In this note, with the help of the boundary classification of diffusions, we
derive a criterion of the convergence of perpetual integral functionals of
transient real-valued diffusions. In the particular case of transient Bessel
processes, we note that this criterion agrees with the one obtained via
Jeulin's convergence lemma

Salminen, Paavo

Yor, Marc

English

arXiv

In this note, with the help of the boundary classification of diffusions, we derive a criterion of the convergence of perpetual integral functionals of transient real-valued diffusions. In the particular case of transient Bessel processes, we note that this criterion agrees with the one obtained via Jeulin's convergence lemma

Salminen, Paavo,

Hal-Diderot

A note on a.s. finiteness of perpetual integralfunctionals of diffusionsPaavo Salminen, Marc YorTo cite this version:Paavo Salminen, Marc Yor. A note on a.s. finiteness of perpetual integral functionals ofdiffusions. 2005. <hal-00013789>HAL Id: hal-00013789https://hal.archives-ouvertes.fr/hal-00013789Submitted on 14 Nov 2005HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00013789, version 1 - 14 Nov 2005A note on a.s. niteness of perpetual integralfuntionals of diusionsPaavo SalminenÅbo Akademi University,Mathematial Department,Fänriksgatan 3 B,FIN-20500 Åbo, Finland,email: phsalminabo.Mar YorUniversité Pierre et Marie Curie,Laboratoire de Probabilitéset Modèles aléatoires ,4, Plae Jussieu, Case 188F-75252 Paris Cedex 05, FraneAbstratIn this note, with the help of the boundary lassiation of diu-sions, we derive a riterion of the onvergene of perpetual integralfuntionals of transient real-valued diusions.In the partiular ase of transient Bessel proesses, we note thatthis riterion agrees with the one obtained via Jeulin's onvergenelemma.Keywords: Brownian motion, random time hange, exit boundary,loal time, additive funtional, stohasti dierential equation.AMS Classiation: 60J65, 60J60.1. Consider a diusion Y on an open interval I = (l, r) determined bythe SDEdYt = σ(Yt) dWt + b(Yt) dt,whereW is a standard Wiener proess dened in a omplete probability spae(Ω,F , {Ft},P). It is assumed that σ and b are ontinuous and σ(x) > 0 forall x ∈ I. We assume also that Y is transient andlimt→ζYt = r a.s., (1)1where ζ is the life time of Y. Hene, if ζ <∞ thenζ = Hr(Y ) := inf{t : Yt = r}.For the speed and the sale measure of Y we usemY (dx) = 2 σ2(x)eBY (x) dx and SY (dx) = e−BY (x) dx, (2)respetively, whereBY (x) = 2∫ x b(z)σ2(z)dz. (3)Let f be a positive and ontinuous funtion dened on I, and onsiderthe perpetual integral funtionalAζ(f) :=∫ ζ0f(Ys) ds.We are interested in nding neessary and suient onditions for a.s. nite-ness of Aζ(f).When Y is a Brownian motion with drift µ > 0 suh a onditionis that the funtion f is integrable at +∞ (see Engelbert and Senf [4℄ andSalminen and Yor [9℄). This ondition is derived in [9℄ via Ray-Knight theo-rems and the stationarity property of the loal time proesses (whih makesJeulin's lemma appliable). In this note a ondition (see Theorem 2) validfor general Y is dedued by exploiting the fat that AHx(f) for x < r an, viarandom time hange, be seen as the rst hitting time of a point for anotherdiusion.2. The next proposition presents the key result onneting perpetualintegral funtionals to rst hitting times. The result is a generalization ofProposition 2.1 in [8℄ disussed in Propositions 2.1 and 2.3 in [2℄.Proposition 1. Let Y and f be as above, and assume that there exists atwo times ontinuously dierentiable funtion g suh thatf(x) =(g′(x)σ(x))2, x ∈ I. (4)Set for t > 0At :=∫ t0f(Ys) ds. (5)and let {at : 0 ≤ t < Aζ} denote the inverse of A, that is,at := min{s : As > t}, t ∈ [0, Aζ).2Then the proess Z given byZt := g (Yat) , t ∈ [0, Aζ),is a diusion satisfying the SDEdZt = dW˜t +G(g−1(Zt)) dt, t ∈ [0, Aζ).where W˜t is a Brownian motion andG(x) =1f(x)(12σ(x)2 g′′(x) + b(x) g′(x)).Moreover, for l < x < y < rAHy(Y ) = inf{t : Zt = g(y)} =: Hg(y)(Z) a.s. (6)with Y0 = x and Z0 = g(x).3. To x ideas, assume that the funtion g as introdued in Proposition 1is inreasing. We dene g(r) := limx→r g(x), and use the same onvention forany inreasing funtion dened on (l, r). The state spae of the diusion Zis the interval (g(l), g(r)) and a.s. limt→ζ(Z)Zt = g(r). Clearly, letting y → rin (6) it follows thatAHr(Y ) = inf{t : Zt = g(r)} a.s., (7)where both sides in (7) are either nite or innite. Now we haveTheorem 2. For Y, A, f and g as above it holds that Aζ is a.s. nite if andonly if for the diusion Z the boundary point g(r) is an exit boundary, i.e.,∫ g(r)SZ(dα)∫ αmZ(dβ) <∞, (8)where the sale SZ and the speed mZ of the diusion Z are given bySZ(dα) = e−BZ (α) dα and mZ(dβ) = 2 eBZ (β) dβwithBZ(β) = 2∫ βG ◦ g−1(z) dz.The ondition (8) is equivalent with the ondition∫ r (SY (r)− SY (v))f(v)mY (dv) <∞. (9)3Proof. As is well known from the standard diusion theory, a diusion hitsits exit boundary with positive probability and an exit boundary annot beunattainable (see [5℄ or [1℄). This ombined with (7) and the haraterizationof an exit boundary (see [1℄ No. II.6 p.14) proves the rst laim. It remainsto show that (8) and (9) are equivalent. We haveBZ(α) = 2∫ g−1(α)G(u) g′(u) du= 2∫ g−1(α)(12g′′(u)g′(u)+b(u)σ2(u))du= log(g′(g−1(α))) +BY (g−1(α)).Consequently,SZ(dα) = e−BZ (α) dα =1g′(g−1(α))exp(−BY (g−1(α)))dαandmZ(dα) = 2 eBZ(α) dα = 2 g′(g−1(α)) exp(BY (g−1(α)))dα.Substituting rst α = g(u) in the outer integral in (8) and after this β = g(v)in the inner integral yield∫ g(r)SZ(dα)∫ αmZ(dβ) = 2∫ rdu e−BY (u)∫ udv (g′(v))2 eBY (u)= 2∫ rdv (g′(v))2 eBY (v)∫ rvdu e−BY (u)by Fubini's theorem. Using the expressions given in (2) for the speed andthe sale of Y and the relation (4) between f and g omplete the proof.4. It is easy to derive a ondition that the mean of Aζ(f) is nite. Indeed,Ex (Aζ(f)) =∫∞0Ex (f(Ys)) ds=∫ rlGY0 (x, y) f(y)mY (dy) <∞, (10)where GY0 is the Green kernel of Y w. r. t. mY . Under the assumption (1)we may take for x ≥ yGY0 (x, y) = SY (r)− SY (x).4Consequently, the ondition (9) may be viewed as a part of the ondition(10).5. Sine the exit ondition (8) plays a ruial rle in our approah wedisuss here shortly two proofs of this ondition, thus making the paper asself-ontained as possible.Let Y be an arbitrary regular diusion living on the interval I with theend points l and r. The sale funtion of Y is denoted by S and the speedmeasure by m. It is also assumed that the killing measure of Y is identiallyzero. Reall the denition due to Fellerr is exit ⇔∫ rS(dα)∫ αm(dβ) <∞. (11)Note that by Fubini's theorem∫ rS(dα)∫ αm(dβ) =∫ rm(dβ)(S(r)− S(β)),and, hene, S(r) < ∞ if r is exit. Moreover, if r is exit then Hr < ∞ withpositive probability.5.1. We give now some details of the proof of (11) following losely Kallen-berg [7℄ (see also Breiman [3℄). For l < a < b < r let Hab := inf{t : Yt =a or b}. Then for a < x < bEx (Hab) =∫ baĜY0 (x, z)m(dz), (12)where ĜY0 is the (symmetri) Green kernel of Y killed when it exits (a, b),i.e.,ĜY0 (x, z) =(S(b)− S(x))(S(y)− S(a))S(b)− S(a)x ≥ y.If r is exit there exists h > 0 suh that Px(Hr < h) > 0 for any xedx ∈ (a, r). Using this property it an be dedued (see [7℄ p. 377) that for anya ∈ (l, r)Ex (Har) <∞,whih, from (12), is seen to be equivalent with (11).5.2. Another proof of (11) an be found in It and MKean [5℄ p. 130. Topresent also this briey reall rst the formulaEx(exp(−λHb)) =ψλ(x)ψλ(b), (13)5where λ > 0 and ψλ is an inreasing solution of the generalized dierentialequationddmddSu = λu. (14)Letting b→ r in (13) it is seen thatr is exit ⇔ limb→rψλ(b) <∞.Let ψ+λ denote the (right) derivative of ψλ with respet to S. Sine ψλ isinreasing it holds that ψ+λ > 0. The fat that ψλ solves (14) yields for z < rψ+λ (r)− ψ+λ (z) = λ∫ rzψλ(a)m(da).In partiular, ψ+λ is inreasing and ψ+λ (r) > 0. Hene, assuming now thatψλ(r) < ∞ we obtain S(r) <∞, and, further,λψλ(z)∫ rzS(dα)∫ αzm(dβ) ≤ λ∫ rzS(dα)∫ αzψλ(β)m(dβ)=∫ rzS(dα)(ψ+λ (α)− ψ+λ (z))= ψλ(r)− ψλ(z)− ψ+λ (z) (S(r)− S(z)) <∞,whih yields the ondition on the right hand side of (11). Assume next thatthe ondition on the right hand side of (11) holds, and onsider for z < β0 ≤ (ψλ(β))−1 (ψ+λ (β)− ψ+λ (z)) = (ψλ(β))−1∫ βzψλ(α)m(dα).Integrating over β giveslog(ψλ(r))− log(ψλ(z))− ψ+λ (z)∫ rz(ψλ(β))−1 S(dβ)=∫ rzS(dβ) (ψλ(β))−1∫ βzψλ(α)m(dα)≤∫ rzS(dβ)∫ βzm(dα) <∞,whih implies that ψλ(r) <∞, thus ompleting the proof.66. As an appliation of Theorem 2, we onsider a Bessel proess withdimension parameter δ > 2. Let R denote this proess. It is well known thatlimt→∞Rt = +∞ and that R solves the SDEdRt = dWt +δ − 12Rtdt,where W is a standard Brownian motion. Here the funtion BR (f. (3))takes the formBR(v) = (δ − 1) log v,and, onsequently,∫∞dv (g′(v))2 eBR(v)∫∞vdu e−BR(u)=∫∞dv (g′(v))2 vδ−1∫∞vdu u−δ+1=∫∞dv (g′(v))2 vδ−11δ − 2v−δ+2leading to ∫∞0f(Rt) dt <∞ ⇔∫∞u f(u) du < +∞.Another way to derive this ondition is via loal times and Jeulin's lemma[6℄. Indeed, by the oupation time formula and Ray-Knight theorem for thetotal loal times of R (see, e.g. [10℄ Theorem 4.1 p. 52) we have∫∞0f(Rs) ds(d)=∫∞0f(a)ρaγγ aγ−1da=1γ∫∞0a f(a)ρaγaγdawhere δ = 2 + γ and ρ is a squared 2-dimensional Bessel proess. Usingthe saling property, it is seen that the distribution of the random variableρaγ/aγdoes not depend on a. Hene, we obtain by Jeulin's lemma that if thefuntion a 7→ a f(a), a > 0, is loally integrable on [0,∞) then∫∞0f(Rs) ds <∞ ⇔∫∞a f(a) da <∞. (15)7The same argument allows us to reover the result in [9℄, that is,∫∞0g(W (µ)s ) ds <∞ ⇔∫∞g(x) dx <∞. (16)where g is any non-negative loally integrable funtion and W (µ) denotes aBrownian motion with drift µ > 0. To see this, write g(x) = f(ex) and useLamperti's representationexp(W (µ)s ) = R(µ)A(µ)s, s ≥ 0,whereA(µ)s =∫ s0du exp(2W (µ)u )and R(µ) is a Bessel proess with dimension d = 2(1+ µ) starting from 1, weobtain (f. [8℄ Remark 3.3.(3))∫∞0f(exp(W (µ)s )) ds =∫∞0(R(µ)u)−2f(R(µ)u ) du a.s.,and, in order to get (16) it now only remains to use the equivalene (15).We wish to underline the fat that in Theorem 2 it is assumed that thefuntion f is ontinuous whereas the approah via Jeulin's lemma, whih wedeveloped above, demands only loal integrability.Referenes[1℄ A.N. Borodin and P. Salminen. Handbook of Brownian Motion  Fatsand Formulae, 2nd edition. Birkhäuser, Basel, Boston, Berlin, 2002.[2℄ A.N. Borodin and P. Salminen. On some exponential integral funtionalsof BM(µ) and BES(3). Zap. Nauhn. Semin. POMI, 311:5178, 2004.Preprint available in http://arxiv.org/abs/math.PR/0408367.[3℄ L. Breiman. Probability. Addison Wesley, Reading, MA, 1968.[4℄ H.J. Engelbert and T. Senf. On funtionals of Wiener proess with driftand exponential loal martingales. In M. Dozzi, H.J. Engelbert, andD. Nualart, editors, Stohasti proesses and related topis. Pro. Win-tersh. Stohasti Proesses, Optim. Control, Georgenthal/Ger. 1990,number 61 in Math. Res., Aademi Verlag, pages 4558, Berlin, 1991.8[5℄ K. It and H.P. MKean. Diusion Proesses and Their Sample Paths.Springer Verlag, Berlin, Heidelberg, 1974.[6℄ T. Jeulin. Sur la onvergene absolue de ertaines intégrales. In J.Azéma and M. Yor, editors, Séminaire de Probabilités XVI, number920 in Springer Leture Notes in Mathematis, pages 248256, Berlin,Heidelberg, New York, 1982.[7℄ O. Kallenberg. Foundations of modern probability. Springer Verlag, NewYork, Berlin, Heidelberg, 1997.[8℄ P. Salminen and M. Yor. Perpetual integral funtionals as hitting timesand oupation times. Elet. J. Prob., 10:371419, 2005.[9℄ P. Salminen and M. Yor. Properties of perpetual integral funtionals ofBrownian motion with drift. Ann. I.H.P., 41(3):335347, 2005.[10℄ M. Yor. Some Aspets of Brownian Motion. Part I: Some speial fun-tionals. Birkhäuser Verlag, Basel, 1992.9

A note on a.s. finiteness of perpetual integral functionals of diffusions

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