This study was inspired by the questions arising in mathematical finance, namely by the questions related to perpetuities containing the liabilities, perpetuities subjected to the influence of economical factors (see, for example, Kardaras, Robertson [23]), and also with the price of Asian options and similar questions (see, for instance, Jeanblanc, Yor, Chesnay [21], Vecer [38] and references there). The study of exponential functionals is also important in the insurance, since the distributions of these functionals appear very naturally in the ruin problem (see, for example, Asmussen [2], Paulsen [29], Kabanov, Pergamentshchikov [22], Spielmann, Vostrikova [37] and the references there).

In mathematical finance exponential functionals of the processes with independent increments (PII in short; in what follows, this abbreviation will also denote the property of being a process with independent increments) arise very often1. This fact is related to the observation that log price is usually not a homogeneous process on a relatively long time interval. For this reason several authors used for the modeling of log price the process $X={({X_{t}})_{t\ge 0}}$ such that where L is a Lévy process and g is a càdlàg random process independent of L for which the integral is well defined. In this case, the conditioned process given σ-algebra generated by g, is a PII. Another important example of X is a Lévy process L with time changed by an independent increasing process ${({\tau _{t}})_{t\ge 0}}$ (cf. Carr, Wu [16]), i.e. Again, conditionally to the process τ, the process $X={({X_{t}})_{t\ge 0}}$ is PII.

In Salminen, Vostrikova [31, 32] we proved the recurrent formulas for the Mellin transform and we used 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 and also where the process $X={({X_{t}})_{t\ge 0}}$ is PII. It is not difficult to see that under the condition we have ${I_{\infty }}<+\infty $ (P-a.s.). As it was shown in Bertoin, Yor [10], this condition is necessary and sufficient in the case of Lévy processes.

Exponential functionals of Lévy processes were studied in a large number of articles, most of them were related to the study of ${I_{\infty }}$. In the celebrated paper by Carmona, Petit, Yor [15] the asymptotic behaviour of exponential functionals ${I_{\infty }}$ was studied, in particular for α-stable Lévy processes. The authors also gave an integro-differential equation for the density of the law of exponential functionals, when this density exists w.r.t. the Lebesgue measure. The questions related to the characterization of the law of exponential functionals by the moments were studied by Bertoin, Yor in [10].

In a more general setting, related to the Lévy case, the functional where $X={({X_{t}})_{t\ge 0}}$ and $\eta ={({\eta _{t}})_{t\ge 0}}$ are independent Lévy processes, was intensively studied. The conditions for finiteness of the integral (2) were obtained in Erickson, Maller [18]. The continuity properties of the law of this integral were studied in Bertoin, Lindner, Maller [9], where the authors gave the conditions for absence of atoms and also the conditions for absolute continuity of laws of integral functionals w.r.t. the Lebesgue measure. The question of smoothness of the density of the law of ${I_{\infty }}$ in the Schwarz sense was considered in Carmona, Petit, Yor [15] and Bertoin, Yor [10]. Under the assumptions on the existence of smooth density of these functionals, the equations for the density are given in Carmona, Petit, Yor [15], Bertoin, Yor [10], Behme [4], Behme, Lindner [5], Kuznetsov, Pardo, Savov [24].

In the papers by Patie, Savov [28], Pardo, Rivero, Van Shaik [27], again for a Lévy process, the properties of exponential functionals ${I_{{\tau _{q}}}}(\eta )$ killed at an independent exponential time ${\tau _{q}}$ of the parameter $q>0$, were investigated. In the article [27] the authors studied the existence of the density of the law of ${I_{{\tau _{q}}}}(\eta )$, they gave an integral equation for the density and the asymptotics of the law of ${I_{\infty }}(\eta )$ at zero and at infinity, when X is a positive subordinator. The results given in [28] involve the analytic Wiener–Hopf factorisation, Bernstein functions and contain the conditions for regularity, semi-explicite expression and asymptotics for the distribution function of ${I_{{\tau _{q}}}}(\eta )$. In Behme, Lindner, Reker, Rivero [6], Behme, Lindner, Reker [7] the authors give sufficient conditions for absolute continuity of the laws of ${I_{{\tau _{q}}}}(\eta )$ as well as the sufficient conditions for the absolute continuity of the laws of ${I_{t}}(\eta )$ and ${I_{\infty }}(\eta )$.

Despite numerous studies, the distributions of ${I_{t}}(\eta )$ and ${I_{\infty }}(\eta )$ are known only in a limited number of cases. When X is a Brownian motion with drift, the distributions of ${I_{t}}$ and ${I_{\infty }}$ were studied in Dufresne [17] and for a large number of specific processes X and η, like a Brownian motion with drift and a compound Poisson process, the distributions of ${I_{\infty }}(\eta )$ were given in Gjessing, Paulsen [19].

The exponential functionals for diffusions stopped at the first hitting time were studied in Salminen, Wallin [30], where authors derive the Laplace transform of the functionals and then, to find their laws, perform numerical inversion of the Laplace transform. The relations between hitting times and occupation times for the exponential functionals were considered in Salminen, Yor [33], where the versions of identities in law such as Dufresne’s identity, Ciesielski–Taylor’s identity, Biane’s identity, LeGall’s identity were given.

In this article we consider a real-valued process $X={({X_{t}})_{t\ge 0}}$ with independent increments and ${X_{0}}=0$, which is a semimartingale with respect to its natural filtration. We denote by $(B,C,\nu )$ a semimartingale triplet of this process, which can be chosen to be deterministic (see Jacod, Shiryaev [20], Ch. II, p.106). We suppose that $B={({B_{t}})_{t\ge 0}}$, $C={({C_{t}})_{t\ge 0}}$ and ν are absolutely continuous with respect to the Lebesgue measure, i.e. with càdlàg functions $b={({b_{s}})_{s\ge 0}}$, $c={({c_{s}})_{s\ge 0}}$, $K={({K_{s}}(A))_{s\ge 0,A\in \mathcal{B}(\mathbb{R})}}$. We assume that the compensator of the measure of jumps ν verifies the usual relation: for each $t\in {\mathbb{R}^{+}}$ For the main result we will assume an additional technical condition: The last condition implies that $\mathbf{E}(|{X_{t}}|)<+\infty $ for $t>0$ (cf. Sato [34], Th. 25.3, p.159) so the truncation of jumps is no more necessary.

We recall that the characteristic function of ${X_{t}}$, is expressed as follows: for $\lambda \in \mathbb{R}$ We recall also that X is a semimartingale if and only if for all $\lambda \in \mathbb{R}$ the characteristic function of ${X_{t}}$ is of finite variation in t on finite intervals (cf. Jacod, Shiryaev [20], Ch.2, Th. 4.14, p.106). Moreover, the process X always can be written as a sum of a semimartingale and a deterministic function which is not necessarily of finite variation on finite intervals.

The article is organized as follows. Part 2 is devoted to the Kolmogorov type equation for the law of ${I_{t}}$. It is known that the exponential functional ${({I_{t}})_{t>0}}$ is not a Markov process with respect to the filtration generated by the process X. It is a continuous increasing process, what prevents the use of stochastic calculus in an efficient way. For these reasons we fix t and introduce a family of stochastic processes ${V^{(t)}}={({V_{s}^{(t)}})_{0\le s\le t}}$ 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 a fixed time t and gives a Generalised Ornstein-Uhlenbeck process process with a PII noise (GOU process in short). We give the Kolmogorov type equations for ${V^{(t)}}$ (see Theorem 1). Assuming 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 ${V_{s}^{(t)}},0<s<t$. The density of the law of ${V_{t}^{(t)}}$ can be obtained just by integration of the right-hand side of the equation for the density of the law of ${V_{s}^{(t)}}$ in s on the interval $]0,t[$.

In Part 3 we consider the question of existence of the smooth density of the process ${V^{(t)}}$. The question of existence of the density of the law of ${V_{s}^{(t)}}$, $0\le s\le t$, of the class ${C^{1,2}}(]0,t[\times {\mathbb{R}^{+,\ast }})$, where ${\mathbb{R}^{+,\ast }}=\mathbb{R}\setminus \{0\}$, is a rather difficult one, which was an open question in all cited papers on exponential functionals (see [15, 10, 4, 5, 24]). Since the processes ${V_{t}^{(t)}}$ and ${I_{t}}$ coincide (P-a.s.), the results on the density of ${V_{t}^{(t)}}$ give the answer for the density of ${I_{t}}$. In Proposition 2 we give sufficient conditions for the existence of the density of the class ${C^{\infty }}(]0,t[\times {\mathbb{R}^{+,\ast }})$ of the law of ${I_{t}}$ when X is a Lévy process. For a non-homogeneous PII we give a partial answer to this question in Corollary 1.

Part 4 is devoted to Lévy processes. When X is a Lévy 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 ${I_{\infty }}$, since for these equations we have the explicit boundary conditions (cf. Corollary 2). In Corollary 3 we consider the well-known Brownian case. In Corollary 4 we give the equations for the case of Lévy processes with integrable jumps, and in Corollary 5, we consider the case of exponential jumps. In the particular case of ${I_{\infty }}$ and integrable jumps the equations coincide with the known ones from [15].

We introduce, for fixed $t>0$, a time reversal process $Y={({Y_{s}})_{0\le s\le t}}$ with Of course, this process depends on the parameter t, but we will omit this parameter for the simplicity of notations.

For convenience of the readers we present here Lemma 1 and Lemma 2 proved in Salminen, Vostrikova [31]. The first result establishes the relation between ${I_{t}}$ defined by (1) and the process $Y={({Y_{s}})_{0\le s\le t}}$.

In the following lemma we claim that Y is PII and we precise its semimartingale triplet. For that we introduce the functions $\bar{b}={({\bar{b}_{u}})_{0\le u\le t}}$, $\bar{c}={({\bar{c}_{u}})_{0\le u\le t}}$ and $\bar{K}={({\bar{K}_{u}})_{0\le u\le t}}$ putting where ${\mathbf{1}_{\{\cdot \}}}$ is the indicator function and $A\in \mathcal{B}(\mathbb{R})$. It means, for instance for $\bar{b}={({\bar{b}_{u}})_{0\le u\le t}}$, that So, the function $\bar{b}$ can have a discontinuity at t, since in general ${b_{0}}\ne {b_{t}}$. These functions are constructed to correspond, in a way, to the derivatives in time of the semimartingale characteristics of the process Y.

To obtain an integro-differential equation for the density, we introduce two important processes related to the process Y, namely the process $V={({V_{s}})_{0\le s\le t}}$ and $J={({J_{s}})_{0\le s\le t}}$, with We underline that both processes depend on the parameter t, since it is so for the process Y.

We notice that according to Lemma 1, ${I_{t}}={V_{t}}$ (P-a.s.), and then they have the same laws. As we will see, the process $V={({V_{s}})_{0\le s\le t}}$ is a Markov process with respect to the natural filtration ${\mathbb{F}^{Y}}={({\mathcal{F}_{s}^{Y}})_{0\le s\le t}}$ of the process Y and this fact will help us very much to find the equation for the density of the law of ${I_{t}}$.

We define the set of functions such that $\hspace{0.1667em}f(0)={f^{\prime }}(0)=0$. Here we use the usual notation ${\mathcal{C}_{b}^{2}}({\mathbb{R}^{+}})$ for the set of twice continuously differentiable functions with bounded derivatives on ${\mathbb{R}^{+}}$.

For $0\le s\le t$ we put We notice that the assumptions (3) and (4) imply that (λ-a.s.) so that ${\overline{a}_{s}}$ is (λ-a.s.) well defined. We introduce also for $f\in \mathcal{C}$ the generator ${({\mathcal{A}_{s}^{V}})_{0\le s<t}}$ of the process V via

The proof of Theorem 1 will be divided in four parts. In Lemma 4 we give a semimartingale decomposition of the process ${({e^{-{Y_{s}}}})_{0\le s\le t}}$, then in Lemma 5 we prove a semimartingale decomposition of ${(f({V_{s}}))_{0\le s\le t}}$ with f belonging to $\mathcal{C}$ and in Lemma 6 we prove (13), and finally in Lemma 7 we obtain the equation (14). Then we combine all results together to get (15).

In the case of Lévy processes, the question of existence of a smooth density of ${I_{\infty }}$ in the Schwarz sense was considered in Carmona, Petit, Yor [15] and Bertoin, Yor [10]. Sufficient conditions for the smoothness of the one-dimensional law of an Ornstein–Uhlenbeck process driven by a pure discontinuous Lévy process was obtained in Bondarchuk, Kulik [14], namely under the Kallenberg condition the density of the one-dimensional distribution of an Ornstein–Uhlenbeck process is of the class ${C_{b}^{\infty }}(\mathbb{R})$.

The question of existence of the density of the law of ${V_{s}}$, $0<s<t$, of the class ${C^{1,2}}(]0,t[,\mathbb{R})$ is a rather difficult one, which was, at our knowledge, an open question in all cited works on exponential functionals. We give here a partial answer to this question via the known result on the Malliavin calculus from Bichteler, Gravereaux, Jacod [11]. For the convenience of the readers we present this result here in the one-dimensional setting.

We consider the stochastic differential equation where $x\in \mathbb{R}$, $a,b,c$ are measurable functions on $\mathbb{R}$ and ${\mathbb{R}^{2}}$ respectively, W is the standard Brownian motion, and μ and ν are the jump measure and its compensator of ${Z^{x}}$. It is assumed that the solution of this equation exists and is unique, and also that the following assumptions hold.

Assumption (A-r):

Assumption (SB): there exist $\gamma >0$ and $\delta >0$ such that

Assumption (SC-bis): there exists $\zeta >0$ such that for all $u\in [0,1]$

Theorem 2.29 (cf. [11], p. 15) Suppose that the assumptions (A-(2r+10)), (SB) and (SC-bis) are satisfied. Then for $t>0$ the law of ${Z_{t}^{x}}$ has a density ${p_{t}}(x,y)$ w.r.t. the Lebesgue measure and the map $(t,x,y)\to {p_{t}}(x,y)$ is of the class ${C^{r}}(]0,t]\times \mathbb{R}\times \mathbb{R})$.

To apply this theorem, let us write a stochastic differential equation for ${({V_{s}})_{0\le s\le t}}$. For that we put for $0\le s\le t$

To apply Theorem 2.29 from [11] we suppose that the process X, and hence also the process Y, are Lévy processes. We introduce a supplementary process with $\epsilon >0$. We see that ${V_{s}^{\epsilon }}-\epsilon ={V_{s}}$, and ${V_{s}^{0}}={V_{s}}$, and also that the density ${p_{s}}(\epsilon ,y)$ of the law of ${V_{s}^{\epsilon }}$ w.r.t. the Lebesgue measure and the density ${p_{s}}(y)$ of the law of ${V_{s}}$ exist or not at the same time and they are related: for all $x>0$ and $y>0$ So, both densities are of the same regularity w.r.t. $(s,y)$. In addition we have: for all $s\ge 0$ and all $\omega \in \Omega $, ${V_{s}^{\epsilon }}(\omega )\ge \epsilon $.

For the PII case we obtain the following partial result.

In this section we consider a particular case of Lévy processes. Namely, let X be a Lévy process with the parameters $({b_{0}},{c_{0}},{K_{0}})$. As before, we suppose that and we put

Due to the homogeneity of a Lévy process, the equation for the density can be simplified as we can see from the following proposition.

Let us denote by ${\nu ^{+}}$ and ${\nu ^{-}}$ the Lévy measures of positive and negative jumps, respectively; namely, for $x>0$ To simplify the notations, we put also Let us suppose in addition that