Tanaka formula for symmetric Lévy processes

Transcription

1 Tanaka formula for symmeric Lévy processes Paavo Salminen 1 and Marc Yor 2 1 Åbo Akademi Universiy, Mahemaical Deparmen, FIN-25 Åbo, Finland phsalmin@abo.fi 2 Universié Pierre e Marie Curie, Laboraoire de Probabiliés e Modèles aléaoires, 4, Place Jussieu, Case 188, F Paris Cedex 5, France Summary. Saring from he poenial heoreic definiion of he local imes of a Markov process when hese exis we obain a Tanaka formula for he local imes of symmeric Lévy processes. The mos ineresing case is ha of he symmeric α-sable Lévy process (for α (1, 2] which is sudied in deail. In paricular, we deermine which powers of such a process are semimaringales. These resuls complee, in a sense, he works by K. Yamada [19] and Fizsimmons and Geoor [8]. Keywor: resolven, local ime, sable Lévy process, addiive funcional. AMS Classificaion: 6J65, 6J6, 6J7. 1 Inroducion and main resuls I is well known ha here are differen consrucions and definiions of local imes corresponding o differen classes of sochasic processes. For a large panorama of such definiions, see Geman and Horowiz [12]. The mos common definiion of he local imes L = {L x : x, } of a given process {X : } is as he adon Nikodym derivaive of he occupaion measure of X wih respec o he Lebesgue measure in ; precisely L saisfies f(x s = f(xl x dx (1 for every Borel funcion f : +. There is also he well known sochasic calculus approach developed by Meyer [16] in which one works wih a general semimaringale {X : }, and defines Λ = {λ x : x, } wih respec o he Lebesgue measure from he formula f(x s d <X c > s = f(xλ x dx. (2

2 2 Paavo Salminen and Marc Yor Of course, in he paricular case when d <X c > s =, i.e., X c is a Brownian moion, hen he definiions of L and λ coincide. In oher cases, e.g., if X c, hey will differ. In his paper we focus on he poenial heoreic approach applicable in he Markovian case in which he local imes are defined as addiive funcionals whose p-poenials are equal o p-resolven kernels of X. Local imes can hereby be inerpreed as he increasing processes in he Doob-Meyer decomposiions of cerain submaringales. Considering he p-resolven kernels and passing o he limi, in an adequae manner, as p, we obain a formula (3, which clearly exen Tanaka s original formula for he local imes of Brownian moion o hose of he symmeric α-sable processes, α (1, 2], already obained by T. Yamada [2] and furher developed in K. Yamada [19]. Our approach may be simpler and may help o make hese resuls beer known o probabiliss working wih Lévy processes. The formula (3 below and is counerpars abou decomposiions of powers of symmeric α-sable Lévy processes show a he same ime similariies and differences wih he well known formulae for Brownian moion (see, in paricular, Chaper 1 in [23] concerning he principal values of Brownian local imes. We hope ha he Tanaka represenaion of he local imes in (3 may be useful o gain some beer undersanding for he ay-knigh heorems of he local imes of X as presened in Eisenbaum e al. [6], since in he Brownian case, Tanaka s formula has been such a powerful ool for his purpose, see, e.g., Jeulin [15]. We now sae he main formulae and resuls for he symmeric α-sable Lévy process X = {X }. To be precise, we ake X o saisfy E (exp(iλx = exp( λ α, λ, in paricular, for α = 2, X equals 2 imes a sandard BM. General crieria can be applied o verify ha X possesses a joinly coninuous family of local imes {L x } saisfying (1. The consans c i appearing below and laer in he paper will be compued precisely in Secion 5; clearly, hey depend on he index α and/or he exponen γ. 1 For all and x X x α 1 = x α 1 + N x + c 1 L x, (3 where N x is a maringale such ha for γ < α/(α 1, especially for γ = 2, ( E sup Ns x γ <. (4 s Moreover, he coninuous increasing process associaed wih N x is <N x > := c 2. (5 X s x 2 α

3 Tanaka formula for symmeric Lévy processes 3 2 For α 1 < γ < α he submaringale { X x γ } has he decomposiion X x γ = x γ + N (γ + A (γ, (6 where N (γ is a maringale and A (γ is he increasing process given by A (γ := c 3. (7 X s x α γ 3 For < γ < α 1 he process { X x γ } is no a semimaringale bu for (α 1/2 < γ < α 1 i is a Dirichle process wih he canonical decomposiion X x γ = x γ + N (γ + A (γ, (8 where N (γ is a maringale and A (γ, which has zero quadraic variaion, is given by he principal value inegral A (γ := c 4 p.v. X s x α γ := c dz ( 4 L x+z z α γ L x z. (9 The paper is organized so ha in Secion 2 some preliminaries abou symmeric Lévy processes including heir generaors and some varians of he Iô formula are presened. In Secion 3 we derive he Tanaka formula for general symmeric Lévy processes admiing local imes. The above saed resuls for symmeric sable Lévy processes are proved and exended in Secion 4. In Secion 5 we compue explicily he consans c i feaured above and also furher ones appearing especially in Secion 4. This is done by exhibiing some close relaions beween hese consans and he known expressions of he momens E( X 1 γ where X 1 denoes a sandard symmeric α-sable variable. In Secion 6, we consider, insead of X x γ, he process {(X x γ, }, where a γ, := sgn(a a γ, is he symmeric power of order γ, and we deermine he parameer values for which hese processes are semimaringales or Dirichle processes, hus compleing resuls 1, 2 and 3 above. 2 Preliminaries on symmeric Lévy processes Throughou his paper, we consider a real-valued symmeric Lévy process X = {X } and, if nohing else is saed, we assume X =. The Lévy exponen Ψ of X is a non-negaive symmeric funcion such ha

4 4 Paavo Salminen and Marc Yor E (exp(iξx = E (cos(ξx = exp ( Ψ(ξ. (1 The Lévy measure ν of X saisfies, as is well known, he inegrabiliy condiion (1 z 2 ν(dz <. By symmery, ν(a = ν( A for any A B, he Borel σ-field on ; hence, Ψ(ξ = 1 2 σ2 ξ 2 ( e i ξ z 1 i ξ z 1 { z 1} ν(dz = 1 2 σ2 ξ (1 cos(ξz ν(dz. (11 ecall also (see, e.g., Ikeda and Waanabe [14] p. 65 ha X admis he Brownian-Poisson represenaion X = σ B + z Π(, dz + z (Π π(, dz, (12 (,] { z 1} (,] { z <1} where he Brownian moion B and he Poisson random measure Π wih he inensiy π(, dz := E(Π(, dz = ν(dz are independen. Due o he symmery of ν, he generaor of X can be wrien as Gf(x := G B f(x + G Π f(x := 1 2 σ2 f ( (x + f(x + y f(x f (x y 1 { y <1} ν(dy = 1 2 σ2 f (x + (f(x + y f(x f (x y ν(dy. (13 where G acs on regular funcions f in paricular hose in he Schwarz space S( of rapidly decreasing funcions. Given a smooh funcion f, he predicable form of he Iô formula (see Ikeda and Waanabe [14] and K. Yamada [19] wries f(x f(x = σ Gf(X s (14 f (X s db s + (f(x s + z f(x s (Π π(, dz. The formula (14 connecs wih he Iô formula for semimaringales, as developed by Meyer [16], and displayed as

5 Tanaka formula for symmeric Lévy processes 5 f(x = f(x + f (X s dx s + σ2 f (X s 2 + (f(x s f(x s f (X s X s. (15 The sum of jumps <s (... may be compensaed by <s hence, we have recovered he inegraed form of (13: Gf(X s = G B f(x s + G Π f(x s. G Π f(x s, and, We record also a more general compensaor formula employed laer in he paper. For his, le Φ : + be a Borel measurable funcion. Then E Φ(X s, X s 1 { Xs } <s ( = E \{} π(, dzφ(x s, X s + z. (16 3 Local imes for symmeric Lévy processes From now on, we assume ha 1 dξ <. ( Ψ(ξ From sandard Fourier argumens (see Beroin [1] and, e.g., Borodin and Ibragimov [2] p. 67 one can show he exisence of a joinly measurable family of local imes {L x : x, } saisfying for every Borel-measurable funcion f : + he occupaion ime formula f(x s = f(xl x dx. For he condiion (expressed in erms of he funcion v in (22 under which (, x L x is coninuous, see Beroin [1] p In paricular, he condiion hol for symmeric α-sable Lévy processes; in fac i was shown by Boylan [3], see also Geoor and Kesen [13], ha L x+y L x K y θ (18 for any θ < (α 1/2 and some random consan K.

6 6 Paavo Salminen and Marc Yor Our approach oward a Tanaka formula for hese local imes is based on he poenial heoreic consrucion which we now develop. I is well known, see Beroin [1] p. 67, ha for any p > u (p (x = 1 π cos(ξx dξ (19 p + Ψ(ξ is a coninuous version of he densiy of he resolven ( U (p (, dx = E e p 1 {X dx} d. Moreover, for every x he local ime {L x } can be chosen as a coninuous addiive funcional such ha ( u (p (y x = E y e p d L x. (2 From (2 we deduce he Doob-Meyer decomposiion given in he nex Proposiion 1. For every fixed x u (p (X x = u (p (X x + M (p,x + p u (p (X s x L x, (21 where M (p,x is a maringale wih respec o he naural filraion {F } of X. Moreover, for every fixed, boh he maringale {M s (p,x : s } and he random variable L x belong o BMO; in paricular, L x has some exponenial momens. Proof. Sraighforward compuaions using he Markov propery show ha for y = X ( s E y e p d L x F s = e p d L x + e p s u (p (X s x, which ogeher wih an inegraion by pars yiel (21. We leave he proofs of he remaining asserions o he reader. A varian of he Tanaka formula shall now be obained by leing p in (21. The resul is saed in Proposiion 2 bu firs we need an imporan ingredien. Lemma 1. For every x ( lim u (p ( u (p (x = 1 p π 1 cos(ξx Ψ(ξ dξ =: v(x. (22

7 1 Tanaka formula for symmeric Lévy processes 7 Proof. The saemen follows from (19 by dominaed convergence because (cf. ( dξ <, Ψ(ξ and ξ 2 dξ <. Ψ(ξ Noice also ha v is coninuous. The formula (23 below generalizes in a sense he Tanaka formula for Brownian moion o symmeric Lévy processes. In he nex secion we sudy he paricular case of symmeric sable processes. Proposiion 2. Le v be he funcion inroduced in (22 and M (p,x he maringale defined in Proposiion 1. Then where Ñ x := lim p M (p,x v(x x = v(x + Ñ x + L x, (23 defines a maringale. emark 1. Sandard resuls abou maringale addiive funcionals of X yield he following represenaions Ñ x = σ v (X s x db s + (v(x s x + z v(x s x (Π π(, dz, (,] and < Ñ x > = σ 2 (v (X s x 2 + where v is a weak derivaive of v. (v(x s x + z v(x s x 2 π(, dz, Proof. Consider he ideniy (21. Le herein p and use Lemma 1 o obain ( v(x x = v(x lim M (p,x + p u (p (X s x + L x. (24 p From (19 u (p (y u (p (, and, consequenly, Nex we show ha p u (p (X s x p u (p (. (25 lim p p u(p ( =. (26

8 8 Paavo Salminen and Marc Yor Indeed, using (19 again, p u (p ( = 1 π 1 π 1 p dξ p + Ψ(ξ p dξ p + Ψ(ξ + p π 1 dξ Ψ(ξ, (27 and (26 resuls by dominaed convergence. Hence, (24 yiel (23 wih Ñ x as claimed. I remains o prove ha Ñ x is a maringale. For his i is enough o show ha ( E Ñ x M (p,x as p. (28 To prove (28 consider Ñ x M (p,x p u (p (X s x + v(x (u (p ( u (p (x + v(x x (u (p ( u (p (X x. From (25 and (26, he inegral erm goes o as p. Nex, by Fubini s heorem and (1 ( v(x E x (u (p ( u (p (X x = 1 π p 1 E(cos(ξ(X x dξ Ψ(ξ(p + Ψ(ξ = 1 π p 1 cos(ξ x exp( Ψ(ξ dξ Ψ(ξ(p + Ψ(ξ 1 ( π p 1 cos(ξ x Ψ(ξ(p + Ψ(ξ dξ + Ψ(ξ Ψ(ξ(p + Ψ(ξ dξ. Applying he dominaed convergence heorem for he firs erm above and (27 for he second one give ( lim E v(x x (u (p ( u (p (X x =, p compleing he proof. Example 1. For sandard Brownian moion B we have Consequenly, u (p (x = 1 2p e 2p x. ( v(x := lim u (p ( u (p (x = x. p and he formula (23 akes he familiar form

9 Tanaka formula for symmeric Lévy processes 9 where N x = lim = p B x = x + N x + Lx e 2 p B s x sgn(b s x db s sgn(b s x db s. 4 Symmeric α-sable Lévy processes Le X = {X }, X =, denoe he symmeric α-sable process wih he Lévy exponen Ψ(ξ = ξ α, α (1, 2. We remark ha he condiion (17 is saisfied, and also ha he local ime of X has a joinly coninuous version, as is discussed in Secion 3. For clariy, we have excluded he Brownian moion from our sudy. However, he corresponding resuls for Brownian moion may be recovered by leing α 2. ecall also ha E( X γ < for γ < α, and ha he Lévy measure is ν(dz = c 5 (α z α 1 dz, α (1, 2. (29 The funcion v inroduced in Lemma 1 is in he presen case given by v(x = c 6 (α x α 1. (3 The resuls announced in he Inroducion are now presened again and proven in a more complee form hrough he following hree proposiions. The firs one reas he claim 1 in he Inroducion. Proposiion 3. a For fixed x c 6 (α ( X x α 1 x α 1 = Ñ x + L x (31 where {Ñ x } is a square inegrable maringale. In fac, for all γ < α/(α 1, especially for γ = 2, ( E sup Ñ s x γ <. (32 s Moreover, he coninuous increasing process associaed wih Ñ x is <Ñ x > := c 7 (α. (33 X s x 2 α b For every and x he variable L x belongs o BMO; in fac, for all s E (L x L x s F s K α, (34 for some consan K α, which does no depend on s.

10 1 Paavo Salminen and Marc Yor Proof. The fac ha Ñ x is a maringale is clear from Proposiion 2. Because L x has some exponenial momens (cf. Proposiion 1, i is seen easily from(31 ha for γ > E( Ñ x γ < if ( E X x γ(α 1 <, which is rue for γ(α 1 < α. Consequenly, an exension of he Doob- Kolmogorov inequaliy, gives (32. The maringale Ñ x has no coninuous maringale par. Hence, leing [Ñ x ] := s ( Ñ x s 2 := (c 6 (α 2 s ( Xs x α 1 X s x α 1 2 i hol ha { (Ñ x 2 [Ñ x ] } is a maringale. Consequenly, <Ñ x > can be obained as he dual predicable projecion of [Ñ x ], and from he Lévy sysem of X, e.g., (16, we ge <Ñ x > = (c 6 (α 2 c 5 (α dy ( Xs y α+1 x + y α 1 X s x α 1 2. Puing z = X s x and inroducing y = zu he laer inegral akes he form dy ( z + y α 1 y α+1 z α = z 2 α du u α+1 ( 1 + u α Consequenly, <Ñ x > is as claimed. To prove he second par of he proposiion, noice ha by he maringale propery E (L x L x s F s = c 6 (α E ( X x α 1 X s x α 1 F s c 6 (α E ( X X s α 1 F s c 6 (α E ( X s α 1 K α (α 1/α, where also he scaling propery and he inequaliy are used. x p y p x y p, < p 1, The following corollary plays he same rôle for X as he classical Iô- Tanaka formula plays for Brownian moion. In fac, a large par of his paper discusses for which funcions he ideniy (35, or some varian of i is valid.

11 Tanaka formula for symmeric Lévy processes 11 Corollary 1. Le f be a bounded Borel funcion wih compac suppor and define F (y := dxf(x y x α 1. Then F (X = F ( + dxf(x N x + c 1 (α f(x s (35 expresses he canonical semimaringale decomposiion of {F (X } wih { dxf(x N x } a maringale. Proof. I suffices o inegrae boh sides of (31 (or raher (3 wih respec o he measure f(x dx. emark 2. a In K. Yamada [19] he represenaion (31 of he local ime (or Tanaka s formula for symmeric α-sable processes is derived using he so called mollifier approach as in Ikeda and Waanabe [14] in he Brownian moion case. In his case he maringale is given by ( = c 6 (α Xs x + z α 1 X s x α 1 (Π π(, dz, Ñ x (,] where Π and π are he Poisson random measure and he corresponding inensiy measure, respecively, associaed wih X. b The inequaliy (34 hol for all symmeric Lévy processes having local imes. Indeed, i is proved in Beroin [1] p. 147 Corollary 14 ha he funcion v defined in (22, Lemma 1, induces a meric on, and, in paricular, he riangle inequaliy hol. Consequenly, E (L x L x s F s E(v(X X s = E(v(X s E(v(X < because E(v(X = 1 π 1 π 1 exp( Ψ(ξ dξ Ψ(ξ ( dξ + <. Ψ(ξ 1 c We leave i o he reader o esablish a version of Corollary 1 for general symmeric Lévy processes. Proposiion 4. For a given x and α 1 < γ < α he submaringale { X x γ : } has he decomposiion X x γ = x γ + N (γ + A (γ, (36 where N (γ is a maringale and A (γ is he increasing process given by

12 12 Paavo Salminen and Marc Yor A (γ = c 3 (α, γ. (37 X s x α γ Moreover, when α 1 γ α/2 he increasing process <N (γ > is of he form <N (γ > = c 8 (α, γ. (38 X s x α 2γ Proof. Formula (36 is obained by inegraing boh sides of equaion (31 (or (3 aken a level z wih respec o he measure dz/ z x α γ. The form of he lef hand side is obained from he scaling argumen. Because A (γ is coninuous he compuaion for finding < N (γ > is very similar o he compuaion of <Ñ x > in he proof of Proposiion 3. We have <N (γ > = which easily yiel (38. ν(dy ( X s x + y γ X s x γ 2, (39 For he nex proposiion, we recall he noion of Dirichle process, ha is a process which can decomposed uniquely as he sum of a local maringale and a coninuous process wih zero quadraic variaion (see, e.g., Föllmer [1], Fukushima [11]. Proposiion 5. a For < γ < α 1 he process X x γ is no a semimaringale. b For (α 1/2 < γ < α 1 he process X x γ is a Dirichle process wih he canonical decomposiion X x γ = x γ + N (γ + A (γ, (4 where N (γ is a maringale and A (γ is given by he principal value inegral A (γ = c 4 (α, γ p.v. = c 4 (α, γ X s x α γ dz ( L x+z z α γ Moreover, he increasing process <N (γ > is as given in (38. L x z. (41 Proof. a We ake x = and adap he argumen in Yor [21] applied herein for coninuous maringales. Assume ha Y := X γ, γ < α 1, defines a semimaringale. Then X α 1 = Y θ wih θ = γ/(α 1 > 1, and Iô s formula for semimaringales (noice ha Y c gives

13 where Σ := Tanaka formula for symmeric Lévy processes 13 Y θ = θ Ys θ 1 dy s + Σ, (42 <s ( Y θ s Y θ s θ Y θ 1 s Y s. The argumen of he proof is ha under he above assumpion he local ime L 1 {Xs =} d X s α 1 (43 would be equal o zero. To derive his conradicion noice from (42 and (43 ha L = 1 {Ys =} dy θ s = 1 {Ys =} dσ s. Bu because Σ is a purely disconinuous increasing process and L is coninuous his is possible only if L, which canno be he case; hus proving ha Y is no a semimaringale. b To prove (4 we consider formula (3 a levels x + z and x z and wrie dz ( X z α γ (x + z α 1 X (x z α 1 (44 = dz ( N x+z z α γ N x z + c1 (α dz ( L x+z z α γ The inegral on he lef hand side is well defined since by scaling dz ( X z α γ (x + z α 1 X (x z α 1 = X x γ r(α, γ wih r(α, γ := dz z α γ ( 1 z α z α 1, L x z. which is an absoluely convergen inegral. Nex noice ha he principal value inegral on he righ hand side of (44 is well defined by he Hölder coninuiy in x of he local imes (cf. (18. I also follows ha he firs inegral on he righ hand side of (44 is meaningful and, by Fubini s heorem, i is a maringale. In Fizsimmons and Geoor [8] i is proved ha H := dz z α γ ( L z L. has zero p-variaion for p > p o := (α 1/γ (noice 1 + γ in [8] correspon ours α γ. Since p o < 2 i is now easily seen ha also {A (γ } has zero quadraic variaion and he claimed Dirichle process decomposiion follows wih c 4 (α, γ = c 1 (α/r(α, γ. (45

14 14 Paavo Salminen and Marc Yor 5 Explici values of he consans An imporan ingredien in he compuaion of he explici values of he consans is he formula for absolue momens of symmeric α-sable, α (1, 2, random variables due o Shanbhag and Sreehari [18] (see also Sao [17] p. 163, Chaumon and Yor [4] p. 11. To discuss his briefly le Z be an exponenially disribued r.v. wih mean 1, U a normally disribued r.v. wih mean and variance 1, X (α a symmeric α-sable r.v. wih characerisic funcion exp( ξ α, Y (α/2 a posiive α/2-sable r.v. wih Laplace ransform exp( ξ α/2. Assume also ha hese variables are independen. Then i is easily checked ha (Z/Y (α/2 α/2 d = Z (46 and X (α d = 2 U (Y (α/2 1/2. (47 From (46 we obain for γ < α/2 E ((Y (α/2 γ = Γ (1 2γ α Γ (1 γ, and, furher, from (47 for 1 < γ < α ( m γ := E X (α γ = 2 γ Γ ( 1 + γ 2 Γ ( α γ α / ( π 2 γ Γ (. (48 2 The consans wih he associaed reference numbers of he formulae where hey appear in he paper are summarized in he following able.

15 Tanaka formula for symmeric Lévy processes 15 Consan Value ef. c 1 (α ((α 1 π m α 1 /Γ (1/α (3, (56, (57 c 2 (α (2(α 1 m 2(α 1 /(α m α 2 (5 c 3 (α, γ (γ m γ /(α m γ α (7, (37 c 4 (α, γ c 1 (α/r(α, γ (41, (45 c 5 (α α/(2 Γ (1 α cos(απ/2 (29 c 6 (α (c 1 (α 1 = (2πc 5 (α 1 1 (3 c 7 (α c 2 (α (c 6 (α 2 (33 c 8 (α, γ c 3 (α, 2γ 2c 3 (α, γ (38 We consider firs he consan c 3 (α, γ and, for clariy, recall formula (36: wih α 1 < γ < α and X x γ = x γ + N (γ A (γ = c 3 (α, γ Noice ha leing γ α 1 yiel, in a sense, + A (γ, (49 X s x α γ. A (α 1 = c 1 (α L x, (5 alhough, using he value in he able, c 3 (α, γ. From (49 i is seen ha f(y = y x γ belongs o he domain of he exended generaor G, and, by scaling we obain he following inegral represenaion c 3 (α, γ = ν(dy ( 1 + y γ 1 γy. On he oher hand, aking x = in (49, and using scaling again ogeher wih (48, we ge

16 16 Paavo Salminen and Marc Yor which is equivalen wih hence, E ( X γ = c 3 (α, γ γ/α m γ = c 3 (α, γ α γ/α γ E ( X s γ α, c 3 (α, γ = γ m γ /α m γ α. m γ α A similar argumen lea o an expression for c 1 (α. From (5 we ge E ( X α 1 = c 1 (α E ( L. (51 We derive from (51 he exisence of a consan c (α such ha and i follows from (5 ha E ( d L = c (αd 1/α, m α 1 = α c 1 (α c (α/(α 1. (52 We now compue c (α o obain c 1 (α from (52. For his consider he ideniy (2 for x = y = ( u (p ( = E e p s d s L s, which in erms of c (α rea 1 π dξ p + ξ α = c (α An elemenary compuaion reveals ha hence, c (α = 1 Γ ((α + 1/α, π e p s s 1/α. c 1 (α = ((α 1 π m α 1 /Γ (1/α. Nex we find from formula (31 ha c 6 (α = 1/c 1 (α. (53 To compue c 8 (α, γ for α 1 γ α/2 and he limiing case c 2 (α = c 8 (α, α 1 noice from (39 ha c 8 (α, γ = ν(dy ( 1 + y γ 1 2.

17 Tanaka formula for symmeric Lévy processes 17 Comparing he inegral represenaions of c 3 and c 8 i is seen ha 2c 3 (α, γ + c 8 (α, γ = c 3 (α, 2γ (54 which can also be deduced from he following formulae (i E ( X 2 γ ( = 2 E X s γ d s A (γ s + E (<N (γ > ( = (2c 3 (α, γ + c 8 (α, γ E (ii E ( X 2 γ ( = c 3 (α, 2γ E X s α 2γ. X s α 2γ, The firs one of hese is an easy applicaion of he Iô formula for semimaringales and he second one follows (49 because γ α/2. From equaion (54 we ge c 8 (α, γ = c 3 (α, 2γ 2c 3 (α, γ = 2γ ( m2γ α m 2γ α m γ m γ α The consan c 2 is now obained by leing here γ α 1 and using m 1 = +. Consequenly 2(α 1 m 2(α 1 c 2 (α =. α m α 2 To find he consan c 5 (α, we use he relaionship (11 beween Ψ and ν which yiel afer subsiuion y = ξz c 5 (α = ( 1 1 cos y 2 y α+1 dy. Inegraing by pars and using he formulae 2.3.(1 p. 68 in Erdelyi e al. [7] lead us o he explici value of he inegral. 1 cos y Γ (1 α y α+1 dy = cos(απ/2. α The consan c 6 (α can also clearly be expressed in erms of c 5 c 6 (α = (2πc 5 (α 1 1 = 1 π = 1 π Γ (2 α α 1 cos((α 1π/2. 1 cos ξ ξ α I can be verified by he duplicaion formula for he Gamma funcion ha his agrees wih (53. I hol also ha c 6 (α 1/2 as α 2. dξ

18 18 Paavo Salminen and Marc Yor The consan c 7 is obained by simply comparing he definiions of N x in (3 and Ñ x in Proposiion 3. We have N x = 1 c 6 (α Ñ x implying c 7 (α = c 2 (α (c 6 (α 2. 6 Symmeric principal values of local imes Our previous resuls may be summarized as follows 1. for α 1 γ < α he process { X x γ } is a submaringale whose Doob-Meyer decomposiion is given by (36, 2. for (α 1/2 < γ < α 1 he process { X x γ } is a Dirichle process whose canonical decomposiion is given by (4. These resuls do no discuss wheher {(X x γ, }, he symmeric power of order γ, i.e., (X x γ, := sgn(x x X x γ, (55 is or is no a semimaringale or a Dirichle process. In he presen secion i is seen ha his quesion can be answered compleely relying on some resuls in Fizsimmons and Geoor [8] and [9], see also K. Yamada [19]. Le x = in (55 and inroduce he principal value inegral (cf. (9 p.v. X θ, s := dz ( L z z θ L z, where by he Hölder coninuiy (18 he inegral is well defined for θ < (α 1/2. Proposiion 6. a For α 1 < γ < α he process {X γ, } is a semimaringale. b For (α 1/2 < γ α 1 he process {X γ, } is a Dirichle process and no a semimaringale. c In boh cases he unique canonical decomposiion of he process can be wrien as where and X γ, r + (α, γ = N γ, + c 1 (α p.v. r + (α, γ = X α γ, s dx x α γ ( 1 x α 1 (1 + x α 1 N γ, dx ( = N x x α γ N x., (56

19 In paricular, for γ = α 1 Tanaka formula for symmeric Lévy processes 19 X α 1, r + (α, 1 = N α 1, + c 1 (α p.v.. (57 X s Proof. Because {X } is a maringale, i follows from he Io formula for semimaringales (15 ha for 1 γ α he process {X γ, } is a semimaringale. The oher saemens in a and b are derived from he decomposiion (56 which we now verify similarly as (4 in Proposiion 5. Hence, we sar again from he ideniy (3 considered a x and x, and wrie, informally dx ( X x α γ x α 1 X + x α 1 (58 = dx x α γ ( N x N x + c1 (α To analyze he inegral on he lef hand side noice ha (a; α, γ = is absoluely convergen and dx ( L x x α γ L x, dx x α γ ( a x α 1 a + x α 1. (a; α, γ = a γ, r + (α, γ. Now he res of he proof is very similar o ha of Proposiion 5 b, and is herefore omied. emark 3. a The increasing process associaed wih N γ, is given by <N γ, > = (r + (α, γ 2 = (r + (α, γ 2 ν(dz ( (X s + z α γ, Xs α γ, 2 X s α 2γ ν(dz ( (1 + z α γ, 1 2. We also have by scaling ( E X s α 2γ = s (2γ α/α E ( X 1 2γ α = α 2γ 2γ/α E ( X 1 2γ α. b Since and X γ = (X + γ + (X γ X γ, = (X + γ (X γ

20 2 Paavo Salminen and Marc Yor i is sraighforward o derive he decomposiion formulae for {(X + γ } and {(X γ }, and we leave his o he reader. c Noe how differen (57 is in he Brownian case α = 2, for which on one hand {B } is a maringale, and on he oher hand ϕ(b = log B s db s p.v. wih ϕ(x = x log x x. For principal values of Brownian moion and exensions of Iô s formula, see Yor [22], [23] and Cherny [5]. B s eferences 1. J. Beroin. Lévy Processes. Cambridge Universiy Press, Cambridge, UK, A.N. Borodin and I.A. Ibragimov. Limi heorems for funcionals of random walks. Proc. Seklov Ins. Mah., 195(2:576 59, English ransl. AMS E.S. Boylan. Local imes for a class of Markov processes. Illinois J. Mah., 8:19 39, L. Chaumon and M. Yor. Exercises in probabiliy; a guided our from measure heory o random processes via condiioning. Cambridge Univ. Press, Cambridge, A.S. Cherny. Principal values of he inegral funcionals of Brownian moion: exisence, coninuiy, and an exension of Iô s formula. In J. Azéma, M Émery, M. Ledoux, and M. Yor, ediors, Séminaire de Probabiliés XXXV, number 1755 in Springer Lecure Noes in Mahemaics, pages , Berlin, Heidelberg, New York, N. Eisenbaum, H. Kaspi, M.B. Marcus, J. osen, and Z. Shi. A ay-knigh heorem for symmeric Markov processes. Ann. Probab., 28: , A. Erdélyi, W. Magnus, F. Oberheinger, and F.G. Tricomi. Tables of Inegral Transforms. McGraw-Hill, New York, P. Fizsimmons and.k. Geoor. Limi heorems and variaion properies for fracional derivaives of he local ime of a sable process. Annales de l I.H.P., 28: , P. Fizsimmons and.k. Geoor. On he disribuion of he Hilber ransform of he local ime of a symmeric Lévy process. Ann. Probab., 2: , H. Föllmer. Dirichle processes. In D. Williams, edior, Sochasic Inegrals, volume 851 of Springer Lecure Noes in Mahemaics, pages , Berlin, Heidelberg, Springer Verlag. 11. M. Fukushima. Dirichle forms and Markov processes. Norh-Holland and Kodansha LTD, Amserdam, Tokyo, D. Geman and J. Horowiz. Occupaion densiies. Ann. Probab., 8:1 67, K. Geoor and H. Kesen. Coninuiy of local imes for Markov processes. Composiio Mah., 24:277 33, N. Ikeda and S. Waanabe. Sochasic Differenial Equaions and Diffusion Processes, 2nd ediion. Norh-Holland and Kodansha, Amserdam, Tokyo, 1989.

21 Tanaka formula for symmeric Lévy processes T. Jeulin. ay-knigh s heorem on Brownian local imes and Tanaka s formula. In E. Cinlar, K.L. Chung, and.k. Geoor, ediors, Seminar on Sochasic Processes, 1983, Boson, Basel, Birkhauser Verlag. 16. P.A. Meyer. Un Cours sur les Inégrales Sochasiques. In C. Dellacherie and P.A. Meyer, ediors, Séminaire de Probabiliés X, number 511 in Springer Lecure Noes in Mahemaics, Berlin, Heidelberg, New York, K. Sao. Lévy processes and infiniely divisible disribuions. Cambridge Press, Cambridge, D.N. Shanbhag and M. Sreehari. On cerain self-decomposable disribuions. Z. Wahrscheinlichkeisheorie verw. Gebiee, 38: , K. Yamada. Fracional derivaives of local imes of α-sable Levy processes as he limis of occupaion ime problems. In I. Berkes, E Csáki and M. Csörgö, ediors, Limi Theorems in Probabiliy and Saisics II, J. Bolyai Sociey Publicaions, pages , Budapes, T. Yamada. Tanaka formula for symmeric sable processes of index α, 1 < α < 2. Unpublished manuscrip, M. Yor. Un exemple de processus qui n es pas une semimaringale. Asérisque; Temps locaux, 52 53: , M. Yor. Sur la ransformée de Hilber des emps locaux browniens e une exemsion de la formule d Iô. In J. Azéma and M. Yor, ediors, Séminaire de Probabiliés XVI, number 92 in Springer Lecure Noes in Mahemaics, pages , Berlin, Heidelberg, New York, M. Yor. Some Aspecs of Brownian Moion. Par II: Some ecen Maringale Problems. Birkhäuser Verlag, Basel, 1997.