On exponential functionals of processes with independent increments

arXiv:1610.08732v2 [math.PR] 17 Jul 2017

ON EXPONENTIAL FUNCTIONALS OF PROCESSES WITH INDEPENDENT INCREMENTS

˚bo P. Salminen, Department of Natural Sciences A Akademi University FIN-20500 ˚ Abo Finland and L. Vostrikova 1 , LAREMA, D´epartement de Math´ematiques, Universit´e d’Angers, 2, Bd Lavoisier 49045, Angers Cedex 01

Abstract. In this paper we study the exponential functionals of the processes X with independent increments , namely Z t It = exp(−Xs )ds,, t ≥ 0, 0

and also

I∞ =

Z

∞

exp(−Xs )ds. 0

When X is a semi-martingale with absolutely continuous characteristics, we derive necessary and sufficient conditions for the existence of the Laplace exponent of It , and also the sufficient conditions of finiteness of the Mellin transform E(Itα ) with α ∈ R. We give a recurrent integral equations for this Mellin transform. Then we apply these recurrent formulas to calculate the moments. We present also the corresponding results for the exponential functionals of Levy processes, which hold under less restrictive conditions then in [7]. In particular, we obtain an explicit formula for the moments of It and I∞ , and we precise the exact number of finite moments of I∞ .

MSC 2010 subject classifications: 60G51, 91G80

11

This work is supported in part by DEFIMATHS project of the Reseach Federation of ”Math´ematiques de Pays de la Loire” and PANORisk project of Pays de la Loire region. 1

2

ON EXPONENTIAL FUNCTIONALS

1. Introduction The exponential functionals arise in many areas : in the theory of selfsimilar Markov processes, in the theory of random processes in random environments, in the mathematical statistics, in the mathematical finance, in the insurance. In fact, self-similar Markov processes are related with exponential functionals via Lamperti transform, namely self-similar Markov process can be written as an exponential of Levy process time changed by the inverse of exponential functional of the same Levy process (see [21]). In the mathematical statistics the exponential functionals appear, for exemple, in the study of Pitman estimators (see [23]). In the mathematical finance the question is related to the perpetuities containing the liabilities, the perpetuities subjected to the influence of economical factors (see, for example, [18]), and also with the prices of Asian options and related questions (see, for instance, [16] and references therein). In the insurance, this connection is made via the ruin problem, the problem in which the exponential functionals appear very naturally (see, for exemple [25], [1], [17] and references therein). In the case of Levy processes, the asymptotic behaviour of exponential functionals was studied in [9], in particular for α-stable Levy processes. The authors also give an integro-differential equation for the density of the law of exponential functionals, when this density w.r.t the Lebesgue measure exists. In [7], for Levy subordinators, the authors give the formulas for the positive and negative moments, the Mellin transform and the Laplace transform. The questions related with the characterisation of the law of exponential functionals by the moments was also studied. General information about Levy processes can be find in [29], [5], [20]. In more general setting, related to the L´evy case, the following functional (1)

Z

∞

exp(−Xs )dηs

0

was studied by many authors, where X = (Xt )t≥0 and η = (ηt )t≥0 are independent L´evy processes. The interest to this functional can be explained by a very close relation between the distribution of this functional and the stationary density of the corresponding generalized

ON EXPONENTIAL FUNCTIONALS

3

Ornshtein-Uhlenbeck process. We recall that the generalized OrnshteinUhlenbeck process Y = (Yt )t≥0 verify the following differential equation (2)

dYt = Yt− dXt + dηt

When the jumps of the process η are strictly bigger than -1, and X0 = 0, the solution of this equation is   Z t dηs (3) Yt = E(X)t Y0 + 0 E(X)s where E(X) is Dol´ean-Dade exponential (see [15] for the details). Fiˆ such that for all t > 0 nally, if we introduce another L´evy process X ˆ t ), E(X)t = exp(X then the integral of the type (1) appear in (3). The conditions for finiteness of the integral (1) was obtained in [13]. The continuity properties of the law of this integral was studied in [6], where the authors give the condition for absence of the atoms and also the conditions for absolute continuity of the laws of integral functionals w.r.t. the Lebesgue measure. Under the assumptions which ensure the existence of the density of these functionals, the equations for the density are given in [3], [4], [19]. It should be noticed that taking the process η being only drift, one can get I∞ as a special case of the L´evy framework. In the papers [24] and [22], again for Levy process, the properties of the exponential functionals Iτq was studied where τq is independent exponential random variable of the parameter q > 0. In the article [22] the authors studied the existence of the density of the law of Iτq , they give an integral equation for the density and the asymptotics of the law of I∞ at zero and at infinity, when X is a positive subordinator. The results given in [24] involve analytic Wiener-Hopf factorisation, Bernstein functions and contain the conditions for regularity, semiexplicite expression and asymptotics for the distribution function of the exponential functional killed at the independent exponential time τq . Despite numerous studies, the distribution properties of It and I∞ are known only in a limited number of cases. When X is Brownian motion with drift, the distributions of It and I∞ was studied in [11] and for a big number of specific processes X and η, like Brownian motion with drift and compound Poisson process, the distributions of I∞ was given in [14].

4

ON EXPONENTIAL FUNCTIONALS

Exponential functionals for diffusions was studied in [26]. The authors considered exponential functionals stopped at first hitting time and they derive the Laplace transform of these functionals. To find the laws of such exponential functionals, the authors perform a numerical inversion of the corresponding Laplace transform. The relations between the hitting times and the occupation times for the exponential functionals was considered in [28], where the versions of identities in law such as Dufresne’s identity, Ciesielski-Taylor’s identity, Biane’s identity, LeGall’s identity was considered. Howerever, the exponential functionals involving non-homogeneous processes with independent increments (PII in short) has not been studied sufficiently up to now. Only a few results can be found in the literature. Some results about the moments of the exponential functional of this type are given in [12]. At the same time PII models for logarithme of the prices are quite natural in the mathematical finance, it is the case of the non-homogeneous Poisson process, the Levy process with deterministic time change, the integrals of L´evy processes with deterministic integrands, the hitting times for diffusions and so on (see for instance [30], [10], [2]). The aim of this paper is to study the exponential functionals of the processes X with independent increments, namely Z t (4) It = exp(−Xs )ds, t ≥ 0, 0

and also I∞ =

Z

∞

exp(−Xs )ds,

0

and give such important characteristics of these exponential functionals as the moments and the Laplace transforms and the Mellin transforms of mentioned functionals. For that we consider a real valued process X = (Xt )t≥0 with independent increments and X0 = 0, which is a semi-martingale with respect to its natural filtration. We denote by (B, C, ν) a semi-martingale triplet of this process, which can be chosen deterministic (see [15], Ch. II, p.106). We suppose that B = (Bt )t≥0 , C = (Ct )t≥0 and ν are absolutely continuous with respect to the Lebesgue measure in t, i.e. that X is an Ito process such that Z t Z t Bt = bs ds, Ct = cs ds, ν(dt, dx) = dtKt (dx) 0

0

ON EXPONENTIAL FUNCTIONALS

5

with measurable functions b = (bs )s≥0 , c = (cs )s≥0 , and the kernel K = (Kt (A))t≥0,A∈B(R\{0}) . For more information about the semimartingales and the Ito processes see [15]. We assume that the compensator of the measure of the jumps ν verify the usual condition: for each t ∈ R+ Z tZ (5) (x2 ∧ 1)Ks (dx) ds < ∞. 0

R\{0}

We recall that the characteristic function of Xt φt (λ) = E exp(iλXt ) is defined by the following expression: for λ ∈ R Z tZ 1 2 (eiλx −1−iλx1{|x|≤1} ) Ks (dx) ds} φt (λ) = exp{iλBt − λ Ct + 2 0 R\{0} which can be easily obtained by the Ito formula for semimartingales. We recall also that X is a semi-martingale if and only if for all λ ∈ R the characteristic function of Xt is of finite variation in t on finite intervals (cf. [15], Ch.2, Th. 4.14, p.106 ). Moreover, the process X always can be written as a sum of a semi-martingale and a deterministic function which is not necessarily of finite variation on finite intervals. From the formula for the characteristic function we can easily find the Laplace transform of Xt , if it exists: E(e−αXt ) = e−Φ(t,α) Putting λ = iα in the previous formula, we get that Z tZ 1 2 Φ(t, α) = αBt − α Ct − (e−αx − 1 + αx1{|x|≤1} ) Ks (dx) ds. 2 0 R\{0} As known, in the case when X is a Levy process with parameters (b0 , c0 , K0 ), it holds E(e−αXt ) = e−tΦ(α) with Z 1 2 Φ(α) = αb0 − α c0 − (e−αx − 1 + αx1{|x|≤1} ) K0 (dx). 2 R\{0} In this article we establish necessary and sufficient conditions of the existence of the Laplace exponent, and also the sufficient conditions for finiteness of E(Itα ) with fixed t > 0 (see Proposition 1). Next, we use time reversal of the process X at fixed time t > 0 to introduce a

6

ON EXPONENTIAL FUNCTIONALS (t)

(t)

new process Y (t) = (Ys )0≤s≤t with Ys = Xt − X(t−s)− . In Lemma 1 we show that Z t (t) (t) −Yt eYs ds. It = e 0

Then, we prove that Y (t) is a process with independent increments and we identify its semi-martingale characteristics. This time reversal plays an important role and permits to replace the process (It )t≥0 which is (t) not Markov, by a family of Markov processes V (t) = (Vs )0≤s≤t indexed R (t) (t) s (t) by t > 0, where Vs = e−Ys 0 eYu du. In the sequel we omit (t) to simplify the notations. In Theorem 1 we consider the case of α ≥ 0 and we give a recurrent integral equation for the Mellin transform of It . In the Corollaries 2 and 3 we consider the case when X is a Levy process. We give the formulas for positive moments of It and I∞ and also the formula for the Laplace transform. The results for I∞ coincide, of course, with the ones given in [7] for Levy subordinators. But it holds under less restrictive integrability conditions on I∞ and less restrictive condition on Levy measure at zero. We can also precise the number of finite moments of I∞ (cf. Corollary 3 and Corollary 5). In Theorem 2 and Corollaries 4 and 5 we present analogous result for the case α < 0.

2. Finiteness of E(Itα ) for fixed t > 0. Now, we fix t > 0. In the following proposition we give necessary and sufficient conditions for the existence of the Laplace transform of Xt , and also sufficient conditions for the existence of the Mellin transform of It . In what follows we assume that K verify the following stronger condition then (5): for t > 0 Z tZ (6) (x2 ∧ |x|)Ks (dx)ds < ∞. 0

R\{0}

This condition says, roughly speaking, that the ”big” jumps of the process X are integrable, and it ensures that finite variation part of semi-martingale decomposition of X remains deterministic. Moreover, the truncation of the jumps is no more necessary. In addition, Kruglov theorem can be applied to show that (6) is equivalent to E(|Xt |) < ∞ (cf. [29], Th. 25.3, p.159).

ON EXPONENTIAL FUNCTIONALS

7

Proposition 1. Let α ∈ R and t > 0. Under (6), the condition Z tZ (7) (e−αx − 1 + αx)Ks (dx)ds < ∞, 0

R\{0}

is equivalent to E(e−αXt ) < ∞.

(8)

Moreover, the condition (7) implies for α ≥ 1 and α ≤ 0 that E(Itα ) < ∞.

(9)

In addition, if (7) is valid for α = 1, then for 0 ≤ α ≤ 1 we have (9). Remark 1. Assume that X has only positive jumps, then for α > 0 the condition (7) is always satisfied. In the same way, if X has only negative jumps, then, of course, for α < 0 the condition (7) is satisfied. If X has bounded jumps, the condition (7) is also satisfied. In general case, the condition (7) is equivalent to the one of the following conditions: if α > 0, then Z tZ (10) e−αx Ks (dx)ds < ∞, 0

x1

e−αx Ks (dx)ds < ∞.

In the case of Levy processes these conditions coincide with the ones given in [20], p. 79. Remark 2. It should be noticed that if the condition (7) is verified for α′ = α + δ with δ > 0 and α > 0, then it is verified also for α. To see this, apply H¨older inequality to the integrals in (10) and (11) with and q = α+δ . In fact, since p1 + 1q = 1 and the parameters p = α+δ α δ ν([0, t]×] − ∞, −1[) < ∞ we have for (10): Z tZ e−αx Ks (dx)ds ≤ 0

Z t Z 0

x 0, a new process Y (t) = (Ys )0≤s≤t with (t) Ys = Xt − X(t−s)− . To simplify the notations we anyway omit the (t) index (t) and write Ys instead of Ys . First of all we establish the relation between It and the process Y . Lemma 1. For t > 0 the following equality holds: Z t −Yt eYs ds It = e 0

12

ON EXPONENTIAL FUNCTIONALS

Proof. Using the definition of the process Y and the assumption that X0 = 0 we have Z t Z t Z t −Yt Ys −Yt +Ys e e ds = e ds = e−X(t−s)− +X0 ds 0

=

Z

0 t

e−Xt−s ds = 0

Z

0 t

e−Xs ds = It , 0

since the integration of the both versions of the process w.r.t. the Lebesgue measure gives the same result.  We will show that the process Y is PII and we will give its semimartingale triplet with respect to its natural filtration. For that we put  ¯bu = bt−u for 0 ≤ u < t, bt for u = t, which can be written also as ¯bu = 1{t} (u)(bt − b0 ) + bt−u ,

where 1{t} is indicator function of the set {t}. We do the similar defi¯ nitions for c¯ and K: c¯u = 1{t} (u)(ct − c0 ) + ct−u ,

¯ u (A) = 1{t} (u)(Kt (A) − K0 (A)) + Kt−u (A) K for all A ∈ B(R \ {0}). Lemma 2. The process Y is a process with independent increments, it is a semi-martingale with respect to its natural filtration, and its ¯ C, ¯ ν¯) is given by : semi-martingale triplet (B, Z s Z s ¯ ¯ ¯ ¯ s (dx) ds , (16) Bs = bu du, Cs = c¯u du, ν¯(ds, dx) = K 0

0

where 0 ≤ s ≤ t.

Proof Let us take 0 = s0 < s1 < s2 < · · · < sn = t with n ≥ 2. Then the increments (Ysk − Ysk−1 )1≤k≤n of the process Y are equal to (X(t−sk−1 )− −X(t−sk )− )1≤k≤n and 0 = t−sn < t−sn−1 < · · · < t−s0 = t. From the fact that X is the process with independent increments, the characteristic function of the vector (Xt−sk−1 −h − Xt−sk −h )1≤i≤n with small h > 0, can be written as a product of the corresponding characteristic functions. Namely, for any real constants (λk )0≤k≤n we get: n n Y X E exp(iλk (Xt−sk−1 −h −Xt−sk −h )) λk (Xt−sk−1 −h −Xt−sk −h )) = E exp(i k=1

k=1

ON EXPONENTIAL FUNCTIONALS

13

Then, passing to the limit as h → 0+, n n X Y E exp(i λk (X(t−sk−1 )− −X(t−sk )− )) = E exp(iλk (X(t−sk−1 )− −X(t−sk )− )) k=1

k=1

Hence, Y is a process with independent increments.

We know that we can identify the semi-martingale characteristics w.r.t. the natural filtration of the process with independent increments from the characteristic function of this process. We notice that Ys = Xt − X(t−s)− and by the independence of Ys and X(t−s)− E(eiλXt ) = E(eiλYs )E(eiλX(t−s)− ) Then, E exp(iλYs ) = E exp(iλXt )/E exp(iλX(t−s)− ) = Z Z t Z 1 2 t exp{iλ bu du− λ cu du+ (eiλx −1−iλx) Ku (dx) du} 2 t−s t−s t−s R\{0} We substitute u by u′ = t − u in the integrals to obtain Z

t

E exp(iλYs ) = Z Z sZ 1 2 s bu du − λ cu du + (eiλx − 1 − iλx) K u (dx) du} exp{iλ 2 0 0 R\{0} 0 Therefore, the characteristics of Y are as in (16) and the proof is complete.✷ Z

s

4. Recurrent formulas for the Mellin transform of It with α ≥ 0. Let us consider two important processes related with the process Y , namely the process V = (Vs )0≤s≤t and J = (Js )0≤s≤t defined via Z s −Ys V s = e Js , Js = eYu du. 0

We underline that the both processes depend of the parameter t.

We remark that according to Lemma 1 , It = Vt for each t ≥ 0. For α ≥ 0 and t ≥ 0 we introduce the Mellin transform of It of the parameter α: (α) mt = E(Itα ) = E(e−αYt Jtα ) and the Mellin transform for shifted process: Z t α  (α) −(Xu −Xs− ) ms,t = E e du s

14

ON EXPONENTIAL FUNCTIONALS (α)

(α)

Notice that m0,t = mt . Notice also that Z t α  (α) −(Xu −Xs ) ms,t = E e du s

In fact, Xu − Xs = Xu − Xs− − ∆Xs and Z t Z t −(Xu −Xs− ) −∆Xs e du = e e−(Xu −Xs ) du s

s

Since ∆Xs and (Xu − Xs )u≥s are independent, and E(e−α∆Xs ) = 1, we (α) get the equality of two expressions for ms,t . We introduce also two functions: for 0 ≤ s ≤ t Z 1 2 (α) (e−αx − 1 + αx)Ks (dx) (17) Hs = αbs − α cs − 2 R\{0}

and

(α) ¯ (α) = 1{t} (s)(Ht(α) − H (α) ) + Ht−s H s 0

(18)

These functions represent the derivatives w.r.t. s, of the Laplace ex¯ α). We recall that ponents Φ(s, α) and Φ(s, Z sZ α2 Φ(s, α) = αBs − Cs − (e−αx − 1 + αx) ν(du, dx) 2 0 R\{0} and

2 ¯ α) = αB ¯s − α C¯s − Φ(s, 2

Z sZ 0

R\{0}

(e−αx − 1 + αx)¯ ν (du, dx)

where ν and ν¯ are the compensators of the jump measure of X and Y respectively. We notice, that these functions are well-defined under condition (7). We also notice that ¯ α) = Φ(t, α) − Φ(t − s, α) Φ(s, Our aim now is to obtain a recurrent integral equation for the Mellin transform of It . For condition (19) below see Remarks 1 and 2. Theorem 1. Let α ≥ 1 be fixed and assume that for t > 0 there exists δ > 0 such that Z tZ (19) e−(α+δ)x Ks (dx) ds < ∞. 0

Then, holds (20)

(α) mt

x 0 Z t (α) −Φ(α) t (21) mt = αe m(α−1) eΦ(α) s ds s 0

Moreover,

d h (α) i (α) (α−1) = −mt Φ(α) + αmt mt dt

(22)

Proof. From Lemma 2 we know that Y = (Ys )0≤s≤t is a process with ¯s + independent increments which is a semi-martingale, i.e. Ys = B ¯ s , where B ¯ is a deterministic process of finite variation on finite M ¯ is a local martingale. But the local martingales with intervals and M independent increments are always the martingales (see [31]). For n ≥ 1 we introduce the stopping times

τn = inf{0 ≤ s ≤ t : Vs ≥ n or exp(−Ys ) ≥ n}

with inf{∅} = +∞. For fixed s, 0 < s < t, we write the Ito formula α for Vs∧τ : n Z s∧τn Z s∧τn 1 α−1 α α−2 Vu− dVu + α(α − 1) Vs∧τn = α Vu− d < V c >u 2 0 0 Z s∧τn Z
α α−1 (Vu− + x)α − Vu− − αVu− x µV (du, dx) (23) + 0

R\{0}

where µV is the measure of the jumps of V . Using integration by part formula, we have: (24)

dVu = du + Ju d(e−Yu )

Now, again by the Ito formula, we get Z Z u 1 u −Yv− −Yu −Y0 −Yv− (25) e =e − e d < Y c >v e dYv + 2 0 0 Z uZ + e−Yv− (e−x − 1 + x)µY (dv, dx) 0

R\{0}

Then, putting (25) into (24), we obtain (26)

dVuc = −e−Yu− Ju dYuc = −Vu− dYuc , 2 d < V c >u = Vu− d < Y c >u

and (27)

∆Vu = e−Yu− Ju (e−∆Yu − 1) = Vu− (e−∆Yu − 1),

16

ON EXPONENTIAL FUNCTIONALS

where ∆Vu = Vu − Vu− and ∆Yu = Yu − Yu− . The previous relations imply that Z s∧τn α α−1 (28) Vs∧τn = α Vu− du 0

+α

Z

0

s∧τn

α−1 Ju Vu− d(e−Yu )

+

Z

s∧τn

Z

0

R\{0}

1 + α(α − 1) 2

Z

s∧τn

0

α Vu− d < Y c >u


α Vu− e−αx − 1 − α(e−x − 1) µY (du, dx)

To use in efficient way the Ito formula for e−Yu given before, we introduce the processes A = (Au )0≤u≤t and N = (Nu )0≤u≤t via Z u Z uZ 1 ¯ −Yv− ¯ Au = [−dBv + dCv ] + e e−Yv− (e−x − 1 + x)¯ ν (dv, dx) 2 0 0 R\{0} Z u Z uZ −Yv− ¯ Nu = − e dMv + e−Yv− (e−x −1+x)[µY (dv, dx)−¯ ν (dv, dx)] 0

0

R\{0}

We notice that A is a process of locally bounded variation and N is ¯ C¯ are of a local martingale with localizing sequence (τn )n≥1 , since B, bounded variation on bounded intervals and Z uZ ¯ s (dx)ds < ∞. (e−x − 1 + x)K 0

R\{0}

From (25) we get that e−Yu = e−Y0 + Au + Nu .

(29)

We incorporate this semi-martingale decomposition into (28) and we consider its martingale part. This martingale part is represented by the term Z s∧τn

α

0

α

Z

0

s∧τn

¯u + Vuα [dM

Z

R\{0}

Vuα−1 Ju dNu =

(e−x − 1 + x)(µY (dv, dx) − ν¯(dv, dx))]

which is a local martingale. Let (τn′ )n≥0 be a localizing sequence for this local martingale and let τ¯n = τn ∧ τn′ . Then we do additional stopping with τn′ in previous expressions, and we take mathematical expectation. Using the fact that the expectations of martingales starting from zero are equal to zero and also applying the projection theorem, we obtain:  Z s∧¯τn α−1 α Vu du (30) E(Vs∧¯τn ) = αE 0

ON EXPONENTIAL FUNCTIONALS

+αE +E

Z

Z

s∧¯ τn α−1 Vu− Ju

0 s∧¯ τn

Z

0

and, hence, (31)

α E(Vs∧¯ τn )

dAu

α Vu− R\{0}

= αE

Z



Z

1 + α(α − 1)E 2

17 s∧¯ τn

Vuα

0

dC¯u

 −αx  e − 1 − α(e−x − 1) ν¯(du, dx)

s∧¯ τn

Vuα−1 du

0



−E

s∧τn

Z

α Vu−

0





¯ α) dΦ(u,



We remark that τ¯n → +∞ (P − a.s.) as n → +∞. To pass to the limit as n → ∞ in r.h.s. of the above equality, we use the Lebesgue monotone convergence theorem for the first term and the Lebesgue dominated convergence theorem for the second term. In fact, for the second term we have using (17) and (18) : Z t Z s∧¯τn α α ¯ α) ≤ ¯ u(α) |du V d Φ(u, Vu− |H u− 0

0

In addition,

E

Z

t

0

α Vu−

¯ (α) |du |H u



≤ sup

0≤u≤t

E(Vuα )

Z

0

t

¯ (α) |du |H u

¯ u(α) )0≤u≤t is deterministic function, integrable on finite The function (H intervals. Hence, it remains to show that sup E(Vsα ) < ∞.

(32)

0≤s≤t

By the Jensen inequality Vsα

≤s

α−1

Z

s

eα(Yu −Ys ) du

0

We remark that Yu − Ys = X(t−s)− − X(t−u)− , and then Z s α α−1 E(eα(Xt−s −Xt−u ) ) du (33) E(Vs ) ≤ s 0

Since the process X is a process with independent increments, we have for 0 ≤ u ≤ s ≤ t E(eα(Xt−s −Xt−u ) ) = E(e−αXt−u )/E(e−αXt−s ) = Z t−u Z t−s Z t (α) (α) exp{− Hr dr + Hr dr} ≤ exp{ |Hr(α) |dr} 0

0

0

18

ON EXPONENTIAL FUNCTIONALS

Due to the Remark 1 and the condition (19), H (α) is integrable function on finite intervals, and hence, Z t α α sup E(Vs ) ≤ t exp{ |Hr(α) |dr} < ∞ 0≤s≤t

0

To pass to the limit in the l.h.s. of (31), we show that the family of α ) is uniformly integrable, uniformly in 0 ≤ s ≤ t. For that we (Vs∧τ n n≥1 ¯s + M ¯ s where B ¯ is a drift part and M ¯ is a martingale recall that Ys = B part of Y , and we introduce Z s ¯ ¯s −M ¯ eMu du Vs = e 0

Then,

−Ys

Vs = e

Z

s

¯ s −B ¯s −M

Yu

e du = e 0 ¯ u −B ¯s B

≤ sup (e 0≤u≤s

Z

s

¯

¯

eMu +Bu du 0

¯ ) V¯s ≤ eV ar(B)t V¯s

¯ t is the variation of B ¯ on the interval [0, t]. This quantity where V ar(B) is deterministic and bounded, hence, it is sufficient to prove uniform α ) . integrability of (V¯s∧τ n n≥1 Next, we show that (V¯s )0≤s≤t is a submartingale w.r.t. a natural filtration of Y . Let s′ > s, then Z s′ ¯ ′ ¯ −M ¯ E(Vs′ | Fs) = E(e s eMu du | Fs) = 0

¯ s) ¯ ′ −M −(M s

E(e

¯s M

[V¯s + e

Z

s′

s ¯ ′ −M ¯ s) ¯ −(M s E(e Vs ¯ ′ −M ¯ s) −(M s

¯

eMu du] | Fs) ≥

¯ ¯ | Fs) = V¯s E(e−(Ms′ −Ms ) ) The expression for E(e ) can be find from the expression of the characteristic exponent of Y without its drift part: ¯

¯

E(e−(Ms′ −Ms ) = ! Z ′ Z s′ Z 1 s −x exp cu du + (e − 1 + x)¯ ν (du, dx) ≥ 1. 2 s s R\{0} Then, (V¯sα )0≤s≤t is a submartingale, and by Doob stopping theorem (P -a.s.) α . E(V¯sα | Fs∧τn ) ≥ V¯s∧τ n Hence, for all n ≥ 1, c > 0 and I(·) indicator function E(V¯ α I{V¯ >c} ) ≤ E(E(V¯ α | Fs∧τn )I{E(V¯ α | F )>c} ) ≤ s∧τn

s∧τn

s

s

s∧τn

ON EXPONENTIAL FUNCTIONALS

19

α+δ δ δ c− α E(E(V¯sα | Fs∧τn ) α ) ≤ c− α E(V¯sα+δ ) It remains to show that sup0≤s≤t E(V¯sα+δ ) < ∞. The last inequality can be proved in the same way as (32).

After limit passage, we get that Z s Z s α α ¯ (34) E(Vs ) = − E(Vu ) dΦ(u, α) + α E(Vuα−1 )du 0

0

We see that each term of this equation is differentiable w.r.t. s for s < t. In fact, the family (Vsα )0≤s≤t is uniformly integrable and the ¯ α))0≤s 0, i.e. Xt = Lτt for t ≥ 0. Then, by change of variables u = r ln(1 + s) we get Z Z Z t 1 r ln(1+t) −L˜ u 1 r ln(1+t) −(Lu −u/r) −Lr ln(1+s) e du = e du It = e ds = r 0 r 0 0

20

ON EXPONENTIAL FUNCTIONALS

˜ is Levy process with generating triplet (b0 − 1 , c0 , K0 ). We where L r ˜ the Laplace exponent of L. ˜ Then for k ≥ 0, Φ(k) ˜ denote by Φ = k Φ(k) − r and  1−(1+t)−rΦ(1)+1 Z r ln(1+t)  if rΦ(1) − 1 6= 0, rΦ(1)−1 1 ˜ −Φ(1)u e du = E(It ) =  r 0 ln(1 + t) otherwise. (s)

For shifted process Xu = Lτu − Lτs , u ≥ s the corresponding moments Z t n  Z t n  (n) −(Lτu −Lτs ) −L(τu −τs ) ms,t = E e du =E e du s

s

L

since L is homogeneous process and (Lτu − Lτs )u≥s = (Lτu −τs )u≥s . We change the variable putting τu −τs = r ln(1 + u) −r ln(1 + s) = v −s 1+t ) and we remark where v is new variable. We denote v(s, t) = r ln( 1+s that 1 + u = (1 + s) exp((v − s)/r). Then, !n # " Z t n  Z s+v(s,t) 1 + s ˜ e−Lv−s dv E e−(Lτu −τs ) du =E r s s (1 + s)n = E rn

" Z

Finally, we get for n ≥ 0: (n)

ms,t =

v(s,t)

˜

e−Lu du 0

!n #

(1 + s)n (n) m ˜ v(s,t) rn

(n) ˜ on where m ˜ v(s,t) is n-th moment of the exponential functional of L ˜ is strictly monotone on [0, v(s, t)]. We suppose for simplicity that Φ the interval [0, n + 1]. Then, using the integral equation of Theorem 1 and the expression for the moments of Levy processes given in Corollary 1 below, and the fact that

e−Φ(s,n+1) = e−Φ(n+1)τs = (1 + s)−rΦ(n+1) we get for n ≥ 1: n−1

(n+1) mt

(n + 1)! X = rn k=0

Z

0

˜

t

(1 + s)

q(n)

˜

e−Φ(k)v(s,t) − e−Φ(n)v(s,t) Y ds ˜ − Φ(k)) ˜ (Φ(i)

0≤i≤n, i6=k

ON EXPONENTIAL FUNCTIONALS

21

˜ where q(n) = n − rΦ(n + 1) and Φ(k) is the Laplace exponent of Levy k ˜ ˜ process L, Φ(k) = Φ(k) − r , 1 ≤ k ≤ n. To express the final result we put ρ(k) = rΦ(k) − k, γ(n, k) = n − k − r(Φ(n + 1) − Φ(k)) and Qt (n, k) =

(1 + t)γ(n,k)+1 − 1 (γ(n, k) + 1)(1 + t)ρ(k)

Then, after the integration we find that for n ≥ 1 (n+1)

E(Itn+1) = mt

= (n + 1)!

n−1 X k=0

Qt (n, k) − Qt (n, n) Y (ρ(i) − ρ(k))

0≤i≤n, i6=k

Remark 4. It is clear that the explicit formulas for the moments in non-homogeneous case will be rather exceptional. For numerical calculus the following formula could be useful: for all 0 ≤ s ≤ t Z t (α) (α−1) ms,t = α mu,t eΦ(u,α)−Φ(s,α) du s

To obtain this formula it is sufficient to find the Laplace exponent of (s) (s) shifted process (Xu )s≤u≤t with Xu = Xu − Xs . Since X is a PII, the variables Xs and Xu − Xs are independent, and E(e−αXu ) = E(e−αXs ) E(e−α(Xu −Xs ) ) and then, the Laplace exponent of shifted process X (s) is given by: Φ(s) (u, α) = Φ(u, α) − Φ(s, α). 5. Positive moments of It and I∞ for Levy processes Now we suppose that X is Levy process with the parameters (b0 , c0 , K0 ). In this case the condition (6) become: Z (35) (x2 ∧ |x|)K0 (dx) < ∞. R\{0}

It should be noticed that in [7] the condition on Levy measure was Z (|x| ∧ 1)K0 (dx) < ∞ R\{0}

22

ON EXPONENTIAL FUNCTIONALS

and this condition is stronger at zero and weaker at infinity then (35). When X is Levy process, the condition (19) become: there exists δ> 0 Z (36) e−(α+δ)x K0 (dx) < ∞. x 0 and diffusion coefficient σ > 0, i.e. Xt = µt + σWt where W = (Wt )t≥0 is 2 2 is not standard Brownian motion. Then, Φ(α) = αµ − α 2σ and if 2µ σ2 an integer, we get: E(Itn )

= n!

n−1 X k=0

2 2

2 2

e−(kµ−k σ /2)t − e−(nµ−n σ /2)t Y (i − k)(µ − (i + k)σ 2 /2)

0≤i≤n, i6=k

Example 6. Let X be compound Poisson process such that Xt = PNt k=1 Uk where (Uk )k≥0 is a sequence of independent random variables with distribution function F and N is a homogeneous Poisson process R with intensity λ > 0. Then, Φ(α) = λ R\{0} (1 − e−αx )F (dx). In particular, if the Uk ’s are standard normal variables, we get that Φ(α) = 2 λ(1 − eα /2 ) and E(Itn )

2 2 n−1 X exp(λt(1 − ek /2 )) − exp(λt(1 − en /2 )) Y = n! 2 2 λn (ek /2 − ei /2 ) k=0

0≤i≤n, i6=k

(α)

(α)

We introduce the Laplace-Carson transform m ˆ q of mt eter q > 0: Z ∞

m ˆ (α) = q

of the param-

(α)

qe−qt mt dt

0

This integral is always well-defined in general sense, since the integrand is positive.

ON EXPONENTIAL FUNCTIONALS

23

Corollary 2. (cf. [7]) Let X be a Levy process which verifies (35) and (α) (36), and such that for fixed α ≥ 1, m∞ < ∞. Then the Laplace(α) (α−1) (α) (α−1) Carson transforms m ˆ q and m ˆq of mt and mt respectively, are well-defined and we have a recurrent formula: m ˆ (α) ˆ (α−1) q q (q + Φ(α)) = αm

(37)

(n)

In particular, for integer n ≥ 1 such that m∞ < ∞ we get: n! (38) m ˆ (n) q = Qn k=1 (q + Φ(k))

As a consequence,

n! k=1 Φ(k) Moreover, if all positive moments of I∞ exist and the series below converges, then the Laplace transform of I∞ of parameter β > 0 is given by: ∞ X (−1)n β n −β I∞ Q (39) E(e )= n k=1 Φ(k) n=0 n E(I∞ ) = Qn

Proof. The first equality for the Laplace-Carson transforms follows directly from (22). The second equality can be obtained as particular case from the first one, by recurrence. (n)

(n)

For the third one we prove that m ˆ q → m∞ as q → 0. In fact, (n) (n) mt → m∞ as t → +∞ and Z 1 t (n) lim ms ds = m(n) ∞ . t→+∞ t 0 Z t (n) Let us denote Mt = m(n) s ds. By integration by part formula we 0

have for each t > 0 : Z t Z t (n) t −qs (n) −qt qe ms ds = [q e Mt ]0 + q 2 e−qs Ms(n) ds 0

0

(n)

Then, since M0

m ˆ (n) q Since q

2

R∞ 0

(n)

(n)

= 0 and Mt /t → m∞ as t → +∞, Z ∞ Z ∞ −qs (n) 2 = e−qs Ms(n) ds qe ms ds = q 0

0

−qs

se

ds = 1, we get Z (n) (n) 2 m ˆ q − m∞ = q

0

∞

e−qs (Ms(n) − s m(n) ∞ )ds

24

ON EXPONENTIAL FUNCTIONALS (n)

For each ǫ > 0 there exists tǫ such that for s ≥ tǫ , | Mss Then, Z tǫ (n) (n) 2 e−qs |Ms(n) − s m(n) |m ˆ q − m∞ | ≤ q ∞ |ds+

(n)

− m∞ | ≤ ǫ.

0

q2

Z

Z tǫ M (n) ∞ s (n) 2 −qs (n) e−qs |Ms(n) − s m∞ |ds + ǫ se − m∞ ds ≤ q s 0 tǫ

We notice that

lim q

q→0

2

Z

tǫ

e−qs | Ms(n) − s m(n) ∞ | ds = 0

0

Then, taking limǫ→0 limq→0 in the previous inequality we get that (n) m ˆ (n) q → m∞

as q → 0. Finally, we take the limit as q → 0 in second equality, to obtain the third one. The formula for the Laplace transform of I∞ can be proved by using Taylor expansion with remainder in Lagrange form.  Example 7. Let X be homogeneous Poisson process of intensity λ. Then all positive moments of I∞ exist, and we have for 0 ≤ β < λ E(e−β I∞ ) =

∞ X n=0

(−1)n β n Q . λn nk=1 (1 − e−k )

Corollary 3. Let α0 = inf{α > 0 | Φ(α) ≤ 0} with inf{∅} = +∞. n Then, E(I∞ ) < ∞ if and only if 1 ≤ n < α0 . In particular, for Brownian motion with drift coefficient b0 and diffusion coefficient c0 6= 0, 1 Φ(n) = nb0 − n2 c0 2

and the moment of I∞ of order n ≥ 1 will exist if n < 2bc00 . If X is a subordinator with non-zero Levy measure K0 such that (35) holds, then Z Z Φ(n) = n[b0 − xK0 (dx) ] − (e−nx − 1)K0 (dx), R+ \{0}

R+ \{0}

and under the condition b0 − all moments of I∞ exist.

Z

R+ \{0}

xK0 (dx) ≥ 0,

ON EXPONENTIAL FUNCTIONALS

25

k Proof. Let n = sup{k ≥ 1 : E(I∞ ) < ∞}. If n = +∞, then Φ(k) > 0 for all k ≥ 1 and α0 = +∞. If 1 ≤ n < +∞, from Corollary 2 we get that Φ(n) > 0. Since Φ(α) is concave function such that Φ(0) = 0, we conclude that n < α0 . Conversely, substituting t − s by s in (21) of Theorem 1 we get: Z t n n−1 E(It ) ≤ n E(I∞ ) e−Φ(n)s ds 0

If 1 ≤ n < α0 , Φ(n) > 0, and the integral on the r.h.s. of this inequality n converge as t → ∞. By induction, it gives that E(I∞ ) < ∞. Moreover, the results for continuous case and the case when X is a subordinator, follow immediately from the expression of Φ(k).  Example 8. Let X be time changed Brownian motion, namely Xt = µτt + σWτt where W = (Wt )t≥0 is standard Brownian motion, µ ∈ R, σ > 0 and τt is first hitting time of the level t of the independent (from W ) standard Brownian motion B = (Bt )t≥0 with the drift coefficient b > 0. Then, as known, Φ(α) = (b2 +2αµ−α2 σ 2 )1/2 −b with b2 +2αµ− n α2 σ 2 > 0 (see for instance [8], formula 2.0.1,p.295). Then, E(I∞ ) 0. Example 9. Let X be pure discontinuous Levy process with Levy measure c exp(−Mx) K0 (dx) = I]0,+∞[(x)dx x1+β where c > 0, M > 0, 0 < β < 1. Then, Φ(α) =

c Γ(1 − β) ((M + α)β − M β − αM β−1 β). β

Then, Φ(α) > 0 for α ≥ 1, and all moments of I∞ exist. 6. Recurrent formulas for the Mellin transform of It with α < 0. In the following Theorem 2, we derive the integro-differential equation (α) for the Mellin transform mt of It with α < 0. Theorem 2. Let α < 0 be fixed and Z tZ (40) e(|α|+1)x Ks (dx) ds < ∞. 0

x>1

26

ON EXPONENTIAL FUNCTIONALS (α)

(α−1)

Then, for s > 0, ms,t is well-defined as well as ms,t following recurrent differential equation:   d (α) 1 (α) (α−1) (α) ms,t Hs − ms,t (41) ms,t = α ds

and we get the

In the case of Levy process X we have:   1 d (α) (α−1) (α) (42) ms = ms Φ(α) + ms α ds Proof. The proof of this result is similar to the proof of Theorem 1. For n ≥ 1 we introduce the stopping times

1 or exp(−Yu ) ≥ n} n with inf{∅} = +∞. Then from the Ito formula similarly to (4) we get: for 0 < s < t Z t∧τn Z t∧τn 1 α−1 α−2 α α Vu− dVu + α(α − 1) Vu− d < V c >u Vt∧τn = Vs + α 2 s s τn = inf{u ≥ s : Vu ≤

(43)

+

Z

t∧τn

s

Z

R\{0}


α α−1 (Vu− + x)α − Vu− − αVu− x µV (du, dx)

where µV is the measure of the jumps of V . Using (24), (26), (27) we have Z t∧τn α−1 α α du Vu− (44) Vt∧τn = Vs + α s

+α

Z

s

t∧τn α−1 Ju Vu− d(e−Yu )

+

Z

s

t∧τn

Z

R\{0}

1 + α(α − 1) 2

Z

s

t∧τn α Vu− d < Y c >u


α Vu− e−αx − 1 − α(e−x − 1) µY (du, dx)

where µY the measure of the jumps of Y . Taking in account (29) we, finally, find that (45)   Z t∧τn Z t∧τn α α−1 α α ¯ Vu− dΦ(u, α) Vu− du − E E(Vt∧τn ) = E(Vs ) + αE s

s

We remark that τn → +∞ (P − a.s.) as n → +∞. To pass to the limit as n → ∞ in the r.h.s. of the above equality, we use the Lebesgue monotone convergence theorem for the first term and the Lebesgue

ON EXPONENTIAL FUNCTIONALS

27

dominated convergence theorem for the second term. In fact, for second term we have: Z t Z t∧τn α α ¯ α) ≤ ¯ (α) |du V d Φ(u, Vu− |H u− u s

s

In addition,

E

Z

s

t

α Vu−

¯ (α) |du |H u



≤ sup

s≤u≤t

E(Vuα )

Z

s

t

¯ (α) |du |H u

¯ u(α) )0≤u≤t is deterministic function, integrable on finite The function (H intervals. Hence, it remains to show that sup E(Vuα ) < ∞.

(46)

s≤u≤t

By the Jensen inequality Vuα

≤u

α−1

Z

u

eα(Yv −Yu ) dv

0

Then, in the same way as in Theorem 1 we get that Z t α α sup E(Vu ) ≤ s exp{ |Hr(α) |dr} s≤u≤t

0

To pass to the limit in the l.h.s. of (45), we prove that the family α of (Vu∧τ ) is uniformly integrable, uniformly in u ∈ [s, t] in the n n≥1 same way as in Theorem 1. For that we introduce a submartingale (α) (V¯u )s≤u≤t with Z ¯ V¯u(α) = e−αMu

such that

u

¯

eαMv dv

0

¯ Vuα ≤ e|α| V ar(B)t V¯u(α) . (α)

V¯ (α) n |α| On the set {V¯u∧τn > c} we get that 1 ≤ ( u∧τ ) . Then c 1

(α) V¯u∧τn I{V¯ (α)

u∧τn >c}

1

1

(α) (α) α−1 ≤ (V¯u∧τn )1+ |α| c− |α| = (V¯u∧τn ) α

By Jensen inequality we get that (α)

(V¯u∧τn )

α−1 α

1

(α−1)

≤ V¯u∧τn u |α|

Hence, we proved that for all n ≥ 1, c > 0 and s ≤ u ≤ t (α)

E(V¯u∧τn I{V¯ (α)

u∧τn

1

1

1

) ≤ c− |α| max(s |α| , t |α| ) E(V¯u(α−1) ) >c}

To prove uniform integrability it remains to prove sup E(V¯ (α−1) ) < ∞ s≤u≤t

u

28

ON EXPONENTIAL FUNCTIONALS

in the same manner as before. After limit passage we get that Z t Z t α α α ¯ (47) E(Vt ) = E(Vs ) − E(Vu ) dΦ(u, α) + α E(Vuα−1 )du s

s

We differentiate each term of this equality w.r.t. s to obtain that d ¯ s(α) − αE(Vsα−1 ) = 0 E(Vsα ) + E(Vsα ) H ds (α)

(α−1)

We take in account that E(Vsα ) = mt−s,t , E(Vsα−1 ) = mt−s,t and that (α) ¯ s(α) = Ht−s H . This gives us that d (α) (α) (α) (α−1) mt−s,t + mt−s,t Ht−s − α mt−s,t = 0 ds Finally, replacing t − s by s we get (41). In the case of Levy processes (α) (α) (α) mt−s,t = ms due to homogeneity, and Ht−s = Φ(α), and this gives (42).  To present the results about negative moments of Levy process X with the parameters (b0 , c0 , K0 ), we introduce the condition: Z (48) e(|α|+1)x K0 (dx) < ∞. x>1

Corollary 4. (cf. [7])Let X be Levy process verifying (35) and (48), (α−1) and let α ≤ −1 be fixed. Suppose that m∞ < ∞. Then the Laplace(α) (α−1) Carson transforms of m∞ and m∞ are well-defined and we have a recurrent formula:
1 m ˆ (α) ˆ (α) (49) m ˆ (α−1) = q Φ(α) + q m q q α and, hence, 1 (α) m ˆ (α−1) = m ˆ (q + Φ(α)) q α q In particular, under above conditions, for integer n ≥ 2 and α = −n we get: (50)

m ˆ (−n) q

=

m ˆ (−1) q

n−1 (−1)n−1 Y · (q + Φ(−k)) (n − 1)! k=1

As a consequence, (51)

−n E(I∞ )

=

−1 E(I∞ )

n−1 (−1)n−1 Y · Φ(−k) (n − 1)! k=1

ON EXPONENTIAL FUNCTIONALS

29

Proof. Two first equalities given above follow directly from Theorem 2. The third one can be proved in the same way as in Corollary 2 by letting q → 0.  Example 10. For compound Poisson process presented in Example 5 with Uk ’s which follows standard normal distribution, we get for n ≥ 1 n−1 −1 E( I∞ ) Y k2 (e 2 − 1). (n − 1)! k=1

−n E(I∞ )=

Corollary 5. Let β = sup{k ≥ 1 | − ∞ < Φ(−l) < 0 for 1 ≤ l ≤ k} −(n+1) with sup{∅} = 1. Then E(I∞ ) < ∞ if and only if n ≤ β and −1 E(I∞ ) < ∞. In particular, for Brownian motion with the drift coefficient b0 and the −(n+1) diffusion coefficient c0 6= 0, E(I∞ ) < ∞ if and only if 2bc00 > −1. R If X is a subordinator with R+ \{0} xK0 (dx) < ∞, then Z Z Φ(−k) = −k[b0 − xK0 (dx)] − (ekx − 1)K0 (dx), R+ \{0}

R+ \{0}

and under the condition

b0 −

(52) −(n+1)

E(I∞

Z

R+ \{0}

) < ∞ if and only if −(n+1)

xK0 (dx) ≥ 0

R

R+ \{0}

(enx − 1)K0 (dx) < ∞.

Proof. Suppose that E(I∞ ) < ∞ for some n > 0. Then by Cauchy−k Schwartz inequality we get that for all k, 1 ≤ k ≤ n, E(I∞ ) < ∞. −1 Then the formula (51) yields that and E(I∞ ) < ∞ and −∞ < −1 Φ(−k) < 0 for 1 ≤ k ≤ n. Hence, n ≤ β and E(I∞ ) < ∞. −1 Conversely, if n ≤ β and E(I∞ ) < ∞, then −∞ < Φ(−k) < 0 for 1 ≤ k ≤ n. Then from (41) m−(k+1) = s since

Φ(−k) (−k) 1 d (−k) Φ(−k) (−k) |Φ(−k)| (k) ms − m ≤ ms ≤ m∞ −k k ds s −k k

(−k) d m ds s

≥ 0. Hence,

−(n+1) E(I∞ )

=

m−(n+1) ∞

≤

n Y |Φ(−k)|

k=1

k

−1 E(I∞ ) < ∞.

In the case of Brownian motion we conclude that Φ(−k) = −kb0 − 1 2 k c0 > −∞ for all k ≥ 1. Since Φ is concave function with Φ(0) = 0, 2 the condition Φ(−1) < 0 ensures the existence of all negative moments. In the case when X is a subordinator, and under mentioned condition

30

ON EXPONENTIAL FUNCTIONALS

(52), all Φ(−k) < 0 and only the condition of finiteness of Φ(−k) play a role in the existence of the negative moments.  Example 11. Let us apply the Corollary 5 to time changed Brownian motion considered in Example 8. We get that Φ(α) < 0 whenever −b2 ≤ 2αµ − α2 σ 2 < 0. Hence, all negative moments of I∞ exists if −1 E(I∞ ) < ∞ and 2µ + σ 2 > 0. Acknowledgement We are grateful to our referees for very useful remarks and comments. References [1] S. Asmussen. Ruin probabilities, World Scientific, 2000. [2] O. Barndorff-Nielsen, A.N. Shiryaev. Change of Time and Change of Measure. World Scientific, 2010. [3] A. Behme (2015) Exponential functionals of L´evy Processes with Jumps, ALEA, Lat. Am. J. Probab. Math. Stat. 12 (1), 375-397. [4] A. Behme, A. Lindner (2015) On exponential functionals of Levy processes, Journal of Theoretical Probability 28, 681-720. [5] J. Bertoin. L´evy processes, Cambridge University Press, 1996, p.266. [6] J. Bertoin, A. Lindler, R. Maller (2008) On continuity Properties of the Law of Integrals of Levy Processes, In S´eminaire de probabilit´es XLI, 1934, 137–159. [7] J. Bertoin, M. Yor (2005) Exponential functionals of Levy processes, Probability Surveys, 191-212. [8] A. Borodin, P. Salminen. Handbook of Brownian motion - Facts and Formulae, Birkh¨ auser Verlag, Basel-Boston-Berlin, 2002, 672p. [9] P. Carmona, F. Petit, M. Yor (1997) On the distribution and asymptotic results for exponential functionals of Levy processes, In ”Exponential functionals and principal values related to Brownian motion”, 73-130. Biblioteca de la Revista Matematica IberoAmericana. [10] P. Carr, L. Wu. (2004) Time-changed L´evy processes and option pricing, Journal of Financial Economics 71, 113–141. [11] D. Dufresne (1990) The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J., 1-2, 39-79. [12] I. Epifani, A. Lijoi, I Pr¨ unster (2003) Exponential functionals and mean of neutral-to-the-right priors, Biometrika, 90, 4, 791-808. [13] K.B. Erickson, R. Maller (2004) Generalised Ornstein-Uhlenbeck processes and the convergence of L´evy integrals., p. 70-94. In : S´eminaire de probabilit´es, Lect. Notes Math. 1857, Springer, Berlin. [14] H.K. Gjessing, J. Paulsen (1997) Present value distributions with applications to ruin theory and stochastic equations, Stochastic Process. Appl. 71 (1), 123144. [15] J. Jacod, A. Shiryaev. Limit theorems for Stochastic Processes, SpringerVerlag, 1987, 606p. [16] M. Jeanblanc, M. Yor, M. Chesnay. Mathematical Methods for Financial Markets, Springer Finance Textbook, 2009, 332p.

ON EXPONENTIAL FUNCTIONALS

31

[17] Yu. Kabanov, S. Pergamentshchikov (2016) In the insurance business risky investment are dangerous: the case of negative risk sums, Finance and Stochastics, 20, 2, 355-379. [18] C. Kardaras, S. Robertson (2014) Continuous time perpetuities and time reversal of diffusions ARXIV: 1411.7551v1. [19] A. Kuznetsov, J.C. Prado, M.Savov (2012) Distributional properties of exponential functionals of Levy processes, Electron. J. Probab. 8, 1-35. [20] A. Kyprianou (2014) Fluctuations of L´evy processes with applications, SpringerVerlag, Berlin, Heildelberg, second addition, 2014. [21] J. Lamperti (1972) Semi-stable Markov Processes. Z. Wahrscheinlichkeitstheorie verv. Gebiete, 22, 205-225. [22] J. C. Pardo, V. Rivero, K. Van Schaik (2013) On the density of exponential functionals of L´evy processes, Bernoulli, 1938-1964. [23] A. A. Novikov, N. E. Kordzahiya (2012 ) Pitman estimators: an asymptotic variance revisited, Teor. Veroyatnost. i Primenen., Volume 57, Issue 3, 603–611. [24] P. Patie, M. Savov (2016) Bernstein-Gamma functions and exponential functionals of L´evy processes, arXiv:1604.05960v2. [25] J. Paulsen (2008) Ruin models with investment income, Probability Surveys, vol. 5, 416-434. [26] P. Salminen, O. Wallin. (2005) Perpetual integral functional of diffusions and their numerical computations, Dept. of Math, Univ. of Oslo, Pure Mathematics, 35. [27] P. Salminen, L. Vostrikova (2016) On moments of exponential functionals of additive processes, Peprint. [28] P. Salminen, M. Yor. (2005) Perpetual Integral Functionals as Hitting and Occupation Times, Electronic Journal of Probability, Vol. 10, Issue 11, 371-419. [29] K. Sato. L´evy Processes and Infinitely Divisible Distributions, Cambridge University Press, second edition, 2013. [30] A. N. Shiryaev. Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific, 1999, p. 834. [31] A. N. Shiryaev, A. S. Cherny (2002) Vector Stochastic Integrals and the Fundamental Theorems of Asset Pricing, Proc. Steklov Inst. Math. 237, 6-49.