Distributional properties of exponential functionals of Lévy processes

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.17(2012), no. 8, 1–35.

ISSN:1083-6489 DOI:10.1214/EJP.v17-1755

Distributional properties of exponential functionals of Lévy processes

A. Kuznetsov

J. C. Pardo

M. Savov

Abstract

We study the distribution of the exponential functional I(ξ, η) = R∞

0 exp(ξt−)dηt, where ξ andη are independent Lévy processes. In the general setting, using the theory of Markov processes and Schwartz distributions, we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in [9]. In the special case whenηis a Brownian motion with drift, we show that this integral equation leads to an important functional equation for the Mellin trans- form ofI(ξ, η), which proves to be a very useful tool for studying the distributional properties of this random variable. For general Lévy processξ (ηbeing Brownian motion with drift) we prove that the exponential functional has a smooth density on R\ {0}, but surprisingly the second derivative at zero may fail to exist. Under the additional assumption thatξhas some positive exponential moments we establish an asymptotic behaviour ofP(I(ξ, η)> x)asx→+∞, and under similar assumptions on the negative exponential moments ofξwe obtain a precise asymptotic expansion of the density ofI(ξ, η) asx →0. Under further assumptions on the Lévy process ξ one is able to prove much stronger results about the density of the exponential functional and we illustrate some of the ideas and techniques for the case whenξhas hyper-exponential jumps.

Keywords:Lévy processes; exponential functional; integral equations; Mellin transform; asymp- totic expansions.

AMS MSC 2010:60G51.

Submitted to EJP on June 30, 2011, final version accepted on December 1, 2011.

SupersedesarXiv:1105.6365v1.

1 Introduction

In this paper, we are interested in studying distributional properties of the random variable

I(ξ, η) :=

Z ∞ 0

e^ξt−dηt, (1.1)

∗AK supported by Natural Sci. and Engineering Research Council, Canada. JCP supported by CONACYT.

†Dep. of Math. and Stat., York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada.

E-mail:kuznetsov@mathstat.yorku.ca.

‡Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n. 36240 Guanajuato, México.

E-mail:jcpardo@cimat.mx

§New College, Holywell Street, Oxford, OX1 3BN, UK.

E-mail:savov@stat.ox.ac.ukE-mail:mladensavov@hotmail.com

whereξandηare independent real-valued Lévy processes such thatξdrifts to−∞and E[|ξ₁|]<∞andE[|η₁|]<∞.

The exponential functionalsI(ξ, η)appear in various aspects of probability theory.

They describe the stationary measure of generalized Ornstein-Uhlenbeck processes and the entrance law of positive self-similar Markov processes, see [6, 9]. They also play a role in the theory of fragmentation processes and branching processes, see [4, 22].

Besides their theoretical value, the exponential functionals are very important objects in Mathematical Finance and Insurance Mathematics. They are related to Asian op- tions, present values of certain perpetuities, etc., see [10, 17, 14] for some particular examples and results.

In general, the distribution of exponential functionals is difficult to study. It is known explicitly only in some very special cases, see [8, 14, 19]. Properties of the distribution ofI(ξ, η)are also of particular interest. Lindner and Sato [26] show that the density of I(ξ, η)doesn’t always exist, and in the special case whenξandηare specific compound Poisson processes, distributional properties ofI(ξ, η)can be related to the problem of absolute continuity of the distribution of Bernoulli convolutions, which dates back to Erd˝os, see [12]. The distribution ofI(ξ, η), whenξ_s=−sand in some other instances, is known to be self-decomposable and hence absolutely continuous, see [5, 18]. When η is a subordinator with a strictly positive drift, the law of the exponential functional I(ξ, η) is absolutely continuous, see Theorem 3.9 in Bertoin et al. [5]. Some further results are obtained in [24, 29, 30, 35].

The asymptotic behaviour P(I(ξ, η)> x), as x → ∞, is a question which has at- tracted the attention of many researchers. In the general case, but under rather strin- gent requirements on the existence of exponential moments forξand absolute moments forη, it has been studied in [25]. The special case whenηt = t has been considered in [27, 31, 32] and properties of the density of the law of I(ξ, η) at zero and infinity have been studied by [19, 21, 28] and results such as asymptotic and convergent series expansions for the density have been obtained.

The first objective of this paper is to develop a general integral equation for the law ofI(ξ, η)under the assumptions that E[|ξ1|] < ∞, E[ξ₁] < 0, E[|η1|] < ∞and ξ being independent of η. Using the fact that in general I(ξ, η)is a stationary law of a gen- eralized Ornstein-Uhlenbeck process, Carmona et al. [9] show that if ξ has jumps of bounded variation andηt = t then the law ofI(ξ, η)satisfies a certain integral equa- tion. We refine and strengthen their approach and using both stationarity properties ofI(ξ, η)and Schwartz theory of distributions, we show that in the general setting the law of I(ξ, η) satisfies a certain integral equation. This equation is important on its own right, as demonstrated by Corollary 2.5, but it is also amenable to different useful transformations as can be seen from the discussion below.

The second main objective of the paper is to study some properties ofIµ,σ:=I(ξ, η) in the specific case when ηs = µs+σBs, where Bs is a standard Brownian motion.

Quantities of this type have already appeared in the literature, see [14], but have not been thoroughly studied. The latter, as it seems to us, is due to the lack of suitable tech- niques, which are available in the case whenηs=s, and in particular due to the lack of any information about the Mellin transform ofI(ξ, η), which is the key tool for studying the properties ofI(ξ, η), see [19, 21, 27]. We use the integral equation (2.3) and combine techniques from special functions, complex analysis and probability theory to study the Mellin transform ofI_µ,σ, which is defined asM(s) =E

(I_µ,σ)^s−11_{Iµ,σ>0}

. In particular we derive an important functional equation forM(s), see (3.13), and study the decay of M(s)as Im(s)→ ∞. These results supply us with quite powerful tools for studying the properties of the density ofIµ,σ via the Mellin inversion. Furthermore, the functional equation (3.13) allows for a meromorphic extension of M(s) when ξ has some expo-

nential moments. This culminates in very precise asymptotic results for P(I_µ,σ> x), asx → ∞, see Theorem 4.3, and asymptotic expansions fork(x), the density ofI_µ,σ, asx→0, see Theorem 4.1. The latter results show us that while k(x)∈ C^∞(R\ {0}), rather unexpectedlyk⁰⁰(0)may not exist. Finally, we would like to point out that while the behaviour ofP(Iµ,σ> x), asx→ ∞, might be partially studied via the fact thatIµ,σ

solves a random recurrence equation, see for example [25], the behaviour ofk(x), as x→0, seems for the moment to be only tractable via our approach based on the Mellin transform.

As another illustration of possible applications of our general results, we study the density ofIµ,σ whenξ has hyper-exponential jumps (see [7, 8, 20]). This class of pro- cesses is quite important for applications in Mathematical Finance and Insurance Math- ematics, and it is particularly well suited for investigation using our methods due to the rich analytical structure enjoyed by these processes. In this case we show how to derive complete asymptotic expansions ofk(x)both at zero and infinity. We point out that our methodology is not restricted to this particular case, and can be easily applied to more general classes of Lévy processes.

The paper is organized as follows: in Section 2, we study the law ofI(ξ, η)for general independent Lévy processes ξ and η and derive an integral equation for the law of I(ξ, η); in Section 3, we specialize the results obtained in Section 2 to the case when η_s=µs+σB_sand, employing additionally various techniques from special functions and complex analysis, we study the properties of the density ofIµ,σ. Section 4 is devoted to some applications of the results derived in the previous section. In particular, we study the asymptotic behaviour at infinity of the tail ofIµ,σ and of its density at zero, and in the case of processes with hyper-exponential jumps, we show how these results can be considerably strengthened.

2 Integral equation satisfied by the law of I(ξ, η)

Let us introduce some notation which will be used throughout this paper. The main underlying objects are two independent Lévy processesξandηdefined on a probability space(Ω,F,P). As is standard, we assume that both processes are started from zero under the probability measureP^.

Assumption 2.1. Everywhere in this paper we will assume that

E[|ξ1|]<∞, E[ξ1]<0, E[|η1|]<∞. (2.1) The characteristics of the Lévy processes ξ and η will be denoted by (bξ, σξ,Πξ) and (b_η, σ_η,Π_η). In particular Π_ξ(dx) and Π_η(dx) are the Lévy measures of ξ and η, respectively. We use the following notation for the double-integrated tail

Π⁽⁺⁾_ξ (x) = Z ∞

x

Πξ((y,∞))dy and Π⁽⁻⁾_ξ (x) = Z ∞

x

Πξ((−∞,−y))dy,

and similarly forΠ⁽⁺⁾_η andΠ⁽⁻⁾_η . Using the Lévy-Itô decomposition (see Theorem 2.1 in [23]) it is easy to check that Assumption 2.1 implies that the above quantities are finite for allx >0.

We define the Laplace exponentsψξ(z) = ln E e^zξ1

andψη(z) = ln (E[e^zη1]), where without any further assumptionsψξ andψη are defined at least for Re(z) = 0, see [3, Chapter I]. The Laplace exponentψξ can be expressed in the following two equivalent

ways

ψξ(z) = σ_ξ²

2 z²+bξz+ Z

R

(e^zx−1−zx) Πξ(dx) (2.2)

= σ_ξ²

2 z²+bξz+z² Z ∞

0

Π

(+)

ξ (w)e^xzdx+ Z ∞

0

Π⁽⁻⁾_ξ (x)e^−xzdx

,

with a similar expression for ψ_η. The first equality in (2.2) is essentially the Lévy- Khintchine formula (see Theorem 1 in [3]) with the cutoff functionh(x)≡1. The stan- dard choice for the cutoff function in the Lévy-Khintchine formula would be 1_{|x|<1}, however it is well-known that ifE[|ξ1|]<∞then we can take a simpler cutoff function h(x)≡1. The second equality in (2.2) follows easily by repeated integration by parts.

Note that according to (2.2), we havebξ =ψ⁰_ξ(0) =E[ξ1]and similarlybη =ψ_η⁰(0) =E[η1]. We recall that the exponential functional I(ξ, η) is defined by (1.1), its law will be denoted bym(dx) := P(I(ξ, η)∈ dx). The density ofI(ξ, η), provided it exists, will be denoted byk(x).

Our main result in this section is the derivation of an integral equation for the law of I(ξ, η). This equation will be very useful later, when we’ll derive the functional equation (3.13) for the Mellin transform of the exponential functional in the special case whenη is a Brownian motion with drift. The main idea of this Theorem comes from Proposition 2.1 in [9].

Theorem 2.2. Assume that condition(2.1)is satisfied. Then the exponential functional I(ξ, η)is well defined and its law satisfies the following integral equation: forv >0

bξ

Z ∞ v

m(dx)

dv +σ²_ξ

2 vm(dv) + Z ∞

v

Π⁽⁻⁾_ξ lnx

v

m(dx)

dv+ Z v

0

Π⁽⁺⁾_ξ lnv

x

m(dx)

dv +

bη

Z ∞ v

m(dx) x

dv+σ_η² 2

m(dv) v − σ_η²

2 Z ∞

v

m(dx) x²

! dv +

1 v

Z v 0

Π⁽⁺⁾_η (v−x)m(dx)

dv+ 1

v Z ∞

v

Π⁽⁻⁾_η (x−v)m(dx)

dv (2.3)

− Z ∞

v

1 w²

Z w 0

Π

(+)

η (w−x)m(dx)dw

dv

− Z ∞

v

1 w²

Z ∞ w

Π⁽⁻⁾_η (x−w)m(dx)dw

dv= 0,

where all quantities in (2.3) are a.e. finite. Equation (2.3) for the law of I(ξ,−η)on (0,∞)describesm(dx)on(−∞,0).

The proof of Theorem 2.2 is based on the so-called generalized Ornstein-Uhlenbeck (GOU) process, which is defined as

Ut=Ut(ξ, η) =xe^ξt+e^ξt Z t

0

e^−ξs−dηs

=d xe^ξt+ Z t

0

e^ξs−dηs, fort >0. (2.4) Note that the GOU process is a strong Markov process, see [9, Appendix 1]. Lindner and Maller [25] have shown that the existence of a stationary distribution for the GOU process is closely related to the a.s. convergence of the stochastic integralRt

0e^ξs−dηs, as t→ ∞. Necessary and sufficient conditions for the convergence ofI(ξ, η)were obtained

by Erickson and Maller [13]. More precisely, they showed that this happens if and only if

t→∞lim ξt=−∞ a.s. and Z

R\[−e,e]

"

log|y|

1 +Rlog|y|∨1

1 Πξ(R\(−z, z))dz

#

Πη(dy)<∞.

(2.5) It is easy to see that Assumption 2.1 implies (2.5). HenceI(ξ, η)is well-defined and the stationary distribution satisfiesU∞

=d I(ξ, η). This identity in distribution is the starting point of the proof of Theorem 2.2.

As the proof of Theorem 2.2 is rather long and technical, we will divide it into several steps. We first compute the generator ofU, here denoted byL^(U). This result may be of independent interest, therefore we present it in Proposition 2.3 below. Then we note that the stationary measurem(dx)satisfies the equation

Z ∞ 0

L^(U)f(x)m(dx) = 0, (2.6)

wheref is any infinitely differentiable function with a compact support in (0,∞). In- deed, (2.6) follows from (2.1) in [9] or from the definition of infinitesimal generator and the observation that, for allt≥0,

Z ∞ 0

E[f(U_t)]m(dx) = Z ∞

0

f(x)m(dx).

Finally, an application of Schwartz theory of distributions after rephrasing (2.6) gives (2.3).

We start by working out how the infinitesimal generator of U, i.e. L^(U), acts on functions inK ⊂C0(R), where

K =

f(x) : f(x)∈C_b²(R), f(e^x)∈C_b²(R)∩C0(R)

∩ {f(x) = 0, forx≤0;f⁰(0) =f⁰⁰(0) = 0} (2.7) andC_b²(R)stands for two times differentiable, bounded functions with bounded deriva- tives onR ^andC0(R)is the set of continuous functions vanishing at ±∞. Denote by L^(ξ)andL^(η)( resp.D^ξ andD^η) the infinitesimal generators (resp. domains) ofξandη. Note that

L^(ξ)f(x) = bξf⁰(x) +σ²_ξ

2 f⁰⁰(x) + Z

R

(f(x+y)−f(x)−yf⁰(x)) Πξ(dy) (2.8)

= bξf⁰(x) +σ²_ξ

2 f⁰⁰(x) + Z

R+

f⁰⁰(x+w)Π

(+)

ξ (w)dw+ Z

R+

f⁰⁰(x−w)Π⁽⁻⁾_ξ (w)dw, with a similar expression forL^(η). The first formula in (2.8) is a trivial modification of the form of the generator of Lévy processes for the case when the cutoff function ish(x)≡1, see [3, p. 24], whereas the second expression follows easily by integration by parts, the fact thatf ∈ KandE[|ξ1|]<∞. Finally, we are ready to state our result, which should strictly be seen as an extension of Proposition 5.8 in [9] where the generatorL^(U) has been derived under very stringent conditions.

Proposition 2.3. Assume that condition (2.1)is satisfied. Letf ∈ K, g(x) := (xf⁰(x)) andφ(x) :=f(e^x). Then,f ∈D^η,φ∈D^ξ and

L^(U)f(x) =L^(ξ)φ(lnx) +L^(η)f(x)

=b_ξg(x) +σ²_ξ

2 xg⁰(x) + Z x

0

g⁰(v)Π⁽⁻⁾_ξ lnx

v

dv+ Z ∞

x

g⁰(v)Π⁽⁺⁾_ξ lnv

x

dv +bηf⁰(x) +σ²_η

2 f⁰⁰(x) + Z ∞

0

f⁰⁰(x+w)Π

(+)

η (w)dw+ Z ∞

0

f⁰⁰(x−w)Π⁽⁻⁾_η (w)dw.(2.9)

Proof. The main idea is to use the definition of the infinitesimal generator and Itô’s formula. Letf ∈ Kand note that by definition

L^(U)f(x) = lim

t→0

Ex[f(U_t)]−f(x)

t = lim

t→0

1 t

E

f

xe^ξt+

Z t 0

e^ξs−dηs

−f(x)

. Using the fact that (Ut)_t≥0 is a semimartingale and f ∈ K, we apply Itô’s formula to f(Ut)to obtain

f(U_t)−f(x) = Z t

0

f⁰(U_s−)dU_s +1

2 Z t

0

f⁰⁰(U_s−)d[U, U]^c_s

+X

s≤t

(f(U_s)−f(U_s−)−∆U_sf⁰(U_s−)). (2.10)

Now, let H_t := e^ξt and V_t := x+Rt

0e^−ξs−dη_s, and note that U_t = H_tV_t. Hence by integration by parts

Ut=x+ Z t

0

Hs−dVt+ Z t

0

Vs−dHs+ [H, V]t.

Using the Lévy-Itô decomposition (see Theorem 2.1 in [23]) and Assumption 2.1, we find that the Lévy processesξandη can be written as follows

ξt=σξBt+bξt+Xt, ηt=σηWt+bηt+Yt, (2.11) whereB andW are Brownian motions,XandY are pure jump zero mean martingales, and the processesB, W, X andY are mutually independent. Then we get

Vt=x+bη

Z t 0

e^−ξs−ds+ση

Z t 0

e^−ξs−dWs+Nt, whereN_t=Rt

0e^−ξs−dY_sis a pure jump local martingale. On the other hand using Itô’s formula, we have

Ht=e^ξt = 1 + Z t

0

e^ξs−dξs+1 2

Z t 0

e^ξs−d[ξ, ξ]^c_s+X

s≤t

e^ξs−(e^∆ξs−∆ξs−1)

= 1 + bξ+σ_ξ² 2

!Z t 0

e^ξs−ds+σξ

Z t 0

e^ξs−dBs+Net+X

s≤t

e^ξs−

e^∆ξs−∆ξs−1 ,

whereNes=Rt

0e^ξsdXsis a pure jump local martingale. Therefore, we conclude that [H, V]_t=

σ_ξ

Z t 0

e^−ξs−dB_s, σ_η Z t

0

e^−ξs−dW_s

t

+X

s≤t

∆V_s∆H_s= 0 a.s.,

since∆V_s=e^−ξs−∆η_s,∆H_s=H_s−(e^∆ξs−1)and the fact thatξandη are independent and do not jump simultaneously a.s. This implies that

Ut=x+ Z t

0

Hs−dVs+ Z t

0

Vs−dHs=x+ Z t

0

e^ξs−dVs+ Z t

0

Vs−dHs.

Using the expressions ofH andV, we deduce that

Ut=x+bηt+σηWt+ Z t

0

e^ξs−dNs+ bξ+σ_ξ² 2

!Z t

0

V_s−e^ξs−ds +σξ

Z t 0

Vs−e^ξs−dBs+ Z t

0

Vs−dNes+X

s≤t

Vs−e^ξs−

e^∆ξs−∆ξs−1

=x+Kt+K_t^c+bηt+ bξ+σ_ξ² 2

!Z t

0

U_s−ds+X

s≤t

U_s− e^∆ξs−∆ξs−1 ,

where

Kt = Z t

0

e^ξs−dNs+ Z t

0

Vs−dNes=Yt+ Z t

0

Us−dXs, K_t^c = σ_ηW_t+σ_ξ

Z t 0

U_s−dB_s.

From the definition ofKandK^c, and the mutual independence ofB,W,N andNe, we get for the continuous part of the quadratic variation ofU

[U, U]^c_t= [K^c, K^c]t=σ_η²t+σ²_ξ Z t

0

U_s−² ds.

Putting all the pieces together in identity (2.10), we have

f(Ut)−f(x) =Mt+bη

Z t 0

f⁰(Us−)ds+ bξ+σ_ξ² 2

!Z t 0

f⁰(Us−)Us−ds

+X

s≤t

f⁰(U_s−)U_s−

e^∆ξs−∆ξ_s−1

+σ_η² 2

Z t 0

f⁰⁰(U_s−)ds+σ²_ξ 2

Z t 0

f⁰⁰(U_s−)U_s−² ds

+X

s≤t

(f(U_s)−f(U_s−)−∆U_sf⁰(U_s−))

whereM is a local martingale starting from 0 and M describes the integration with respect to K and K^c in the expressions above. Using the fact that f ∈ K implies f(x) = 0 for x < 0 and x|f⁰(x)|+x²|f⁰⁰(x)| < C(f) < ∞, we deduce that Mt is a proper martingale as all other terms in the expression above have a finite absolute first moment. Furthermore applying the compensation formula to the jump part off(U_t)we get

E



 X

s≤t

f⁰(U_s−)U_s− e^∆ξs−∆ξs−1





=E Z t

0

f⁰(U_s−)U_s−

Z

y∈R

(e^y−y−1)Πξ(dy)

ds

.

Similarly, using the fact that∆Us= ∆ηswhen∆ηs6= 0and∆Us=Us−(e^∆ξs−1)when

∆ξ_s6= 0(see the definition ofU) we get

E



 X

s≤t

(f(Us)−f(Us−)−∆Us−f⁰(Us−))





=E Z t

0

Z

z∈R

(f(U_s−+z)−f(U_s−)−zf⁰(U_s−)) Πη(dz)ds

+E Z t

0

Z

y∈R

f(U_s−e^y)−f(U_s−)− e^y−1

f⁰(U_s−)U_s−

Π_ξ(dy)ds

. Finally, asf ∈ K, we derive

Eh f(Ut)i

−f(x) =bηE Z t

0

f⁰(U_s−)ds

+ bξ+σ_ξ² 2

! E

Z t 0

f⁰(U_s−)U_s−ds

+σ²_η 2 E

Z t 0

f⁰⁰(Us−)ds

+σ_ξ² 2 E

Z t 0

f⁰⁰(Us−)U_s−² ds

+E Z t

0

Z

z∈R

f(U_s−+z)−f(U_s−)−zf⁰(U_s−)

Πη(dz)ds

+E Z t

0

Z

y∈R

f(U_s−e^y)−f(U_s−)−yf⁰(U_s−)U_s−

Π_ξ(dy)ds

. and dividing byt, lettingtgo to0and recalling thatU˜0=xa.s., we obtain forf ∈ Kthe identity

L^(U)f(x) =bηf⁰(x) + bξ+σ²_ξ 2

!

xf⁰(x) +σ_η²

2 f⁰⁰(x) +σ²_ξ

2 f⁰⁰(x)x² +

Z

z∈R

f(x+z)−f(x)−zf⁰(x) Πη(dz) +

Z

y∈R

f(xe^y)−f(x)−yxf⁰(x)

Πξ(dy), (2.12)

and therefore the infinitesimal generator ofU satisfies L^(U)f(x) =L^(ξ)φ(lnx) +L^(η)f(x).

In order to finish the proof one only has to apply integration by parts.

The following Lemma will also be needed for our proof of Theorem 2.2.

Lemma 2.4. Assume that condition(2.1)is satisfied. Letν(dv)denote the measure in the left-hand side of formula (2.3). Then|ν|(dv)and henceν(dv)define finite measures on any compact subset of(0,∞)and for anya >0

z→∞lim z⁻¹|ν|((a, z)) = 0. (2.13) Proof. We only need to prove (2.13), as the finiteness of|ν|(dv)on compact subsets of (0,∞)follows from (2.13). It is sufficient to show the claims for1≥a >0. We integrate every term on the left-hand side of (2.3) fromatozand then divide byz. This shows that the limit goes to zero, asz→ ∞. We first note that

z→∞lim z⁻¹ Z z

a

xm(dx) = 0 and lim

z→∞z⁻¹ Z z

a

m(dx)

x ≤ lim

z→∞(az)⁻¹ Z ∞

a

m(dx) = 0.

Hence,

z→∞lim z⁻¹ Z z

a

Z ∞ v

m(dx)dv≤ lim

z→∞

z⁻¹

Z z a

xm(dx) + Z ∞

z

m(dx)

= 0,

z→∞lim z⁻¹ Z z

a

Z ∞ v

m(dx)

x ^dv= 0 and lim

z→∞z⁻¹ Z z

a

Z ∞ v

m(dx) x^{2 d}v= 0.

So far, we have checked that the terms in (2.3) that do not depend on the tail of the Lévy measure vanish under the transformation we made, asz → ∞. Now, we turn our attention to the terms that involve the Lévy measure ofξ. When we’ll be dealing with these integrals, the main trick that we will use is to change the order of integration.

First, we check that

lim sup

z→∞

z⁻¹ Z z

a

Z ∞ v

Π⁽⁻⁾_ξ lnx

v

m(dx)dv

≤lim sup

z→∞

z⁻¹ Z z

a

Z ev v

Π⁽⁻⁾_ξ lnx

v

m(dx)dv

+ lim sup

z→∞

z⁻¹

Π⁽⁻⁾_ξ (1) Z z

a

m(ev,∞)dv

= lim sup

z→∞

z⁻¹ Z z

a

Z ev v

Π⁽⁻⁾_ξ lnx

v

m(dx)dv≤lim sup

z→∞

z⁻¹ Z ez

a

Z x x/e

Π⁽⁻⁾_ξ lnx

v

dv m(dx)

= Z 1

0

Π⁽⁻⁾_ξ (w)e^−wdw

×lim sup

z→∞

z⁻¹ Z ez

a

xm(dx) = 0

where we have applied Fubini’s Theorem, a change of variables w = ln(x/v) and we have used the finiteness of E[|ξ1|] and henceforth the finiteness of the quantities R1

0 Π⁽⁻⁾_ξ w

exp(−w)dwandΠ⁽⁺⁾_ξ (1).

Next using Fubini’s Theorem and the monotonicity of Π⁽⁺⁾_ξ , we note that for any positive numberb,

lim sup

z→∞

z⁻¹ Z z

a

Z v 0

Π

(+) ξ

lnv

x

m(dx)dv

≤lim sup

z→∞

z⁻¹ Z z

0

Z v 0

Π

(+) ξ

lnv

x

m(dx)dv

= lim sup

z→∞

z⁻¹ Z z

0

x

Z ln(z/x) 0

Π⁽⁺⁾_ξ w

e^wdw m(dx)

≤lim sup

z→∞

z⁻¹ Z b

0

Π

(+)

ξ w

e^wdw Z z

0

x m(dx) + Z z

0

x

Z ln(z/x)∨b b

Π

(+)

ξ w

e^wdw m(dx)

!

≤Π

(+) ξ b

.

SinceΠ

(+)

(b)decreases to zero asbincreases, we see that

z→∞lim z⁻¹ Z z

a

Z v 0

Π

(+) ξ

lnv

x

m(dx)dv= 0.

Sinceηhas a finite mean andmis a finite measure lim sup

z→∞

z⁻¹ Z z

a

1 v

Z v 0

Π⁽⁺⁾_η (v−x)m(dx)dv

≤lim sup

z→∞

z⁻¹

Π⁽⁺⁾_η (a) lnz a

+ Z z

a

1 v

Z v v−a

Π⁽⁺⁾_η (v−x)m(dx)dv

= lim sup

z→∞

z⁻¹ Z z

0

m(dx)

Z (x+a)∧z a∨x

Π⁽⁺⁾_η (v−x)^dv v

!

≤ Z a

0

Π

(+) η (s)ds

× lim

z→∞(az)⁻¹ Z z

0

m(dx) = 0.

Similarly, we estimate the following integral lim sup

z→∞

z⁻¹ Z z

a

Z ∞ v

1 w²

Z w 0

Π

(+)

η (w−x)m(dx)dwdv

≤lim sup

z→∞

z⁻¹

Π⁽⁺⁾_η (a) lnz a

+ Z z

a

Z ∞ v

1 w²

Z w w−a

Π⁽⁺⁾_η (w−x)m(dx)dwdv

= lim sup

z→∞

z⁻¹ Z z

a

Z ∞ v−a

Z x+a v∨x

1 w²Π

(+)

η (w−x)dw m(dx)dv

≤ Z a

0

Π⁽⁺⁾_η (s)ds

×lim sup

z→∞

z⁻¹ Z z

a

1

v²m(v−a,∞)dv= 0.

As for the remaining two integrals, we split the innermost integrals at the point x=v+aso that Π⁽⁻⁾_η (x−v) = Π⁽⁻⁾_η (a)and similarly estimate the resulting two terms to get

lim sup

z→∞

z⁻¹ Z z

a

1 v

Z ∞ v

Π⁽⁻⁾_η (x−v)m(dx)dv

= lim sup

z→∞

z⁻¹ Z z

a

Z ∞ v

1 w²

Z ∞ w

Π⁽⁻⁾_η (x−w)m(dx)dwdv= 0.

Thus, we verify (2.13) and conclude the proof of Lemma 2.4.

Now that we have established Proposition 2.3 and Lemma 2.4, we are ready to complete the proof of Theorem 2.2.

Proof of Theorem 2.2. Take an infinitely differentiable functionf with compact support in(0,∞)and letg(x) :=xf⁰(x). We use (2.6), (2.9), and the identityg(x) =Rx

0 g⁰(v)dvto get,

Z ∞ 0

L^(ξ)φ(lnx)m(dx) =bξ

Z ∞ 0

g(x)m(dx) +σ_ξ² 2

Z ∞ 0

xg⁰(x)m(dx) +

Z ∞ 0

Z x 0

g⁰(v)Π⁽⁻⁾_ξ lnx

v

dvm(dx) +

Z ∞ 0

Z ∞ x

g⁰(v)Π⁽⁺⁾_ξ lnv

x

dvm(dx)

= Z ∞

0

g⁰(v)

bξ

Z ∞ v

m(dx)

dv+ Z ∞

0

g⁰(v) σ²_ξ

2 vm(dv)

!

+ Z ∞

0

g⁰(v) Z ∞

v

Π⁽⁻⁾_ξ lnx

v

m(dx)

dv +

Z ∞ 0

g⁰(v) Z v

0

Π⁽⁺⁾_ξ lnv

x

m(dx)

dv=: (g⁰, F1),

where the interchange of integrals is permitted due to claims of Lemma 2.4.

Next, substitutingf⁰(x) =g(x)/xandf⁰⁰(x) =g⁰(x)/x−g(x)/x², we get Z ∞

0

L^(η)f(x)m(dx) =bη

Z ∞ 0

g(x)

x m(dx) +σ_η² 2

Z ∞ 0

g⁰(x) x −g(x)

x²

m(dx) +

Z ∞ 0

Z ∞ 0

g⁰(x+w)

x+w −g(x+w) (x+w)²

Π⁽⁺⁾_η (w)dwm(dx) +

Z ∞ 0

Z ∞ 0

g⁰(x−w)

x−w −g(x−w) (x−w)²

Π⁽⁻⁾_η (w)dwm(dx).

Again, using the identityg(x) =Rx

0 g⁰(v)dvand the fact thatgis a function with compact support on(0,∞), we get after careful calculations and an appeal again to Lemma 2.4 for interchange of integration

Z ∞ 0

L^(η)f(x)m(dx) =b_η Z ∞

0

g⁰(v) Z ∞

v

m(dx) x ^dv +σ_η²

2 Z ∞

0

g⁰(x)m(dx)

x −

Z ∞ 0

g⁰(v) Z ∞

v

m(dx) x^{2 d}v

+ Z ∞

0

g⁰(v)1 v

Z v 0

Π

(+)

η (v−x)m(dx)dv

− Z ∞

0

g⁰(w) Z ∞

w

1 v²

Z v 0

Π

(+)

η (v−x)m(dx)dvdw +

Z ∞ 0

g⁰(v)1 v

Z ∞ v

Π⁽⁻⁾_η (x−v)m(dx)dv

− Z ∞

0

g⁰(w) Z ∞

w

1 v²

Z ∞ v

Π⁽⁻⁾_η (x−v)m(dx)dvdw := (g⁰, F2).

We arrange the above expressions in the formR

g⁰(x)ν(dx), whereν(dx) := F₁(dx) + F₂(dx)is the same as in Lemma 2.4. From Lemma 2.4, we conclude thatν(dx)defines a finite measure on every compact subset of(0,∞)and henceforth we consider it as a distribution in Schwartz’s sense. Thus we get

0 = Z ∞

0

L^(U)f(x)m(dx) = (g⁰, ν) = (g, ν⁰) = (xf⁰, ν⁰) = (f⁰, xν⁰) = (f,(xν⁰)⁰), for each infinitely differentiable functionf with compact support in(0,∞)and deriva- tives in the sense of Schwartz. Therefore using Schwartz theory of distributions for ν(dx), we get thatxν⁰(dx) =Cdxand therefore

ν(dx) = (Clnx+D)dx.

Next, we show that C = D = 0. Note that from (2.13) with a = 1, we have lim_z→+∞z⁻¹Rz

1 ν(dv) = 0. Comparing this with 0 = lim

z→+∞z⁻¹ Z z

1

(Clnx+D)dx= lim

z→+∞(Clnz−C+D) we verify thatC=D= 0. Thus the proof of Theorem 2.2 is complete.

The next result is an almost immediate corollary of Theorem 2.3, and in particular of formula (2.3). See also Corollary 3.14 for a stronger result in a particular case when ηis a Brownian motion with drift.

Corollary 2.5. Assume that condition (2.1)is satisfied. Ifσ_ξ²+σ²_η>0thenm(dx)has a continuous density onR\ {0}.

Proof. The absolute continuity ofI(ξ, η)and boundedness of its derivative on compact subsets of(0,∞), whenσ_ξ²+σ²_η >0is immediate from (2.3). Letk(x)be the density of m(dx). To show the continuity ofk(x), we investigate all integral terms in (2.3): all of them, except possibly the ones involvingΠ⁽⁺⁾_ξ andΠ⁽⁻⁾_ξ , are clearly continuous. Let us check continuity of these remaining two terms. Fixv >0 andv/4 > a >0. Note that, for any realhsuch that|h|< v/4, we have

Z v+h 0

Π⁽⁺⁾_ξ

lnv+h x

k(x)dx=

Z v+h−a 0

Π⁽⁺⁾_ξ

lnv+h x

k(x)dx +

Z v+h v+h−a

Π⁽⁺⁾_ξ

lnv+h x

k(x)dx.

AsΠ

(+)

ξ is continuous and decreasing we verify the dominated convergence theorem applies, ash→0, by boundingΠ⁽⁺⁾_ξ in the first term andk(x)in the second. This shows that all integral terms in (2.3) are continuous inv and hencek(v) is continuous. The computation forΠ⁽⁻⁾_ξ is the same whereas for v <0 we study I(ξ,−η)with the same effect.

3 Exponential functionals with respect to Brownian motion with drift

In the next two sections, we study the special case whenη_t=µt+σB_tis a Brownian motion with drift, so that the exponential functional is now defined as

I_µ,σ:=

Z ∞ 0

e^ξt−(µdt+σdB_t). (3.1) We still work under Assumption 2.1, note that the conditionE[|η1|]<∞is clearly sat- isfied. From now on, we assume that σ > 0, and in order to simplify notations we will writeψ(s) = ψ_ξ(s). Note that formula (3.1) implies I_µ,σ =^d σI_µ/σ,1, therefore it is sufficient to study the exponential functional withσ= 1.

The following three quantities will be very important in what follows ρ := sup{z≥0 : E

e^zξ1

<∞}, ˆ

ρ := sup{z≥0 : E e^−zξ1

<∞}, (3.2)

θ := sup{z≥0 : E e^zξ1

≤1}.

In view of (2.2), it is clear that ρ= sup

z≥0 :

Z ∞ 1

e^zxΠξ(dx)<∞

, ρˆ= sup

z≥0 : Z ∞

1

e^zxΠξ(−dx)<∞

. Thusρ >0(ρ >ˆ 0) if and only if the measureΠξ(dx)has exponentially decaying positive (negative) tail. In this case the Lévy-Khintchine formula (2.2) implies that the Laplace exponentψ(z)can be extended analytically in a strip−ˆρ <Re(z)< ρ. It is clear from (3.2) that0≤θ≤ρ. At the same time, due to Assumption 2.1 we haveE[ξ₁] =ψ⁰(0)<0, which implies thatθ >0if and only ifρ >0.

In the next Lemma we collect some simple analytical properties of the Laplace ex- ponentψ(z).

Lemma 3.1. Assume that ξ satisfies condition (2.1) and that ρ > 0. Then ψ(s) has no zeros in the strip0 < Re(s) < θ. Moreover if ξ has a non-lattice distribution and ψ(θ) = 0, thenθis the unique zero ofψ(s)in the strip0<Re(s)≤θand the unique real zero in the interval(0, ρ).

Proof. Assume that0<Re(s)< θ. Since e^Re(ψ(s))=

E e^sξ1

≤Eh

e^Re(s)ξ1i

=e^ψ(Re(s))

we conclude that Re(ψ(s))≤ψ(Re(s))<0, thereforeψ(s)6= 0in the strip0<Re(s)< θ. Next, assume thatψ(θ+iy) = 0for somey 6= 0andξhas a non-lattice distribution.

Then the characteristic function of the probability measure e^θvP(ξ1 ∈ dv)is equal to one aty, therefore it has to be a lattice distributed probability measure, see [34, p 306, Theorem 5] which contradicts our assumption.

In order to prove thatθis the unique real zero ofψ(s)on the interval(0, ρ), we note that the first formula in (2.2) implies that

ψ⁰⁰(s) =σ_ξ²+ Z

R

x²e^sxΠξ(dx)>0,

thereforeψ(s)is convex on(0, ρ)and it has at most one positive root atθ. Next, let us introduce two other important objects

Jα:=

Z ∞ 0

e^αξtdt, and V := J₁² J2

. (3.3)

We will frequently use the following result, its proof follows immediately from Lemma 2.1 in [27]:

Proposition 3.2. Assume that ξ satisfies condition (2.1). For all z ∈ C in the strip

−1≤Re(z)< θ/αwe haveE[J_α^z]<∞.

Our main object of interest is the probability density function ofIµ,σ, which we will denote byk(x)(or byk_µ,σ(x)if we need to stress dependence on parameters). In the next Lemma, we collect some simple properties ofk(x).

Lemma 3.3. Assume thatξsatisfies condition(2.1). The law ofI_µ,σhas a continuously differentiable densitykµ,σ(x)which is given by

kµ,σ(x) = Z Z

R²+

1 σ√

2πze^{−(x−µy)22zσ2} P(J1∈dy;J2∈dz). (3.4) Moreover, both functionskµ,σ(x)andk⁰_µ,σ(x)are uniformly bounded onR^{and if} µ≤0 thenk_µ,σ(x)is decreasing onR+.

Proof. Expression (3.4) follows by conditioning onξand the fact that Z ∞

0

e^f(t)(µdt+σdBt)=^d N

µ Z ∞

0

e^f(s)ds;σ² Z ∞

0

e^2f(s)ds

,

where N(a, b) denotes a normal random variable with mean a and variance b. The continuity ofkµ,σ(x) follows from the Dominated Convergence Theorem and the fact thatEh

J₂⁻¹²i

<∞, see Proposition 3.2.

Next, we observe that the function|v|e^−v2 is bounded onRand therefore for some C >0we have

Z Z

R²₊

(x−µy) σ³√

2πz³e^{−(x−µy)22zσ2} P(J₁∈dy;J₂∈dz)

≤CE[J₂⁻¹]<∞,

where the last inequality follows from Proposition 3.2. This shows that we can differen- tiate the right-hand side of (3.4) and obtain

k_µ,σ⁰ (x) =− Z Z

R²₊

(x−µy) σ³√

2πz³e^{−(x−µy)22zσ2} P(J₁∈dy;J₂∈dz), (3.5) and from the above discussion it follows that|k_µ,σ⁰ (x)| ≤ CE[J₂⁻¹] <∞ for allx ∈ R^. Finally, forµ ≤0and x >0 we check thatk_µ,σ⁰ (x) <0 (see (3.5)), thereforekµ,σ(x)is decreasing.

Our main tool for studying the properties of kµ,σ(x)will be the Mellin transform of Iµ,σ, which is defined for Re(s) = 1as

Mµ,σ(s) :=E[(Iµ,σ)^s−11_{Iµ,σ>0}] = Z ∞

0

x^s−1kµ,σ(x)dx. (3.6) Later we will extend this definition for a wider range ofs, but a priori it is not clear why this object should be finite for Re(s) 6= 1. Also, this choice of truncated random vari- able may seem awkward, since we only use the information about the densityk_µ,σ(x) forx ≥ 0. However, it is easy to see that the Mellin transform Mµ,σ(s) uniquely de- termineskµ,σ(x)forx≥0whileM_−µ,σ(s)uniquely determineskµ,σ(x)forx≤0. This follows from the simple fact thatkµ,σ(−x) =k_−µ,σ(x)(clearlyIµ,σ

=d −I_−µ,σ, see (3.1)).

Moreover, later it will be clear that our definition of the Mellin transform is in fact quite natural, sinceMµ,σ(s)satisfies the crucial functional equation (3.13), which will lead to a wealth of interesting information aboutk_µ,σ(x).

As a first step in our study of the Mellin transform Mµ,σ(s)we obtain its analytic continuation into a vertical strip in the complex plane.

Lemma 3.4. Assume that ξ satisfies condition (2.1). The function Mµ,σ(s) can be extended to an analytic function in the strip−1<Re(s)<1 +θ, except for a simple pole ats= 0with residuek(0). Moreover, for allsin the strip−1<Re(s)<1 +θwe have

M_µ,σ(s) = k(0)

s +

Z 1 0

(k(x)−k(0))x^s−1dx+ Z ∞

1

k(x)x^s−1dx, (3.7) and for allsin the strip−1<Re(s)<0it is true that

Mµ,σ(s) =−1 s

Z ∞ 0

x^sk⁰(x)dx. (3.8)

Proof. First of all, sincek(x)is a probability density, it is integrable on[0,∞). Also, due to Lemma 3.3, we know thatk(x) =k(0) +k⁰(0)x+o(x)asx→0⁺, these two facts imply thatMµ,σ(s)exists for allsin the strip0<Re(s)≤1.

Next, one can easily check that identity (3.7) is valid forsin the strip0<Re(s)≤1. Sincek(x)−k(0) =k⁰(0)x+o(x), asx→0⁺ we see that the first integral in the right- hand side of (3.7) extends analytically into the larger strip−1 <Re(s) <1, while the second integral is analytic in the half-plane Re(s)<1. Thus (3.7) provides an analytic

continuation ofMµ,σ(s)into the strip−1<Re(s)<1and it is clear thatMµ,σ(s)has a simple pole ats= 0with residuek(0).

Next, we note that for−1<Re(s)<0we have Z ∞

1

k(x)x^s−1dx= Z ∞

1

(k(x)−k(0))x^s−1dx−k(0) s .

Combining this expression with (3.7) and applying integration by parts we obtain (3.8).

If θ = 0, then the proof is finished. However, if θ > 0 we still have to prove that Mµ,σ(s)<∞for1< s <1 +θ, and this requires a little bit more work. The proof will be based on certain special functions. The confluent hypergeometric function (see section 9.2 in [16] or chapter 6 in [11]) is defined as

1F1(a, b, z) =X

n≥0

(a)n

(b)n

zⁿ

n!, (3.9)

where(a)n=a(a+ 1). . .(a+n−1)is the Pochhammer symbol. Using the ratio test it is easy to see that the series in (3.9) converges for allz ∈C^{, thus1}F1(a, b, z)is an entire function ofz. We will also need the parabolic cylinder function, which is defined as

D_p(z) = 2^p2e^−z

2 4

" √ π Γ ^1−p₂ ¹F₁

−p 2,1

2;z² 2

+

√2πz Γ −^p₂1F₁

1−p 2 ,3

2;z² 2

#

. (3.10) Note that the parabolic cylinder function is analytic function ofpand z. See sections 9.24-9.25 in [16] for more information on the parabolic cylinder function. We will prove thatM_µ,1(s)exists for allsin the strip Re(s) ∈(0,1 +θ)and everywhere in this strip we have

Mµ,1(s) = Γ(s)

√2πE

J

s−1 2

2 e^−µ

2

4VD_−s

−µ√ V

. (3.11)

Let us assume first that Re(s) = 1. Then using (3.4) and (3.6) we conclude that Mµ,1(s) =E

Z ∞ 0

x^s−1 1

√2πJ₂e⁻

(x−µJ1 )2 2J2 dx

= 1

√2πE 1

√J₂e^−µ

2 2 V Z ∞

0

x^s−1e^−2J12x2+

µJ1 J2 x

dx

. Performing the change of variablesx=u√

J2 and using the following integral identity (formula 9.241.2 in [16])

Z ∞ 0

u^s−1e^−u

2

2 −uzdu= Γ(s)e^z

2

4 D_−s(z), Re(s)>0, we obtain equation (3.11).

Thus, we have established that (3.11) is true for allson the vertical line Re(s) = 1. Now, we will perform analytic continuation into a larger domain. Formulas 9.246 in [16], give us the following asymptotic expansions: forz∈R

D_−s(z) =





 O

z^−se^−z42

, z→+∞,

O

z^−se^−z42 +O

z^s−1e^z42

, z→ −∞. ^(3.12) Assume thatµ <0 ands∈(0,1 +θ)orµ >0ands∈(0,1). Then, from (3.12) and the fact thatD_s(z)is a continuous function ofzwe find that there exists a constantC₁ >0 such that

|e^−µ

2

4 zD−s −µ√ z

|< C1