On distributions of exponential functionals of the processes with independent increments

The aim of this paper is to study the laws of the exponential functionals of the processes $X$ with independent increments, namely $$I_t= \int _0^t\exp(-X_s)ds, \,\, t\geq 0,$$ and also $$I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$$ Under suitable conditions we derive the integro-differential equations for the density of $I_t$ and $I_{\infty}$. We give sufficient conditions for the existence of smooth density of the laws of these functionals. In the particular case of Levy processes these equations can be simplified and, in a number of cases, solved explicitly.


Introduction
This study was inspired by the questions arising in mathematical finance, namely by the questions related with the perpetuities containing the liabilities, perpetuities subjected to the influence of the economical factors (see, for example, [19]), and also with the price of Asian options and similar questions (see, for instance, [17], [31], and references there). The study of exponential functionals are also important in insurance, since the insurace companies invest the money on risky assets. Then the distributions of these functionals appear very naturally in ruin problem (see for example [2], [24], [18] and references there).
In mathematical finance the non-homogeneous PII models are more realistic, since price processes are not usually homogeneous processes. For this reason, for example, several authors used for the modelling of log price a process X such that where L is Levy process and g is independent from L cádlàg random process for which the integral is well defined. In this case, the conditional distribution of the process X given σ -algebra generated by g, is a PII. Another important example is a Levy process L time changed by an independent increasing process (τ t ) t≥0 (cf. [12]), i.e.
Again, conditionally to the process τ , the process X is PII.
In [26] we obtained the recurrent formulas for Mellin transform and we use these formulas to calculate the moments of exponential functionals of the processes with independent increments. In this paper we obtain the equations for the densities, when they exist, of the laws of exponential functionals I t and I ∞ of PII processes.
Exponential functionals for Levy processes was studied in a big number of articles, the main part of them was related to the study of I ∞ . The asymptotic behaviour of exponential functionals I ∞ was studied in [11], 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. Lebesgue measure exists. The questions related with the characterisation of the law of exponential functionals by the moments was studied in [8].
In more general setting, related to the Lévy case, the following functional (1) ∞ 0 exp(−X s− )dη s where X = (X t ) t≥0 and η = (η t ) t≥0 are independent Lévy processes, was intensively studied. The conditions for finiteness of the integral (1) was obtained in [14]. The continuity properties of the law of this integral was studied in [7], 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 about the existence of the smooth density of these functionals, the equations for the density are given in [4], [5], [20].
In the papers [23] and [22], again for Levy process, the properties of the exponential functionals I τq killed at the independent exponential time τ q of the parameter q > 0, was investigated. In the article [22] the authors studied the existence of the density of the law of I τq , 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 [23] involve analytic Wiener-Hopf factorisation, Bernstein functions and contain the conditions for regularity, semiexplicite expression and asymptotics for the distribution function of I τq . Despite numerous studies, the distribution properties of I t and I ∞ are known only in a limited number of cases. When X is Brownian motion with drift, the distributions of I t and I ∞ was studied in [13] 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 [15].
Exponential functionals for diffusions stoped at first hitting time was studied in [25], where authors derive Laplace transform of the functionals and then, to find their laws, perform numerical inversion of Laplace transform. The relations between hitting times and occupation times for exponential functionals was considered in [27], where the versions of identities in law such as Dufresne's identity, Ciesielski-Taylor's identity, Biane's identity, LeGall's identity was given.
In this article we consider a real valued process X = (X t ) t≥0 with independent increments and X 0 = 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 [16], Ch. II, p.106). We suppose that B = (B t ) t≥0 , C = (C t ) t≥0 and ν are absolutely continuous with respect to Lebesgue measure, i.e.
We assume that the compensator of the measure of jumps ν verify the usual relation: for each t ∈ R + For the the main result we will suppose an additional technical condition that The last condition implies that E(|X t |) < ∞ for t > 0 (cf [28], Th. 25.3, p.159) so that the truncation of the jumps is no more necessary.
We recall that the characteristic function of X t is defined by the following expression: We recall also that X is a semi-martingale if and only if for all λ ∈ R the characteristic function of X t is of finite variation in t on finite intervals (cf. [16], 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.
The article is organized as follows. The part 2. is devoted to the Kolmogorov type equation for the law of I t . As known, the exponential functional (I t ) t>0 is not a Markov process with respect to the filtration generated by the process X. It is continuous increasing process, which prevent the use of the stochastic calculus in an efficient way. For these reasons we fix t and we introduce a family of stochastic processes V (t) = (V (t) s ) 0≤s≤t indexed by t and such that I t = V (t) t (P -a.s.) (see Lemma 1). The construction of such processes is made via the time reversion of the process X at the fixed time t, and this gives us the process is also PII, and if X is Levy process, then Y (t) is so (see Lemma 2). We prove then that V (t) is a Markov process with respect to the natural filtration of the process Y (t) (see Lemma 3), we find the generator and we give the Kolmogorov-type equations for V (t) (see Theorem 1). Supposing the existence of the smooth density of the law of the process V (t) we derive the integro-differential equation for the density of the laws of V  t− coincide. The last fact follows from the absence of the predictable jumps of the process X, and, hence, the ones of Y (t) .
In the part 3. we consider the question of existence of the smooth density of the process V (t) . The question of the existence of the density of the law of V (t) s , 0 ≤ s ≤ t, of the class C 1,2 (]0, t[, R +, * ) is rather difficult question, which was open question in all cited papers on exponential functionals. We remark that in the case of Levy processes the laws of V (t) t and I t coincide. We will give in Proposition 2 sufficient conditions for the existence of the density of the class C ∞ (]0, t[, R +, * ) when X is Levy process. For non-homogeneous diffusions we give a partial answer on this question in Corollary 1.
The part 4 is devoted to Levy processes. When X is Levy process, the equations for the density of I t can be simplified due to the homogeneity (see Proposition 1). We present also the equations for the distribution functions of the laws of I t and I ∞ , since these versions have the explicit boundary conditions, which gives the advantages from numerical point of view (cf. Corollary 2). In Corollary 3 we consider very known Brownian case. In Corollary 4 we give the equations for the case of Levy processes with integrable jumps, and in Corollary 5, we consider the case of exponential jumps.

Kolmogorov type equation for the density of the law of I t
We introduce, for fixed t > 0, a time reversal Of course, this process depends on the parameter t, but we will omit it for simplicity of the notations.
For convenience of the readers we present here Lemma 1 and Lemma 2 proved in [26]. First result establish the relation between I t and the process Y = (Y s ) 0≤s≤t .
Proof We write, using the definition of the process Y and the assumption that X 0 = 0: The last equality holds after time change noticing that the integration of left-hand version and right-hand version of some càdlàg process w.r.t. Lebesgue measure gives the same result. 2 In the following lemma we claim that Y is PII and we precise its semimartingale triplet. For that we introduce the functionsb = So, the functionb can have a discontinuity at t, since in The process Y is a process with independent increments, it is a semi-martingale with respect to its natural filtration, and its semi-martingale triplet (B,C,ν) is given by To obtain an integro-differential equation for the density, we introduce two important processes related with the process Y , namely the process V = (V s ) 0≤s≤t and J = (J s ) 0≤s≤t with We underline that the both processes depend of the parameter t, since it is so for the process Y .
We notice that according to Lemma 1, I t = V t , and then they have the same laws. As we will see, the process V = (V s ) 0≤s≤t is a Markov process with respect to the natural filtration F Y = (F Y s ) 0≤s≤t of the process Y and this fact will help us very much to find the equation for the density of the law of I t .
Lemma 3. The process V = (V s ) 0≤s≤t is a Markov process with respect to the natural filtration F Y = (F Y s ) 0≤s≤t of the process Y .
Proof. We write that for h > 0 Then for all measurable bounded functions f is a measurable function of V s and we conclude that V is Markov process with respect to the filtration generated by Y .
We define the set of functions We notice that the conditions (3) and (4) imply that ( λ-a.s.) Theorem 1. Let us suppose that the conditions (2), (3) and (4) are verified. Then the infinitesimal generator (A V s ) 0≤s<t of the Markov process V is defined by (11). In addition, for 0 ≤ s ≤ t and f ∈ C If for 0 < s ≤ t the density p s w.r.t. Lebesgue measure λ of the law of V s exists and belongs to the class C 1,2 (]0, t[×R +, * ), then λ-a.s.
and the density p t of the law of I t verify: Remark 1. The existence and the uniqueness of the solution of the integro-differential equation given in Theorem 1 follows from the possibility to identify the characteristics of the corresponding process from the equation. Since the law of PII process, which is a semi-martingale, is uniquely defined by its characteristics, the solution of such equation exists and is unique.
Proof of Theorem 1. The proof of Theorem 1 will be divided in three parts : in the first part we prove a decomposition (20), then using limit passage we prove (12), and finally, in the third part we obtain the equations (13), (14) for the densities.
1) Proof of (20). For f ∈ C and 0 ≤ s ≤ t we write the Ito formula: where µ V is the measure of jumps of the process V . From the definition of the process V we can easily find that and that At the same time, again by the Ito formula we get the following decomposition (17) e −Ys = e −Y 0 + A s + N s .
In this decomposition the process (A s ) s≥0 is defined via and it is a process of locally bounded vatiation on bounded intervals. In fact, let us introduce a sequence of stopping times: for n ≥ 1  (2), (3) and (4) that In (17) the process N = (N s ) s≥0 is defined by In the relation (19), the processM is the martingale component of the semi-martingale decomposition of Y : Y s =B s +M s , and µ Y is the measure of jumps of the process Y . It should be noticed that since Y is a process with independent increments andB is deterministic,M is a martingale (see [30], Th. 58, p. 45) as well as its pure discontinuous partM d . Then, the process (N s∧τn ) 0≤s≤t is a local martingale as a stochastic integral of a bounded function w.r.t. a martingale.
We put (16), (17), (18) and (19) into (15) to obtain a final decomposition for f (V s ). To present this final decomposition we put for y ≥ 0 and and also Finally, we get a decomposition In this decomposition B V is a process with locally bounded variation and N V is a local martingale. In fact, let us use the same sequence of stopping times τ n as previously and let D = sup y∈R max(|f (y)|, |f (y)|, |f (y)y|, |f (y)y 2 |). Then, The first term of the r.h.s. is finite since the functions (B s ) 0≤s≤t and (C s ) 0≤s≤t have finite variation on finite intervals and since (4) holds. Now, using the Taylor-Lagrange formula of the second order, we find that for y > 0 and |x| ≤ 1 where 0 < θ < 1. Since for |x| < 1, 1 + θ(e −x − 1) ≥ 1 e and |e −x − 1| ≤ e|x|, we find that |F (V u− , x)| ≤ 1 2 De 4 x 2 . For |x| > 1 we use Taylor-Lagrange formula of the first order to get with some positive constant C. Then, the conditions (3) and (4) implies that Using above results we see that (N V s∧τn ) 0≤s≤t is a local martingale as an integral of a bounded function w.r.t. a martingale.
2) Proof of (12). Let (τ n ) n∈N be the localising sequence for N V and τ n = τ n ∧ τ n . Let s ∈ [0, t[ and δ > 0 such that s + δ ≤ t. Then, from previous decomposition using the localisation we get: Since f is bounded function and lim n→∞τn = +∞, we can pass to the limit in the l.h.s. by the Lebesgue convergence theorem. The same can be done on the r.h.s. since the process B V = (B V s ) 0≤s≤t is a process of bounded variation on bounded intervals, uniformly in s and n. After taking a limit as n → +∞ we get that We remark that We show that the quantities are uniformly bounded, for small δ > 0, by a constant. In fact, we can write that are uniformly bounded, for small values of δ > 0, by a constant, we deduce that the quantities are uniformly bounded for small δ by a constant, too. Under these conditions we can exchange the limit and the conditional expectation and it gives us the expression for the generator of V at 0 ≤ s < t. As a conclusion, we get that for 0 ≤ s < t Then after the replacement of V s− by V s in (21) and the integration w.r.t. s we obtain (12). For s = t we take lim s→t− in (12). (13) and (14). We denote by P s the law of V s . Then from (12) we get that for 0 ≤ s ≤ t

3) Proof of
Moreover, since for s > 0, the law P s of V s has a density p s w.r.t. Lebesque measure, it gives To obtain the equation for the density, we consider the set of continuously differentiable functions on compact support C 2 K ⊆ C. We differentiate the above equation with respect to s to get Using the integration by part formula we deduce that The mentioned relations together with the equation (24) gives that for all f ∈ C 2 K : and it proves our claim about the equation for p s .
We integrate the equation for p s on the interval ]0, t − δ[ for δ > 0 and we pass to the limit as δ → 0. Since the laws of V t− and V t coincide, we get in this way the equation for p t . 2

Some results about the existence of the smooth density
The question of the existence of the density of the law of V s , 0 ≤ s ≤ t, of the class C 1,2 (]0, t[, R) is rather difficult question, which was open question in all cited works on exponential functionals. We will give here a partial answer on this question via the known result on Malliavin calculus given in [9]. For the convenience of the readers we present this result here in the one-dimensional case.
We consider the following stochastic differential equation: where x ∈ R, a, b, c are measurable functions on R and R 2 respectively, W is standard Brownian motion, and µ and ν are jump measure and its compensator of X x . It is assumed that the solution of this equation exists and is unique, and also that the following assumptions hold.
To apply this theorem let us write stochastic differential equation for (V s ) 0≤s≤t . For that we put for 0 ≤ s ≤ t    a s (y) = y(−b s + 1 2c s + R (e −z − 1 + z)K s (dz) + 1, b s (y) = y √c s , c s (y, z) = y(e −z − 1).
and thatc s > 0 for 0 < s ≤ t. Then the process (V s ) 0≤s≤t satisfy the following stochastic differential equation: where µ Y is the jump measure of Y and W is DDS Brownian motion corresponding to the continuous martingale part Y c of Y .
Proof. We recall that V s is defined by (9). Let us introduce the procesŝ Y via the relation : for 0 ≤ s ≤ t Since (B,C,ν) are absolutely continuous w.r.t. Lebesgue measure with the derivatives (b,c,K) we get that where W is DDS Brownian motion corresponding to the continuous martingale part of Y . Let us put this decomposition into (27) and we obtain (25).
To apply the Theorem 2.29 from [9] we suppose that Y is a Levy process and we introduce the supplementary process We see that V x s − x = V s , and V 0 s = V s , and also that the density p s (x, y) of the law of V x s w.r.t. Lebesgue measure and the density p s (y) of the law of V s exist or not at the same time and are related: for all x > 0 and y > 0 p s (x, y + x) = p s (y).
So, the both densities are the same regularity w.r.t.(s, y).

Proposition 2.
Suppose that Y is Levy process with the triplet (b 0 , c 0 , K 0 ) and the following conditions are satisfied: Then, for s > 0, the law of V s has a density p s and the map (s, y) → p s (y) is of class C ∞ (]0, t], R +, * ).
Proof. When Y is Levy process, the functions a s , b s , c s figured in (25) are independent on s and are equal to: We consider the process V x with x > 0. We see that the Assumption (A-r) is satisfied for all r ≥ 1 with G = K 0 , as well as the Assumption (SB-(ζ, θ)) putting = x 2 c 0 and (SC-bis) taking ζ = 1 2 e −A . Then the map (s, x, y) → p s (x, y) is of class C ∞ (]0, t], R +, * × R +, * ), and the map (s, y) → p s (y) is of class C ∞ (]0, t], R +, * ) For non-homogeneous diffusion we obtain the following partial result. Then, the law of V s has a density p s such that the map y → p s (y) is of class C ∞ (R +, * ).
Proof. We notice that the law of Y s coincide with the law of Levy process with the triplet ( 1 sB s , 1 sC s , 1 s s 0 RK s (dz)du) at the time s. Then the previous proposition can be applied and it gives the claim.

When X is Levy process
In this section we consider a particular case of Levy processes. Namely, let X be Levy process with the parameters (b 0 , c 0 , K 0 ). As before we suppose that (28) R (x 2 ∧ 1)K 0 (dx) < ∞ and |x|>1 e |x| K 0 (dx) < ∞ and we put Due to the homogeneity of Levy process, the equation for the density can be simplified as we can see from the following proposition.
Proposition 3. Suppose that (28) holds and the density of the law of I t exists and belongs to the class C 1,2 (]0, t] × R +, * ). Then this density verify the following equation: In particular case, when I ∞ < ∞ (P -a.s.) and the density p ∞ of the law of I ∞ exists and belongs to the class C 2 (R +, * ), we have Proof. In the case of Lévy processes we write that (P -a.s.) Due to the homogeneity of the Lévy processes we have the following identity in law: Then, L( where the last equality is obtained by time change. As a conclusion, L(V s ) = L(I s ) for 0 ≤ s ≤ t, and, hence, (p s ) 0<s≤t are the densities of the laws of (I s ) 0<s≤t . In addition, again due to the homogeneity, for all Then, from the Theorem 1 we obtain (29).
Again due to the homogeneity, for 0 < s ≤ t, the generator and it does not depend on s. Moreover, L((V s ) 0≤s≤t ) = L((I s ) 0≤s≤t ) and the equality (12) become We divide the both sides of the above equality by s and we let s go to infinity. Since f is bounded, we get zero as a limit on the l.h.s.. Since Then, EA(f )(I ∞ ) = 0 and we obtain (30) in the same way as in Theorem 1, by the integration by parts and the time change.
Remark 2. Similar equation for the density of I ∞ in the case when R (|x| ∧ 1) K 0 (dx) < ∞ was obtained in [11]. We recall that the condition on K 0 of [11] is stronger at zero then our condition. It should be mentioned that the authors of [11] did not obtained the equation for the density of I t .
Corollary 2. Under the conditions of Proposition 3, the distribution function F t of I t verify second order integro-differential equation with following boundary conditions: When I ∞ < ∞, the similar equation is valid for the distribution function F ∞ of the law of I ∞ : with similar boundary conditions: F ∞ (0) = 0, lim y→+∞ F ∞ (y) = 1.
Proof. We integrate each term of the equation of Theorem 1 on [0, y] and we use the fact that y 0 p t (u)du = F t (y) since F t (0) = 0. We take in account the fact that that the map (t, u) → p t (u) is of class C 1.2 (R +, * × R +, * ) to exchange the integration and the derivation. The same we do for F ∞ (y).
Corollary 3. (cf. [13], [10]) Let us consider Brownian motion with drift, i.e. dX t = b 0 dt + √ c 0 dW t where c 0 = 0 and b 0 > 0. Then the law of exponential functional I t associated with X has a density which verify : ∂ ∂t p t (y) = 1 2 c 0 ∂ 2 ∂y 2 (y 2 p t (y)) − ∂ ∂y ((a 0 y + 1) p t (y)) In particular, for I ∞ we get that Remark 3. We recall that the distribution of I t for Brownian motion with drift was obtained in [13] and in [10], formula 1.10.4, p. 264.
Let us denote by ν + and ν − the Levy measure of positive and negative jumps respectively, namely for x > 0 To simplify the notations we put also Let us suppose in addition that R |x| K 0 (dx) < ∞.

Corollary 4.
Suppose that X is a Levy process with integrable jumps.
Proof. We take the equation (31) and we rewrite it in the following form: ∂ ∂t F t (y) = 1 2 c 0 ∂ ∂y (y 2 ∂ ∂y F t (y)) − (r 0 y + 1) Then we divide the integral over R in two parts integrating on ]0, +∞[ and ] − ∞, 0[. We do the integration by parts : and we change the variables z = ye x . We differentiate the result w.r.t. t and this gives us the claim. Then, under the assumptions of Proposition 3, the density p t of I t , verify : ∂ ∂t p t (y) = 1 2 c 0 ∂ 2 ∂y 2 (y 2 p t (y)) − ∂ ∂y ((r 0 y + 1) p t (y)) + y µ µ ∞ y p t (z) z µ dz In particular, for the density p ∞ of I ∞ we have : 1 2 c 0 ∂ 2 ∂y 2 (y 2 p t (y)) − ∂ ∂y ((r 0 y + 1) p t (y)) + y µ µ ∞ y p t (z) z µ dz = 0 Proof. We take in account that ν + (x) = 1 µ e −µx and ν − (x) = 0 for all x > 0 and this gives us the equation for p t in this particular case.