ [ C E H x s dz s R, H s dz s R = C x [ E y R Let Y s s [,t] be the finite variation process defined by s Y s := Hτ y µdy, dτ, s t. y R H s y µdy, ds ] t ] ]. 3.4 By Itô s formula for jump processes and the inequality a+b q a q qba+b q 1, a b, q 1, applied with q = α + p/, we have s Y Yτ s = + Hτ y Y τ µdy, dτ y R α + p s Hτ y Y τ + Hτ y µdy, dτ y R α + p s 4 1 Hτ y Y τ + H τ y µdy, dτ. y R where we used in the last inequality the elementary bound a + b q q 1 1 a q + b q, a, b, q, applied with q = /. Denote D α,p = 4 1. Taking expectations and using Hölder s inequality, we get ] E [Y s 1

13 D α,p α + pc + c + D α,p α + pc + c + R α α R α α s s E ] [H τ Y τ dτ + Rp p H,s E [ H τ ] E [Y τ ] dτ + Rp p H,t. Applying Lemma 3.1 with T = t, gs := E ] [Y s, ψτ := E [ H τ ], K t := D α,p α + pc + c + R p p H,t and ρx := D α,p α + pc + c + R α α x, and by using Hölder s inequality to estimate ψτdτ, we obtain where gt Φ Φ D 1 α + pc + c + R p α,p H,t p + t H,t, Φx := = x dy ρy α D α,p c + c + R x α. Hence we have E ] [Y t D α,p c + c + R α α α αα + p R Dα,p c + c + p + t H,t. Now, choose the truncation level R = Since the assumption on x claims that x H,t > 1. x α α > H,t t Dα,p c + c + p ɛ αα + p, 13

14 we establish the following bound on moments E ] [Y t D α,pc + c + α + p1 + ɛ H α,t x p. p Finally, plugging the latter inequality into 3.4 yields P sup X s R x C D α,p c + c + α + p1 + ɛ H α,t, st px α and together with.7 and the choice of truncation level R = x/ H,t in.9 and 3.3, Theorem 3. is proved. Under further assumptions on Z and H, the process H Z is a time-changed stable process and we get the following maximal inequality, which is asymptotically optimal in terms of L α -norm when H L α [,t] is bounded on Ω for all t : Corollary 3.3 Let Z be a symmetric stable process of index α 1, and Lévy measure νdy = c y α 1 dy, c >. Let H B α with a.s. lim t + H s α ds = +. Let p > α and ɛ >. Then there exists K := Kc, p, ɛ >, independent of α, such that for all t and for all x α > H s α ds we have the estimate P sup Proof. st s L Ω p ɛ αα + p H τ dz τ x Kx t α 4 1 c, H s α ds L Ω. 3.5 By [1, Theorem 3.1], the process H Z is a time-changed process of Z, i.e. we have the identity a.s. H s dz s = Ẑτ t, t, where τ = τ t t given by τ t := H s α ds is a time change process, and Ẑ is a symmetric stable process defined on Ω and having the same distribution as Z. Since the symmetry of Ẑ implies it is self-similar of index α, then so is the supremum process: d sup Ẑ s = skt t k 1 α 14 sup Ẑ s, k >. st t

15 Thus, denoting βt := τ t 1/α L Ω, we have s P sup H τ dz τ x st = P sup st Ẑτ s x P sup Ẑs x sτ t P sup Ẑs x sβt α = P sup Ẑs x. s1 βt Finally, applying Theorem 3. with H s complete. = 1 for all s t = 1, the proof is Remark 3.4 If Z is a non-symmetric strictly stable process and H is positive and satisfies further the hypothesis of Corollary 3.3 resp. that of Theorem 4. below, then the stable stochastic integral H Z is still a time-changed process of Z. Thus, applying in the proof above resp. in the proof of Theorem 4. Theorem 3 in [17] instead of Theorem 3.1 in [1], an estimate somewhat similar to that of Corollary 3.3 resp. Theorem 4. can be established. 4 Small range maximal inequalities In this part, we derive small range estimates in the unilateral case 1.. Recently, Breton and Houdré investigated in [8] small and intermediate range concentration for stable random vectors. In particular, the small range behavior is covered by their Theorem 1, whose small deviation rate is of order exp c α x α/α 1 for some positive c α depending on α. Before proving a similar rate for suprema of stable stochastic integrals, let us establish first the result for symmetric stable processes via Proposition 4.1 below. We point out that using the scaling property, it is sufficient to get the result on the time interval [, 1]. Proposition 4.1 Let Z be a symmetric stable process of index α 1, and Lévy measure νdy = c y α 1 dy, c >. Then for all λ > λ α, where λ α is the unique 15

16 solution of the equation λ log 1 + αλ = 4c c α, there exists x α, λ > such that for all x x α, λ, P sup Z s x c x α λ log 1 + α 1 + exp t1 α λ αλ c x α α λ Proof. As in the proof of inequality.9, we have P sup Z s x P sup Z s R+ > t1 t1 c αr + P α + P sup Z s R x t1 sup Z s R x t1. 4. The Lévy process Z R is a martingale with jumps bounded by R, hence has exponential moments, see e.g. [9, Proposition 3.14]. Moreover, the angle bracket process < Z R, Z R > is computed to be < Z R, Z R > t = y R = ct α R α = v t R. y νdyds Let φz := z e z z 1, z >, and define for all β > the process S β,r by S β,r t = exp βz R t β φβr < Z R, Z R > t, t. By [16, Lemma 3.19], S β,r is a supermartingale for all β >. Thus, the exponential Markov s inequality yields P sup Z s R x t1 inf P sup S β,r t exp βx β v 1 R φβr β> t1 inf exp βx + β v 1 R φβr β> x x = exp exp R x R log 16 R + v 1R R 1 + Rx v 1 R log 1 + Rx v 1 R

17 = exp x R log 1 + αrα 1 x, c where in the latter inequality we used 1 + u log1 + u u u log1 + u, u, which is equivalent to 1 + u/ log1 + u u, u, established by a standard convexity argument. Now, let the truncation level R be such that x = λr 1 α for some λ >. Plugging the last inequality into 4., we get P sup Z s x c x α λ log α 1 + exp t1 α λ x =: F λ 1 + αλ c x λ α α 1 α α A necessary condition for the upper bound in 4.3 to make sense is that the real number F x/λ α/α 1 has to be smaller than 1, which is the case in a neighborhood of + if λ > λ α. Finally, choose x α, λ > such that F x α, λ/λ α/α 1 = 1 to obtain the maximum range of validity for the result. Now, we can establish a small range maximal inequality for stable stochastic integrals: Theorem 4. Let Z be a symmetric stable process of index α 1, and Lévy measure νdy = c y α 1 t dy, c >, and let H B α with a.s. lim t + H s α ds = +. Then for all λ > λ α, where λ α is the unique solution of the equation λ H L Ω,L α [,t] λ log 1 + αλ c = 4c α, there exists x 1 α, λ > such that for all x x 1 α, λ and all t, s P sup H τ dz τ x st c α x α 1 λ log 1 + αλ c x + exp α Proof. sup st λ H L Ω,L α [,t] Following the proof of Corollary 3.3, we have by time change and scaling s x P H τ dz τ x P sup Ẑ s, s1 H L Ω,L α [,t] where Ẑ is a symmetric stable process defined on Ω and having the same law as Z. Finally, Proposition 4.1 applied to Ẑ achieves the proof. 17 α α

18 Remark 4.3 For all ɛ >, let x ɛ be the unique solution of the equation α c x α 1 λ log 1 + αλ c x = ɛ exp α λ H L Ω,L α [,t] Then for all x x ɛ, the inequality 4.4 implies s λ log 1 + αλ c P sup H τ dz τ x 1+ɛ exp st λ H L Ω,L α [,t] x λ H L Ω,L α [,t] α α 1. α α 1. α/α 1 Thus, the order of the upper bound in 4.4 is exp c α x/ H L Ω,L α [,t], and is comparable to that in [8, Theorem 1] for Lipschitz functions of stable random vectors. Remark 4.4 The quantity x 1 α, λ in Theorem 4. can be given explicitly. Indeed, let x α, λ > be the real number where the function F in 4.3 reaches its unique minimum, i.e. x α, λ α λ 1 α 1 = α 1 log αλ log 1 + αλ c < x α, λ α α 1, log 1 + αλ 4c c then choose x 1 α, λ = H L Ω,L α [,t] x α, λ. Remark 4.5 There is no optimal choice for the parameter λ in Theorem 4.: on the one hand, λ = λ α achieves the best maximal inequality 4.4 but in this case the domain for the deviation level x is empty; on the other hand, as λ increases, the domain expands but in this case the maximal inequality 4.4 is the worst. As an application of Theorem 4., let us recover the classical maximal inequality in the Gaussian case, cf. Proposition 1.8 p.55 in []. Corollary 4.6 Let B t t be a standard Brownian motion. Then the following maximal inequality holds P sup B s x exp x, st t x >, t. 18

19 Proof. Let X n n be a sequence of symmetric stable processes of index α n = 1/n and Lévy measure ν n dy = n 1 dy/ y αn+1. Applying Theorem 4. to X n, n, the inequality 4.4 becomes for all x x 1 α n, λ, all λ > λ α n and all t P sup Xs n st where x n 1 1 x n 1 λ log 1 + λ + exp n 1 λt n n 1 x 1 α, λ n 1 tλ n n 1 n 1 = n log 1 + λ log 1 λ log1 + λ, and λ α n is the unique solution of the equation λ log 1 + λ = n 1. x λt n n 1 n 1 n 1, Note that λ α n converges to and x 1 α n, λ to infinity as n goes to infinity. Denoting D[, + the Skorohod space of real-valued càdlàg functions on [, + equipped with the Skorohod topology, the sequence of processes X n n converges weakly in D[, + as n + to a standard Brownian motion B t t say, see e.g. Section 3 of Chapter VII in [15]. Since the supremum functional is continuous on D[, +, then the Continuous Mapping Theorem p. in [7] implies lim P sup Xs n x = P sup B s x, x >, t. n + st st Finally, letting n going to infinity and then λ to in the right-hand-side of 4.5 yield the result Estimates of first passage times of symmetric stable processes above positive continuous curves In [1, 18], the authors investigate functional transformations related to first crossing problems for self-similar diffusions. More precisely, they show via a time change transformation how the distribution of the first passage time of a Gauss-Markov process of Ornstein-Uhlenbeck type can be deduced from the law of the first crossing time of a 19

20 continuous curve by a Brownian motion. In this part, we adapt this method in order to estimate the first passage time of a symmetric stable process above several positive continuous curves, by using the maximal inequalities of Section and 3. To do so, let X φ be a stable-markov process of Ornstein-Uhlenbeck type of index α, and parameter φ, i.e. X φ has the integral representation X φ t := φt dz s, t [, T, T, + ], φs where Z is a symmetric stable process of index α and Lévy measure νdy = c y α 1 dy, c >, and φ is a positive C [, T -function. Let also T φ x := inf{t [, T : X φ t x} be its first exit time of the centered ball of radius x. Given a positive continuous function f such that f, define T f := inf{t : Z t ft} as the first passage time of Z above f. Let us give a first lemma which states an identity in law between first passage times: Lemma 5.1 Let X φ be a stable-markov process of Ornstein-Uhlenbeck type of index α, and parameter φ. Assume that τ t := ds < + for all t [, T and φs α that lim t T τ t = +. Denote by τ 1 the inverse of τ and let h φ,τ be the function defined on, + by h φ,τ t = 1/φ τ 1 t. identity in distribution Proof. Then for all x >, we have the P T φ x dr = P τ 1 T xh φ,τ dr, r [, T. By [1, Theorem 3.1], the process X φ rewrites as a time-changed symmetric stable process, i.e. we have a.s. X φ t = φtẑτ t, t [, T, where Ẑ is a symmetric stable process defined on the same probability space as Z and having the same distribution. Thus, we have for all r [, T P Tx φ r = P inf{t [, T : X φ t x} r

21 { = P inf t [, T : Ẑτ t = P τ 1 T xh φ,τ r. x } r φt Now, we establish via an integration by parts formula several maximal inequalities for stable-markov processes of Ornstein-Uhlenbeck type: Lemma 5. Let X φ be a stable-markov process of Ornstein-Uhlenbeck type of index α, and parameter φ. Let t [, T. Then we have the support estimate P sup Xs φ < y exp ct, y >. 5.1 st α α 1 y α If α, 1], then we have the maximal inequality P sup Xs φ x 4ct α 1 + st αx α φ φ τ φτ dτ, L [,t] x >, 5. whereas if α 1,, then for all x α > tc α α+1 α φ φ τ φτ dτ L [,t], we have P sup Xs φ x K ct st x α α 1 + φ φ τ φτ dτ, 5.3 L [,t] where K c > only depends on c. Proof. Fix t [, T and y >. If a.s. the path of the process X φ lies in the interval y, y up to time t, then there are no jumps of magnitude larger than y before time t, so that we have the set inclusion { sup st X φ s < y } { sup st X φ s < y }. Moreover, the process X φ has the same jumps as the process Z by definition. Thus, if T y 1 denotes the first jump time on the set {z R : z > y} of the Poisson process µ {z R : z > y} [, t] t [,T, then we have P sup Xs φ < y st P sup Xs φ < y st 1

22 = P sup Z s < y st P T y 1 > t = exp tν {z R : z y} = exp ct, αy α where in the second equality we used that T y 1 is exponentially distributed with parameter ν {z R : z > y}. The support estimate 5.1 is proved. Now, we establish 5. and 5.3. By the classical integration by parts formula for semimartingales, cf. [9, Proposition 8.11], we have dz s φs Hence, the process X φ rewrites as = Z t φt = Z t φt + Z s d 1 s φ φ sz s φs ds. X φ φ s t = Z t + φt φs Z s ds, t [, T. 5.4 Denote A t := φ φ τ dτ φτ and let us distinguish two cases: L [,t] if α, 1], then following the proof of inequality.11 but restricted to the symmetric stable process Z yields the inequality P sup Z s x 4ct st αx. α Thus, together with 5.4, we have P sup Xs φ x st P sup Z s x st 1 + A t 4ct1 + A t α ; αx α if α 1,, then Corollary 3.3 applied with e.g. p = 1 and ɛ = α 1/α+1, together with 5.4 show that there exists K c >, which only depends of c, such that P sup Xs φ x P sup Z s x st st 1 + A t K ct1 + A t α for all x α > tc1 + A t α / α α+1/α 1. x α

23 Remark 5.3 The support estimate 5.1 is independent of φ and thus is similar to that of a symmetric stable process. Remark 5.4 No time change techniques are required in the proof of Lemma 5. but just the integration by parts formula which entails 5.4. However, if we assume τ t := ds < +, t [, T, with τ φs α t + as t T and that φ is non-decreasing on [, T, then time change, scaling and Corollary 3.3 entail for sufficiently large x P sup Xs φ x st P sup Ẑτ s st P sup Ẑs s1 K c x α φtα Now, we are able to state the main result of this part: x φt x φtτ 1 α t ds φs α. Theorem 5.5 Let Z be a symmetric stable process of index α, and Lévy measure νdy = c y α 1 dy, c >. Let φ be a positive C [, T -function such that τ t := ds < + for all t [, T and that lim φs α t T τ t = +. Denote by τ 1 the inverse of τ and by h φ,τ the function defined on, + by h φ,τ t := 1/φ τ 1 t. Then for all x >, P T xh φ,τ > r exp cτ 1 r αx α If α, 1], then for all x >, we have P T xh φ,τ r 1 4cτ r 1 + αx α φ φ t φt dt, r >. 5.5 L [,τr 1 ] α, r >, 5.6 whereas if α 1,, then there exists K c >, which only depends of c, such that for all x > and for all r < r α, x, we have P T xh φ,τ r K cτr x α φ φ t φt dt where r α, x is the unique solution of the equation α α+1 α 1 x α = cτr φ φ t φt dt Proof. It is sufficient to apply Lemma 5.1 and Lemma L [,τr 1 ] L [,τr 1 ] α. α, 5.7

24 Thus, given φ, the quantity in the right-hand-side of the inequalities 5.5, 5.6 and 5.7 can be computed explicitly. Let us give two applications of Theorem 5.5. If φt := e λt for λ > and T = +, then X φ is the stable Ornstein-Uhlenbeck process of index α. Therefore, a direct computation in Theorem 5.5 implies the Corollary 5.6 Let Z be a symmetric stable process of index α, and Lévy measure νdy = c y α 1 dy, c >. Letting f α,x,λ t := x1 + λαt 1/α, t, λ >, we have for all x > P inf{t : Z t f α,x,λ t} > r 1 c, r >. 1 + λαr λα α 1 x α If α, 1], then for all x > and all r >, P inf{t : Z t f α,x,λ t} r 4c 1 + λαr 1 α α log1 + λαr λα x α 16cr αx. α Finally, if α 1,, then for all x > and for all r < r α, x, λ, we have the estimate P inf{t : Z t f α,x,λ t} r 1 + λαr 1 α α log1 + λαrk c λαx α 4rK c x, α where K c is the constant of Theorem 5.5 and r α, x, λ is the unique solution of the equation λαx α = c 1 + λαr 1 α α log1 + λαr. α α+1 α 1 Now, we present the case of the stable bridge. Given a symmetric stable process Z = Z t t of index α,, there exists a Markov process X br = X br t tt starting from and ending in at a finite time horizon T, such that its distribution Q is given by dq F t = p T t X t dp F p T t, t, T, where p t is a version everywhere positive of the distribution of the stable random variable Z t, see [6, Chapter VIII]. The process X br is called a stable bridge. By e.g. 4

25 Exercise 1. in [3], X br is the unique solution of the linear equation X br t = Z t X s br ds, t, T, T s which rewrites by the integration by parts formula of Proposition 8.11 in [9] as X br t = T t dz s ds, t, T. T s Hence, the stable bridge X br is a stable-markov process of Ornstein-Uhlenbeck type with parameter φ given by φt = T t, t [, T ]. Thus, using Theorem 5.5, we get the Corollary 5.7 Let Z be a symmetric stable process of index α 1, and Lévy measure νdy = c y α 1 dy, c >. Letting g α,x,t t := xt 1 α + α 1t 1/α 1, t, we have for all x > and all r > c P inf{t : Z t g α,x,λ t} > r exp α α 1 x α = exp ct g α,x,t r x α α 1 g α,x,t rx α whereas for all x > and for all r < r α, x, T, we have T T 1 α + α 1r 1 1 α P inf{t : Z t g α,x,t t} r K ct g α,x,t r xt g α,x,t r x α T α g α,x,t r α+1 x α 4T K c x α, where K c is the constant of Theorem 5.5 and r α, x, λ is the unique solution of the equation α α+1 α 1 T α g α,x,t r α+1 x α = ct g α,x,t r xt g α,x,t r x α. Remark 5.8 In the latter corollary, only the case α 1, is considered, since the time change techniques we use in the proof of Theorem 5.5 are not satisfied when α, 1. Acknowledgments The author would like to thank N. Privault and J.C. Breton for their encouragements and helpful comments during the preparation of this paper. 5,

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