Exponential functionals of Lévy processes

Exponential functionals of Lévy processes Víctor Rivero Centro de Investigación en Matemáticas, México. 1/ 28

Outline of the talk Introduction Exponential functionals of spectrally positive Lévy processes Representation of exponential functionals as integrals of subordinators Estimates of the tail distribution of exponential functionals Quasi-stationary distributions of Ornstein-Uhlenbeck type processes Factorizations of the Pareto distribution Examples 2/ 28

Introduction Let ξ be a real valued Lévy process with characteristics (a, σ 2, Π), and characteristic exponent Ψ(λ) = iaλ + 1 2 σ2 λ 2 ( + 1 e iλx + iλx ) Π(dx), λ R. We assume that R \{} lim ξ t =, t P a.s. The exponential functional associated to ξ is the r.v. defined by I := e ξs ds <, P a.s. 3/ 28

Introduction Applications to Self-similar Markov processes: the first hitting time of zero, limit theorems and entrance laws; Brownian diffusions in random environment; Mathematical finance: perpetuities, computation of price of asian options; self-similar fragmentations. 4/ 28

Introduction There is just a small number of examples where the law of I is explicitly known, or for which the law of or I t := t e ξs ds, t >, I τ, τ exp(λ), independent is known. Bertoin and Yor s paper surveys most of the known results about exponential functionals. We would like to obtain estimates of the tail distribution of I. 5/ 28

Self-decomposability of exponential functionals of spectrally negative Lévy processes Spectrally positive Lévy processes Lemma (Self-decomposable exponential functionals) Let ξ be a spectrally positive Lévy process, that is ξ has no negative jumps, Π(, ) =. The random variable I is self-decomposable. Proof. Let < b < 1 and T log(b) = inf{t > : ξ t < log(b)}. I = Tlog(b) T log(b) <, P a.s. because lim t ξ t =, P a.s.. Tlog(b) e ξs ds + e ξs ds = T log(b) e ξs ds + e ξ T log(b) ξ e e s ds, ( ξ s = ξ Tlog(b)+s ξ Tlog(b), s ) Law = (ξ s, s ), independent of σ(ξ s, s T log(b) ) 6/ 28

Self-decomposability of exponential functionals of spectrally negative Lévy processes By a result by Wolfe (1982) and Sato and Yamazato (1983,1984), the law of I is the invariant measure of a real valued Ornstein-Uhlenbeck process {X t, t } t t ) X t = x + Y t X s ds, X t = e (x t + e s dy s, t, for some subordinator Y, with characteristics (a Y, Π Y ), such that a Y, min{x, 1}Π Y (dx) <, log(x)π Y (dx) <. R + x 2 7/ 28

Self-decomposability of exponential functionals of spectrally negative Lévy processes Is it possible to construct the subordinator Y out from ξ such that e ξs ds Law = e s dy s? If yes, how are the characteristics of Y related to those of ξ? 8/ 28

Self-decomposability of exponential functionals of spectrally negative Lévy processes Some facts from the fluctuation theory for Lévy processes Let ξ = (ξ t, t ), a real valued Lévy process, and define I = (I t = inf{ξ s, s t}, t ), its current infimum, ( L t, t ), the local time at of the reflected process ξ I = (ξ t I t, t ), downward ladder time process ( L 1 t, t ), downward ladder height process (ĥt I bl 1, t ). t The downward ladder process ( L 1, ĥ) is a bivariate subordinator whose Laplace exponent κ is given by for λ, µ, Fristedt s formula κ(λ, µ) = log E(exp{ λ L 1 1 µĥ1}) ( ) dt = c exp (e t e λt µx ) P(ξ t dx). t (,] 9/ 28

Self-decomposability of exponential functionals of spectrally negative Lévy processes Some facts from the fluctuation theory for Lévy processes Let ξ = (ξ t, t ), a real valued Lévy process, and define I = (I t = inf{ξ s, s t}, t ), its current infimum, ( L t, t ), the local time at of the reflected process ξ I = (ξ t I t, t ), Let (e t, t ) be the point process of the excursions out from for ξ I, e t (s) := { ξbl 1 t +s ξ L b 1, t 1 1 s L t L t, if, if L 1 t L 1 t L 1 t >, 1 L t =. 1/ 28

Self-decomposability of exponential functionals of spectrally negative Lévy processes Some facts from the fluctuation theory for Lévy processes Let ξ = (ξ t, t ), a real valued Lévy process, and define I = (I t = inf{ξ s, s t}, t ), its current infimum, ( L t, t ), the local time at of the reflected process ξ I = (ξ t I t, t ), Let (e t, t ) be the point process of the excursions out from for ξ I, e t (s) := { ξbl 1 t +s ξ L b 1, t 1 1 s L t L t, if, if L 1 t L 1 t L 1 t >, 1 L t =. Itô s excursion theory for the process reflected ξ I, ensures that (e t, t ) is a Poisson point process with a characteristic measure n over the space D([, ), R) of paths with finite lifetime. 1/ 28

A new representation of exponential fiunctionals in terms of subordinators Is it possible to construct the subordinator Y out from ξ such that e ξs ds Law = e s dy s? If yes, how are the characteristics of Y related to those of ξ? 11/ 28

A new representation of exponential fiunctionals in terms of subordinators Theorem Let ξ be a real valued Lévy process. We have the equality in law e ξs ds = e b h s dy s, where ĥ is the downward ladder height subordinator, and Y is a subordinator whose characteristics (a Y, Π Y ) satisfy that ( s ) ζ a Y Ls = 1 {ξu=i u}du, Π Y (x, ) = n e ξs ds > x, x >, 12/ 28

A new representation of exponential fiunctionals in terms of subordinators Corollary Let ξ be a spectrally positive real valued Lévy process. We have the equality in law e ξs ds Law = e s dy s, with Y a subordinator as in the previous theorem. 13/ 28

A new representation of exponential fiunctionals in terms of subordinators Proof. Observe that { [, ) = {t : ξ t = I t } [ L 1 s> s, L 1 s ] }, 14/ 28

A new representation of exponential fiunctionals in terms of subordinators Proof. Observe that { [, ) = {t : ξ t = I t } [ L 1 It follows that I = = a = a e ξs 1 {ξs=i s}ds + t> e Is d L s + t> e b h s ds + s> s, L 1 s ] } e I L b 1 L b 1 t t e ξs I b 1 L t ds bl 1 t ζt e b h t exp{e t (s)}ds e hs dỹs,, where Ỹs = ζt t s exp{e t(u)}du, is a subordinator, as the point process of excursions is a Poisson point process. Take Y t = at + Ỹt 14/ 28

A new representation of exponential fiunctionals in terms of subordinators Remark It can be verified that ( L 1, ĥ, Y ) is a trivariate subordinator. Thus, the law of an exponential functional is the invariant measure of a process of the Ornstein-Uhlenbeck type ( (1) Y X t = e t x + t ) e Y (1) s dy (2) s, with (Y (1), Y (2) ) a bivariate subordinator that satisfies that log(x) A Y1 (log(y)) Π Y 2 (dx) <, with (e, ) A Y1 (z) = max{1, Π Y1 (1, )} + z 1 Π Y1 (x, )dx, z 1. See Lindner and Maller (25) for a detailed study of the invariant laws of generalized OU processes. 15/ 28

Asymptotic behaviour of the tail distribution Let ξ be a Lévy process that drifts to, and I = e ξs ds, what is the behaviour of P(I > t) as t? 16/ 28

Asymptotic behaviour of the tail distribution Let ξ be a Lévy process that drifts to, and I = e ξs ds, what is the behaviour of P(I > t) as t? Conjecture P(log I > t) C P(sup ξ s > t), t, s for some constant C (, ). 16/ 28

Asymptotic behaviour of the tail distribution The conjecture holds true in some cases Assume that ξ satisfies Cramér s condition: θ >, E(e θξ1 ) = 1, E(ξ 1 e θξ1 ) <. Then lim x eθx P(sup ξ s > x) = C 1, s lim t θ P (I > t) = C 2, C 1, C 2 (, ) t Bertoin and Doney (1994) + R. (25). If ξ is such that µ = E(ξ 1 ) (, ) and G(x) = min{1, x P(ξ 1 > u)du}, is subexponential then P(log(I) > t) P(sup ξ s > t) 1 G(t), t. s µ Maulik & Zwart, (26). 17/ 28

Asymptotic behaviour of the tail distribution The r.v. I = e b h s dy s satisfies the random affine equation: I Law Law = Q + MĨ, (Q, M) Ĩ = I, with Q = 1 e b h s dy s, M = e b h 1. 18/ 28

Asymptotic behaviour of the tail distribution The r.v. I = e b h s dy s satisfies the random affine equation: I Law Law = Q + MĨ, (Q, M) Ĩ = I, with Q = 1 e b h s dy s, M = e b h 1. According to a result by Grey (1994), given that M 1 a.s., then t P(Q > t) is regularly varying at infinity with index α (RV α) iff P(I > t) RV α, in this case P(I > t) 1 1 E(M α P(Q > t), t. ) 18/ 28

Asymptotic behaviour of the tail distribution Theorem Assume that ξ 1 is in S α for some α >, that is P(ξ 1 > t + s) lim = e αs P(ξ 2 > t), s R, lim t P(ξ 1 > t) t P(ξ 1 > t) = 2 E(eαξ1 ). Then 19/ 28

Asymptotic behaviour of the tail distribution Theorem Assume that ξ 1 is in S α for some α >, that is P(ξ 1 > t + s) lim = e αs P(ξ 2 > t), s R, lim t P(ξ 1 > t) t P(ξ 1 > t) = 2 E(eαξ1 ). Then P(sup s ξ s > t) c () α Π ξ (t, ), t, Kyprianou, Kluppelberg & Maller (24). 19/ 28

Asymptotic behaviour of the tail distribution Theorem Assume that ξ 1 is in S α for some α >, that is P(ξ 1 > t + s) lim = e αs P(ξ 2 > t), s R, lim t P(ξ 1 > t) t P(ξ 1 > t) = 2 E(eαξ1 ). Then P(sup ξ s > t) c () α Π ξ (t, ), t, Kyprianou, Kluppelberg s & Maller (24). P(Y 1 > t) Π Y (t, ) c (1) α E(I 1 α )Π ξ (log(t), ), t, P(I > t) c (2) α E(I 1 α )Π ξ (log(t), ), t, where c (i) α (, ) are constants that can be determined from the characteristics of ξ. In particular, P(log(I) > t) c α P(sup ξ s > t), t, s and log(i) S α. 19/ 28

Asymptotic behaviour of the tail distribution Open problem. Are there other Lévy processes for which P(log(I) > t) c P(sup ξ s > t), t? s Open problem. What can be said for P(I < t) for small values of t? 2/ 28

Factorizations of the Pareto law Factorizations of the Pareto law Let X be a positive self-similar Markov process, that is a strong Markov process with rcll paths and such that there exists α > s.t. for all c > ({cx tc α, t }, IP x ) Law = ({X t, t }, IP cx ) x >. Lamperti s transformation establishes that there exists a Lévy process ξ such that X, issued from x, has the same law as xe ξ τ(tx α ), s τ(t) = inf{s > : e αξu du > t}, t, with inf{ } =, ξ :=. This implies that under IP x, T := inf{t > : X t = } Law = x α e αξs ds 21/ 28

Factorizations of the Pareto law We assume α = 1, T < a.s. and define a process of the Ornstein-Uhlenbeck type associated to X by U t = e t X et 1, t. We denote by T U, the first hitting time of for U. Lamperti s transformation implies that ( ( ) ) ) T U Law, IP x = (log 1 + x e ξs ds, P. Under which conditions we can ensure existence of a quasi-estationary measure for U? Viz. a probability measure, say ν, and an index θ >, such that ν(dx) IE x (f(u t ), t < T U ) = e θt x R + ν(dx)f(x), R + t. for any f continuous and bounded. 22/ 28

Factorizations of the Pareto law If there exists a quasi-stationary measure associated to U, say ν, with index θ, then e θt = ν(dx) IP x (T U > t) x R + = ν(dx) P(xI > e t 1), t. x R 23/ 28

Factorizations of the Pareto law If there exists a quasi-stationary measure associated to U, say ν, with index θ, then e θt = ν(dx) IP x (T U > t) x R + = ν(dx) P(xI > e t 1), t. x R Let J be a random variable such that under P, J follows the law ν, and it is independent of I. We have that, P(JI > t) = (1 + t) θ, t. That is JI Law = Pareto(θ). 23/ 28

Factorizations of the Pareto law Cramér s case Assume that ξ satisfies that θ >, s.t. E(e θξ1 ) = 1. For example ξ has no positive jumps and drifts towards. 24/ 28

Factorizations of the Pareto law Cramér s case Assume that ξ satisfies that θ >, s.t. E(e θξ1 ) = 1. For example ξ has no positive jumps and drifts towards. Let P be the unique measure such that P Gt = e θξt P Gt, G t = σ(ξ s, s t), t. It is known that P is the law of a Lévy process, say ξ, that drifts to. 24/ 28

Factorizations of the Pareto law It has been proved in R. (25) that the measures (η t, t ) ( ( ) ) t η t (dx)f(x) = c θ t θ E f R + I (I ) θ 1, with c θ a normalizing constant, form an entrance law for X, viz. η t (dx) IE x (f(x s ), s < T ) = η t+s (dx)f(x), R + R + for any f continuous and bounded function. Thus, the measure η 1 is such that ( η 1 (dx) IE x f(e t X et 1), e t ) 1 < T = η e t(dx)f(e t x) R + R + = c θ e θt R + η 1 (dx)f(x). 25/ 28

Factorizations of the Pareto law Theorem Let U be a process of the OU type associated to X (hence to ξ), U t = e t X et 1, t. Assume ξ satisfies Cramér s condition. Then the measure ν(dx) = η 1 (dx) defined by ( ( ) ) 1 ν(dx)f(x) = c θ E f R + I (I ) θ 1, is a θ-quasi-stationary law for U. Furthermore, let J be a random variable with law ν and independent of I. We have that JI Law = Pareto(θ). 26/ 28

Factorizations of the Pareto law Example (After Gjessing and Paulsen (1997) and Singh (28)) Let ξ be a Lévy process with drift +1 and no-positive jumps with Lévy measure Π(dx) = b(b (1 a))e x(b (1 a)) dx, x <, with < a < 1 < a + b. The Laplace exponents are ψ(λ) = λ(λ (1 a)) λ + b + a 1, ψ (λ) = λ(λ + (1 a)), λ, λ + b θ = 1 a. Hence, the associated exponential functionals are distributed as I Law = 1 β b+a,1 a β b+a,1 a, I Law = 1 β 1 a,a+b 1, J Law = β 1,a+b 1. We conclude that 1 β b+a,1 a Law β 1,a+b 1 = Pareto(1 a). β b+a,1 a 27/ 28

Factorizations of the Pareto law Example (After Dufresne (1991)) Let ξ = (2(B t bt), t ) with B a Brownian motion. We have that ξ = (2(B t + bt), t ), θ = b. The associated exponential functionals are distributed as I, I Law 1 =, J E 1/2, 2γ b where γ b Γ(b), E 1/2 exponential(1/2). We obtain that 1 2γ b E 1/2 = Pareto(b) 28/ 28