On moments of integral exponential functionals of additive processes

# On moments of integral exponential functionals of additive processes

Paavo Salminen
Faculty of Science and Engineering,
FIN-20500 Åbo, Finland,
phsalmin@abo.fi
Lioudmila Vostrikova
Université d’Angers,
Département de Mathématiques,
F-49045 Angers Cedex 01, France,
vostrik@univ-angers.fr
###### Abstract

Let be a real-valued additive process. In this paper we study the integral exponential functionals of namely, the functionals of the form

 Is,t=∫tsexp(−Xu)du,0≤s

Our main interest is focused on the moments of of order In case the Laplace exponent of exists for positive values of the parameter, we derive a recursive (in ) integral equation for the moments. This yields a multiple integral formula for the entire positive moments of . From these results emerges an easy-to-apply sufficient condition for the finiteness of all the entire moments of . The corresponding formulas for Lévy processes are also presented. As examples we discuss the finiteness of the moments of when is the first hit process associated with a diffusion. In particular, we discuss Bessel processes and geometric Brownian motions.

Keywords: Additive process, process with independent increments, Lévy process, subordinator, first hitting time, diffusion, Bessel process, geometric Brownian motion

AMS Classification: 60J75, 60J60. 60E10

## 1 Introduction

Let be a real valued additive process, i.e., a strong Markov process with independent increments having càdlàg sample paths which are continuous in probability (cf. Sato [15] p.3). Important examples of additive processes are:

(a)   Deterministic time transformations of Lévy processes, that is, if is a Lévy process and is an increasing continuous function such that then is an additive process.

(b)   Integrals of deterministic functions with respect to a Lévy process, that is, if is a Lévy process and is a measurable function then

 Zt:=∫t0g(s)dLs,t≥0,

is an additive process.

(c)   First hit processes of one-dimensional diffusions, that is, if is a diffusion taking values in starting from 0, and drifting to then

 Ha:=inf{t:Yt>a},a≥0,

is an additive process.

Of course, Lévy processes themselves constitute a large and important class of additive processes.

The basic tool in our analysis is the Laplace exponent of The structure of this is revealed in the the Lévy-Khintchine representation of the infinitely divisible distribution of We recall this briefly from Sato [15] Theorem 9.8 p.52. Therein the characteristic function of is given as follows

 E(eiλXt)=eΨ(t,λ),λ∈R, t≥0 (1)

where

 Ψ(t,λ):=exp(iλb(t)−12λ2c(t)+∫R(eiλx−1−iλx1{|x|<1})νt(dx)),

is continuous, is continuous and non-decreasing, and is a measure on such that

for all and

for all and

for all with

 νs(B)→νt(B)ass→t.

Notice that since is assumed to be continuous in probability it follows that is for all continuous.

The aim of this paper is to study integral exponential functionals of i.e., functionals of the form

 Is,t:=∫tsexp(−Xu)du,0≤s

in particular, the moments of We refer also to a companion paper [12] where stochastic calculus is used to study the Mellin transforms of when the underlying additive process is a semi martingale with absolutely continuous characteristics.

The main result of the paper is a recursive equation, see (7) in Theorem 2.3, which generalizes the formula for Lévy processes presented in Urbanik [16] Example 3.4 p.309 and Carmona, Petit and Yor [4] Proposition 3.3 p.87, see also Bertoin and Yor [2] Section 3.1 p.195. This formula for Lévy processes is also displayed below in (18). In Epifani, Lijoi and Prünster [7] an extension of the Lévy process formula to integral functionals up to of increasing additive processes is discussed, and their formula (7) in Proposition 5 p.798 can be seen as a special case of our formula (15) – as we found out after finishing our work. Such an extension is also indicated in [2] on p.196. In spite of these closely related results we feel that it is worthwhile to provide a more thorough discussion of this interesting topic. We also give new (to our best knowledge ) applications of the formulas for first hit processes of diffusions and present explicit results for Bessel processes and geometric Brownian motions.

In mathematical finance the interest in integral exponential functionals is coming from the studies of perpetuities containing liabilities, perpetuities under the influence of economical factors (see, for example, Kardaras and Robertson [11]), and also of pricing Asian options and related questions (see, for instance, Dufresne [6], Carr and Wu [5], Jeanblanc, Yor, Chesney [9] and references therein). We refer to the survey paper [2] for further results, potential applications, and many references, in particular, for Lévy processes but also for additive processes. For applications of increasing additive processes in Bayesian statistics we refer to [7] and references therein. Additive processes can also be found in representations of self-decompasable laws, see Sato [14] and Jeanblanc, Pitman and Yor [10].

The paper is structured as follows. Next section contains the main results of the paper. In particular, the recursive equation for the entire moments is established. We also give formulas for Lévy processes. The paper is concluded with examples on first hitting time processes. It is proved that for Bessel processes drifting to the integral exponential functional of the first hit process has all the moments and for geometric Brownian motion the corresponding functional has only some moments.

## 2 Main results

Let be an additive process and define for and

 m(α)s,t:=E(Iαs,t)=E((∫ts\rm e−Xudu)α),α≥0, (3)

and

 m(α)t:=m(α)0,t,m(α)∞:=m(α)0,∞.

In this section we derive a recursive integral equation for under the following assumption:

(A) has for all a finite Laplace exponent for positive values of the parameter, that is, there exists for all and for all a function such that

 E(e−λXt)=e−Φ(t;λ). (4)

and

 Φ(t;0)=Φ(0;λ)=0. (5)

If is a Lévy process we write (with a slight abuse of the notation) formula (4) as

 E(e−λXt)=e−tΦ(λ). (6)

The functions and are clearly connected as As pointed out in Introduction, is continuous and, therefore also is continuous. This property has an important – obvious – consequence formulated in the next lemma which is easily proved, e.g., with Jensen’s inequality.

###### Lemma 2.1.

Under assumption (A), for it holds for all

Assumption (A) is valid, in particular, for increasing additive processes. Important examples of these are the first hit processes for diffusions (cf. (c) in Introduction). Moments of exponential functionals of some first hit processes are discussed in the next section. Assumption (A) holds also for additive processes of type (a) in Introduction if the underlying Lévy process fullfills Assumption (A).

###### Remark 2.2.

In the case when is a semi-martingale with absolutely continuous characteristics, the sufficient condition (see Proposition 1 in [12]) for the existence of the Laplace exponent as in Assumption (A) in terms of the measure is as follows: for and

 ∫{|x|>1}(e−λx+|x|)νt(dx)<∞⇒E(e−λXt)<∞.

The main result of the paper is given in the next theorem. In the proof we are using similar ideas as in [4].

###### Theorem 2.3.

For and the moments are finite and satisfy the recursive equation

 m(α)s,t=α∫tsm(α−1)u,t\rm e−(Φ(u;α)−Φ(u;α−1))du. (7)
###### Proof.

We start with by introducing the shifted functional via

 ˆIs,t:=∫t−s0e−(Xu+s−Xs)du.

Clearly,

 ˆIs,t=eXsIs,t=eXs∫tse−Xudu, (8)

and we have

 ddsIαs,t=αIα−1s,tddsIs,t=−αIα−1s,te−Xs=−αˆIα−1s,te−αXs.

Consequently,

 Iαs,t−Iα0,t=−α∫s0ˆIα−1u,te−αXudu

The independence of increments implies that and are independent. Hence,

 E(Iαs,t−Iα0,t)=−α∫s0E(ˆIα−1u,t)E(e−αXu)du. (9)

Clearly, a.s. when Hence, applying monotone convergence in (9) yields

 E(Iα0,t)=α∫t0E(ˆIα−1u,t)E(e−αXu)du. (10)

Putting (9) and (10) together results to the equation

 E(Iαs,t)=α∫tsE(ˆIα−1u,t)E(e−αXu)du. (11)

From (8) evoking the independence of and we have

 E(ˆIα−1u,t)=E(Iα−1u,t)/E(e−(α−1)Xu). (12)

Finally, using (12) in (11) and recalling (4) yields (7). The claim that is finite follows by induction from the recursive equation (7) and Lemma 2.1. ∎

###### Remark 2.4.

In [12] the recursive equation (7) is derived via stochastic calculus in case the additive process is a semimartingale with absolutely continuous characteristics.

###### Corollary 2.5.

Let be a Lévy process with the Laplace exponent as in (6). Then the recursive equation (7) for and is equivalent with

 m(α)t=α\rm e−tΦ(α)∫t0m(α−1)u\rm euΦ(α)du. (13)
###### Proof.

Put in (7) to obtain

 m(α)t=α∫t0m(α−1)u,t\rm e−u(Φ(α)−Φ(α−1))du. (14)

Consider

 m(α−1)u,t = E((∫tu\rm e−Xvdv)α−1) = E(\rm e−(α−1)Xu(∫tu\rm e−(Xv−Xu)dv)α−1) = \rm e−uΦ(α−1)E((∫t−u0\rm e−(Xv+u−Xu)dv)α−1) = \rm e−uΦ(α−1)E((∫t−u0\rm e−Xvdv)α−1) = \rm e−uΦ(α−1)m(α−1)t−u.

Subsituting this expression in (14) and changing variables yield the claimed equation. ∎

For positive integer values on the recursive equation (7) can be solved explicitly to obtain the formula (15) in the next proposition. However, we offer another proof highlighting the symmetry properties present in the expressions of the moments of the exponential functional.

###### Proposition 2.6.

For and it holds

 m(n)s,t=n!∫tsdt1∫tt1dt2⋯ (15) ⋯∫ttn−1dtnexp(−n∑k=1(Φ(tk;n−k+1)−Φ(tk;n−k))).

In particular, if and only if the multiple integral on the right hand side of (15) is finite.

###### Proof.

Let and consider

 m(n)s,t = E((∫ts\rm e−Xudu)n) = E(∫ts⋯∫ts\rm e−Xt1−⋯−Xtndt1…dtn) = n!E(∫tsdt1\rm e−Xt1∫tt1dt2\rm e−Xt2…∫ttn−1dtn\rm e−Xtn) = n!∫tsdt1∫tt1dt2…∫ttn−1dtnE(\rm e−(Xt1+⋯+Xtn)),

where, in the third step, we use that is symmetric. By the independence of the increments

 E(\rm e−αXt)=E(\rm e−α(Xt−Xs)−αXs)=E(\rm e−α(Xt−Xs))E(\rm e−αXs).

Consequently,

 E(\rm e−α(Xt−Xs))=E(\rm e% −αXt)/E(\rm e−αXs)=\rm e−(Φ(t;α)−Φ(s;α)).

Since,

 Xt1+⋯+Xtn=n∑k=1(n−k+1)(Xtk−Xtk−1),t0:=0,

we have

 m(n)s,t=n!∫tsdt1∫tt1dt2… ⋯∫ttn−1dtnexp(−n∑k=1(Φ(tk;n−k+1)−Φ(tk−1;n−k+1))).

Using here (5) yields the claimed formula (15). The statement concerning the finiteness of follows by applying the monotone convergence theorem as on both sides of (15). ∎

In the next corollary we give a sufficient condition for to be finite for all .

###### Corollary 2.7.

Variable has all the positive moments if

 ∫∞0e−(Φ(s;n)−Φ(s;n−1))ds<∞ (16)

for all .

###### Proof.

From (15) we have

 m(n)t≤n!n∏k=1∫∞0e−(Φ(s;n)−Φ(s;n−1))ds. (17)

The right hand side of (17) is finite if (16) holds. Let in (17). By monotone convergence, and the claim is proved. ∎

Formula (18) below extends the corresponding formula for subordinators found [16], see also [2] p.195, for Lévy processes satisfying Assumption (A). It is a straightforward implification of Proposition 2.6.

###### Corollary 2.8.

Let be a Lévy process with the Laplace exponent as in (6) and define Then

 m(n)∞:=E(In∞)=⎧⎪⎨⎪⎩n!n∏k=1Φ(k),if  n

.

###### Example 2.9.

A much studied functional is obtained when taking with where is a standard Brownian motion. In the papers by Dufresne [6] and Yor [17] (see also Salminen and Yor [13]) it is proved that

 I∞:=∫∞0e−(σWs+μs)ds∼H(δ)0, (19)

where is the first hitting time of 0 for a Bessel process of dimension starting from and means ”is identical in law with”. In particular, it holds

 ∫∞0exp(−(2Ws+μs))ds∼12Zμ, (20)

where is a gamma-distributed random variable with rate 1 and shape We refer to [6] for a discussion showing how the functional on the left hand side of (19) arises as the present value of a perpetuity in a discrete model after a limiting procedure. Since the Lévy exponent in this case is

 Φ(λ)=λμ−12λ2σ2,

the criterium in Corollary 2.8 yields

 E(In∞)<∞⇔n<2μ/σ2,

which readily can also be checked from (20).

## 3 First hit processes of one-dimensional diffusions

We recall first some facts concerning the first hitting times of one-dimensional (or linear) diffusions. Let now be a linear diffusion taking values in an interval To fix ideas assume that equals or or and that

 limsups→∞Ys=+∞a.s. (21)

Assume and consider for the first hitting time

 Ha:=inf{s:Ys>a}.

Defining it is easily seen – since is a strong Markov process – that is an increasing purely discontinuous additive process starting from 0. Moreover, from assumption (21) it follows that a.s. for all The process satisfies Assumption (A) in Section 2. Indeed, using the well known characterization of the Laplace transform of we have

 Ev(e−βXt)=Ev(e−βHt+v)=ψβ(v)ψβ(t+v),t≥0, (22)

where is the expectation associated with starting from and is a unique (up to a multiple) positive and increasing solution of the ODE

 Gf(x)=βf(x) (23)

satisfying the appropriate boundary condition at in case and is reflecting. In (23) denotes the differential operator associated with In the absolutely continuous case is of the form

 Gf(x)=12σ2(x)f′′(x)+μ(x)f′(x),f∈C2(I),x∈I,

where and are continuous functions. For details about diffusions (and further references), see Itô and McKean [8], and [3]. The Laplace transform of can also be represented as follows

 Ev(e−βXt)=exp(−∫t+vvS(du)∫∞0(1−e−βx)n(u,dx)), (24)

where is the scale function, and is a kernel such that for all and

 ∫t+vv∫∞0(1∧x)n(u,dx)S(du)<∞.

Representation (24) clearly reveals the structure of as a process with independent increments. Comparing with the notation in Introduction, we have

 νt(dx)=∫t+vvn(u,dx)S(du).

From (22) and (24) we may conclude that

 ∫∞0(1−e−βx)n(u,dx)=limw→u−1−Ew(e−βXu)S(u)−S(w). (25)

We now pass to present examples of exponential functionals of first hit processes. Firstly, we study Bessel processes satisfying (21) and show, in particular, that the exponential functional of the first hit process has all the moments. In our second example it is seen that the exponential functional of the first hit process of geometric Brownian motion has only finitely many moments depending on the values of the parameters.

###### Example 3.1.

Bessel processes. Let be a Bessel process starting from . The differential operator associated with is given by

 Gf(x)=12f′′(x)+δ−12xf′(x),x>0,

where is called the dimension parameter. From [3] we extract the following information

for the boundary point is entrance-not-exit and (21) holds,

for the boundary point is non-singular and (21) holds when the boundary condition at is reflection,

for (21) does not hold.

In case when (21) is valid the Laplace exponent for the first hit process is given for and by

 Ev(e−βXt)=ψβ(v)ψβ(t)=v1−δ/2Iδ/2−1(v√2β)t1−δ/2Iδ/2−1((t+v)√2β), (26)

where is the expectation associated with when started from and denotes the modified Bessel function of the first kind. For simplicity, we wish to study the exponential functional of when To find the Laplace exponent when we let in (26). For this, recall that for

 Ip(v)≃1Γ(p+1)(v2)pas v→0. (27)

Consequently,

 E0(e−βXt) = limv→0Ev(e−βXt) = 1Γ(ν+1)(√2β2)δ/2−1tδ/2−1Iδ/2−1(t√2β) =: e−Φ(t;β).

The validity of (16), that is, the finiteness of the positive moments, can now be checked by exploiting the asymptotic behaviour of saying that for all (see Abramowitz and Stegun [1], 9.7.1 p.377)

 Ip(t)≃et/√2πtas t→∞. (28)

Indeed, for

 e−(Φ(t;n)−Φ(t;n−1)) = nδ/2−1Iδ/2−1(t√2n)Iδ/2−1(t√2(n−1))(n−1)δ/2−1 ≃ (nn−1)δ/2−1(nn−1)1/4e−t(√2n−√2(n−1)),

which clearly is integrable at Consequently, by Corollary 2.7, the integral functional

 ∫∞0e−Xtdt

has all the (positive) moments.

###### Example 3.2.

Geometric Brownian motion. Let be a geometric Brownian motion with parameters and , i.e.,

 Ys=exp(σWs+(μ−12σ2)s)

where is a standard Brownian motion initiated at 0. Since a.s. when it follows

a.s if

a.s if

and a.s. if

Consequently, condition (21) is valid if and only if Since we consider the first hitting times of the points Consider

 Ha := inf{s:Ys=a} = inf{s:exp(σWs+(μ−12σ2)s)=a} = inf{s:σWs+(μ−12σ2)s=loga} = inf⎧⎨⎩s:Ws+μ−12σ2σs=1σloga⎫⎬⎭.

We assume now that and Let Then is identical in law with the first hitting time of for Brownian motion with drift starting from 0. Consequently, letting we have for

 E1(e−βXt) = ψβ(0)ψβ(log(1+t)/σ) = exp(−(√2β+ν2−ν)log(1+t)σ) = (1+t)−(√2β+ν2−ν)/σ, =: exp(−Φ(t;β)).

where is the expectation associated with when started from 1 and

 ψβ(x)=exp((√2β+ν2−ν)x)

is the increasing fundamental solution for Brownian motion with drift (see [3] p.132). Notice that the additive process is a deterministic time change of the first hit process of Brownian motion with drift, which is a subordinator. We use now Proposition 2.6 to study the moments of the perpetual integral functional

 I∞=∫∞0e−Xsds.

To simply the notation (cf. (3.2)) introduce

 ρ(β):=(√2β+ν2−ν)/σ.

By formula (15) the th moment is given by

 E1(In∞)=n!∫∞0dt1(1+t1)−(ρ(n)−ρ(n−1))∫∞t1dt2(1+t2)−(ρ(n−1)−ρ(n−2)) ×∫∞t2dt3⋯∫∞tn−1dtn(1+tn)−ρ(1) =⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩n!∏nk=1(ρ(k)−k),if  n

where

 n∗:=min{n∈{1,2,…}:ρ(n)−n≤0}.

Condition (16) in Corollary 2.7 takes in this case the form

 ρ(n)−ρ(n−1)>1. (30)

This being a sufficient condition for the finiteness of we have

 ρ(n)−ρ(n−1)>1⇒ρ(n)−n>0. (31)

Consider now the case Then

 ρ(n)>n⇔n<2σ2, (32)

i.e., smaller the volatility (i.e. ) more moments of exist, as expected. Moreover, in this case

 ρ(n)−ρ(n−1)>1 ⇔√2n+√2(n−1)<2σ (33) ⇔2n−1+√4n(n−1)<2σ2

showing, in particular, that when is “small” there exist ”many” satisfying (32) but not (33).

Acknowledgement. This research was partially supported by Defimath project of the Research Federation of ”Mathématiques des Pays de la Loire”, by PANORisk project ”Pays de la Loire” region, and by the Magnus Ehrnrooth Foundation, Finland.

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