MICHAEL R. TEHRANCHI. e Ws cs ds has a density f given by 22c. e 2/x f(x) = , for x > 0.

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1 THE DISTRIBUTION OF EXPONENTIAL LÉVY FUNCTIONALS MICHAEL R. TEHRANCHI Abstract. The dstrbuto of radom varables of the form R eξ s dη s s cosdered for a Lévy process (ξ, η. I partcular, we gve a smple proof of a result of Carmoa, Pett, ad Yor [3] that the dstrbuto fucto verfes a tegro-dfferetal equato related to the characterstc trple of (ξ, η. Let W be a stadard Weer process ad c a postve costat. The purpose of ths ote s to offer a short, self-cotaed proof of the followg result: Theorem (Dufrese [5]. The radom varable e Ws cs ds has a desty f gve by 22c e 2/x f(x =, for x >. Γ(2c x+2c More geerally, let (ξ,..., ξ, η,..., η be a 2-dmesoal Lévy processes. We am to descrbe the jot dstrbuto fucto ( F (t, x = P e ξ s dη s x,..., e ξ s dη s x. To ths ed, let us troduce a -dmesoal geeralzed Orste Uhlebeck process X = (X,..., X defed by X t = e ξ t ( X e ξ s dη s for each =,...,. The process X has two mportat propertes. The frst s the trval equalty { } {Xt } = e ξ s dη s X. The secod property s that X s a homogeeeous Markov process as see by the detty +u Xt+u = e (ξ t+u (X ξt t e (ξ s ξ t dηs, ad the depedece ad statoarty of the cremets of a Lévy process. Defe the Markov semgroup (P t t by (P t f(x = E[f(X t X = x] for bouded measurable f, ad let L deote ts geerator. These two propertes combe to gve a useful characterzato of the jot dstrbuto fucto: ( F (t, x = P(X t,..., X t X = x = (P t [, (x Now we cosder the stuato as t. Our ma result s the followg Proposto 2. Suppose that for each =,...,, almost surely ( ξt, ad (2 eξ s dη s coverges to a fte lmt. The (2 F (x = P e ξ s dη s x,..., e ξ s dη s x f ad oly f F s the dstrbuto fucto of a probablty measure o R ad LF =. t Date: Aprl 2, 29.

2 Corollary 3. If F s gve by equato (2 ad L s hypoellptc, the F s ftely dfferetable. Proof of Proposto 2. Frst the easy drecto: If F s gve by equato (2 the { ( } F (x = P e (ξ s ξ t dηs e ξ t x e ξ s dη s for all =,..., t = E[F (X t X = x] by the statoarty ad depedece of the cremets of Lévy processes. Hece P t F = F whch mples LF =. For the other drecto, let x R be such that P e ξ s dη s = x = for all =,...,. Note that the set of all such x s s dese R. Now, cosder the Markov process X talzed at X = x. Note that X t almost surely for each by assumptos ( ad (2. Lettg Z t = F (X t, we see Z coverges almost surely to {X t,...,x t } by the assumpto that F s the dstrbuto fucto of a probablty measure. Furthermore, Z s a martgale as F s bouded ad L-harmoc. Hece F (x = lm t E[F (X t ] = P(Xt,..., Xt = P e ξ s dη s x,..., e ξ s dη s x by the bouded covergece theorem. The clam ow follows from the observato that a dstrbuto fucto s specfed by ts values o a dese set of pots. Remark. Carmoa, Pett, ad Yor have derved ths result the cases where = ad ether η t = t [2] or ξ ad η are depedet [3], by calculatg the varat dstbuto of the geeralzed Orste Uhlebeck process U defed by U t = e (u ξt + e ξs dη s. Ther key observato s ( P e ξs dη s x = P(U t x U = whch s the sprt of our equato (. Doat-Mart, Ghomras, ad Yor [4] use the same techque the mult-dmesoal settg. The key dfferece betwee these approaches ad ours s that we are cocered wth the trasto probabltes of our process X from x to, as opposed to the trasto of U from to x. We are ow ready to treat Dufrese s result. Example. Let W be a stadard Weer process ad c a postve costat. The correspodg geeralzed Orste Uhlebeck process X t = e (X Wt+ct e Ws cs ds satsfes the SDE dx t = c + ] x dt X t dw t 2 ad hece has geerator But sce L = c + ] x 2 x + x2 2 2 x 2. F (x = x 2 2c e 2/y dy, x > Γ(2c y+2c 2

3 satsfes LF =, we have prove Dufrese s theorem. We ow gve aother Browa example: Example 2. Let W ad B be depedet Weer processes, c a postve costat, a a real costat. The radom varable e Ws cs (a ds + db s has a desty g gve by where C > s such that whch satsfes the SDE e2a ta x g(x = C ( + x 2 c+/2 g(xdx =. I ths case, the relevat process X s gve by X t = e (X Wt+ct e Ws cs (a ds + db s dx t = c + ] x a dt X t dw t db t. 2 We smply check that the geerator of X s gve by L = c + ] a 2 x + 2 (x2 + 2 x 2 ad that LG = where G(x = x g(xdx. Ths example also appears the paper of Carmoa, Pett, ad Yor [3]. We ote passg that we ca hadle the more geeral case e Ws cs (a ds + dz s where W ad Z are Weer processes wth correlato ρ, by appealg to the detty e Ws cs (a ds + dz s = = + e Ws cs (a ds + ρdw s + ρ 2 db s e Ws cs a + ρc ρ 2 ds + ρ 2 db s ] whch follows from the lmt ews cs [dw s + ( 2 cds] = ewt ct almost surely. Remark 2. Several other proofs of Dufrese s result have appeared the lterature, addto to the Carmoa, Pett, ad Yor papers [2, 3] metoed above. The followg lst s ot exhaustve, but merely dcates the varous techques that have bee proposed. For stace, Yor [8] has show that the dstbuto of the Dufrese tegral s equal to the dstrbuto of the frst passage tme of a Bessel process va Lampert s relato. Matsumoto ad Yor [7] recover the dstrbuto from vertg a Laplace trasform. Fally, Balleul [] has show drectly that f the Dufrese tegral has a smooth desty fucto f, the f must satsfy a certa ODE. The dffcult part of ths last proof s the verfcato, va Mallav calculus, of the exstece of ths smooth desty. The proof gve here bypasses ths techcalty. It s spred by a observato of Goodma [6] the c = /2 case. We ow descrbe descrbe the geerator of the process X the geeral case. Suppose the characterstc fucto of (ξ t, η t s gve by E(e u ξt+v ηt = e tψ(u,v where by the Lévy Khtche formula ψ s of the form ψ(u, v = a u + b v 2 u Au u Bv 2 v Cv + R R (e u p+v q (u p + v q { p + q <} ν(dp, dq ( A B where a ad b are vectors R, A, B, ad C are matrces such that the matrx B T s C symmetrc ad o-egatve defte, ad ν s a Borel measure such that ( p + q ν(dp, dq <. R R 3

4 The the geerator of the Markov semgroup s the tegro-dfferetal operator defed by ] f(x Lf(x = 2 A a x b + (x x j A j + 2x B j + C j 2 f(x x = 2 x,j= x j [ ] + f(exp( px q f(x + (x p + q f(x { p + q <} ν(dp, dq x R R where exp( p the tegral deotes the dagoal matrx wth th dagoal compoet e p. We coclude ths ote by gvg suffcet codtos for the hypotheses of Proposto 2 to hold: Proposto 4. Let ξ be a scalar Lévy process wth trple (a, σ 2, µ. If p µ(dp < ad a + p ν(dp, dq < { p } = { p } the ξ t almost surely. If addto η s a Lévy process wth Lévy measure ν such that q ν(dq < ad log q ν(dq < { q <} the eξs dη s coverges almost surely. Proof. The frst codto s suffcet for E( ξ < ad E(ξ <. Sce t ξ t E(ξ almost surely by the strog law of large umbers, t follows that there exsts a radom tme T < such that ξ t < 2 E(ξ for all t T. Ths proves ξ t. Now, f q ν(dq < the η ca be decomposed to the sum of a Browa moto wth drft { q <} ad a pure jump process η t = bt + cw t + τ t + τt for subordators τ + ad τ. The tegral e ξs ds exsts sce e ξs ds < T e ξs ds + q T e rs ds < almost surely, where r = 2 E(ξ. Smlarly, the Itô tegral e ξs dw s exsts sce e 2ξs ds < almost surely. Now, we eed oly show e ξs dτ s < almost surely for a pure-jump subordator τ. Sce τ s creasg, the tegral s a path-wse Lebesgue Steltjes tegral. Hece, t suffces to show I t = e rs dτ s coverges almost surely to a fte lmt. But sce t I t (ωs o-decreasg, the lmt I always exsts as a radom varable valued [, ]. We ow show that (I t t coverges dstrbuto to a fte-valued radom varable. Lettg φ(u = (euq ν(dq the E(e uit = e R t φ(ue rs ds = e r Takg u > wthout loss, we have the computato x φ(ux dx = R e rt x φ(uxdx ( 2 (uq dx ν(dq x uq log + (uq/2]ν(dq where we have used the boud e z < 2 z. The expresso o the last le s fte for all u > ad coverges to as u. Therefore, the characterstc fucto of I t coverges potwse to g(u = e r R x φ(uxdx. Sce g s cotuous at u =, Lévy s cotuty theorem mples (I t t coverges dstrbuto. 4

5 Ackowledgemet. Ths work was fshed durg a exteded vst to Ecole Polytechque. The author would lke to thak Nzar Touz ad Peter Takov for ther hosptalty durg hs stay. The author would also lke to thak Ismaël Balleul for a terestg dscusso of hs work o Dufrese s result. Refereces [] Ismaël Balleul. Ue preuve smple d u résultat de Dufrese. Sémare de Probabltés, Vol. XLI, (28 [2] Phlppe Carmoa, Frédérque Pett, ad Marc Yor. O the dstrbuto ad asymptotc results for expoetal fuctoals of Lévy processes. Expoetal Fuctoals ad Prcpal Values Related to Browa Moto, M. Yor, edtor. Bbloteca de la Revsta Matemátca Iberoamercaa. (997 [3] Phlppe Carmoa, Frédérque Pett, ad Marc Yor. Expoetal fuctoals of Lévy processes. I Lévy Processes: Theory ad Practce, O.E. Bardorff-Nelse, T. Mkosch, ad S.I. Resck edtors, 4 55, Brkhäuser, Bosto. (2 [4] Cathere Doat-Mart, Raouf Ghomras, ad Marc Yor. O certa Markov processes attached to expoetal fuctoals of Browa moto; applcato to Asa optos. Revsta Matemátca Iberoamercaa 7, (2 [5] Dael Dufrese. The dstrbuto of a perpetuty, wth applcatos to rsk theory ad peso fudg. Scadava Actuaral Joural o. -2, (99 [6] Vctor Goodma. Browa super-expoets. math.pr/626 [7] Hroyuk Matsumoto ad Marc Yor. Expoetal fuctoals of Browa moto. I. Probablty laws at fxed tme. Probablty Surveys 2, (25 [8] Marc Yor. Sur certaes foctoelles expoetelles du mouvemet browe réel. Joural of Appled Probablty 29, o., (992 Statstcal Laboratory, Cetre for Mathematcal Sceces, Cambrdge CB3 WB, U.K. E-mal address: m.tehrach@statslab.cam.ac.uk 5