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 often1¹

See for example the book of E. Eberlein and J. Kallsen Mathematical finance , Springer 2019, and references there.

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 (1) It=∫0texp(−Xs)ds,t≥0, and also I∞=∫0∞exp(−Xs)ds, where the process X=(Xt)t≥0 is PII. It is not difficult to see that under the condition lims→+∞Xss=μ>0(P-a.s.) 0\hspace{0.1667em}\hspace{0.1667em}\hspace{0.1667em}(\mathbf{P}\text{-a.s.})\]]]> we have I∞<+∞ (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∞ . In the celebrated paper by Carmona, Petit, Yor [15] the asymptotic behaviour of exponential functionals I∞ 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 (2) I∞(η)=∫0∞exp(−Xs−)dηs, where X=(Xt)t≥0 and η=(ηt)t≥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∞ 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 Iτq(η) killed at an independent exponential time τq of the parameter q>0 0$]]>, were investigated. In the article [27] the authors studied the existence of the density of the law of Iτq(η) , they gave an integral equation for the density and the asymptotics of the law of I∞(η) 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 Iτq(η) . In Behme, Lindner, Reker, Rivero [6], Behme, Lindner, Reker [7] the authors give sufficient conditions for absolute continuity of the laws of Iτq(η) as well as the sufficient conditions for the absolute continuity of the laws of It(η) and I∞(η) .

Despite numerous studies, the distributions of It(η) and I∞(η) are known only in a limited number of cases. When X is a Brownian motion with drift, the distributions of It and I∞ 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∞(η) 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=(Xt)t≥0 with independent increments and X0=0 , which is a semimartingale with respect to its natural filtration. We denote by (B,C,ν) 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=(Bt)t≥0 , C=(Ct)t≥0 and ν are absolutely continuous with respect to the Lebesgue measure, i.e. (3) Bt=∫0tbsds,Ct=∫0tcsds,ν(dt,dx)=dtKt(dx) with càdlàg functions b=(bs)s≥0 , c=(cs)s≥0 , K=(Ks(A))s≥0,A∈B(R) . We assume that the compensator of the measure of jumps ν verifies the usual relation: for each t∈R+ (4) ∫0t∫R(x2∧1)Ks(dx)ds<+∞. For the main result we will assume an additional technical condition: (5) ∫0t∫|x|>1e|x|Ks(dx)ds<+∞. 1}}{e^{|x|}}\hspace{0.1667em}{K_{s}}(dx)\hspace{0.1667em}ds<+\infty .\]]]> The last condition implies that E(|Xt|)<+∞ for t>0 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 Xt , ϕt(λ)=Eexp(iλXt), is expressed as follows: for λ∈R ϕt(λ)=exp{iλBt−12λ2Ct+∫0t∫R(eiλx−1−iλx)Ks(dx)ds}. We recall also that X is a semimartingale if and only if for all λ∈R the characteristic function of Xt 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 It . It is known that the exponential functional (It)t>0 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)=(Vs(t))0≤s≤t indexed by t and such that It=Vt(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 Vs(t),0<s<t . The density of the law of Vt(t) can be obtained just by integration of the right-hand side of the equation for the density of the law of Vs(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 Vs(t) , 0≤s≤t , of the class C1,2(]0,t[×R+,∗) , where R+,∗=R∖{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 Vt(t) and It coincide (P-a.s.), the results on the density of Vt(t) give the answer for the density of It . In Proposition 2 we give sufficient conditions for the existence of the density of the class C∞(]0,t[×R+,∗) of the law of It 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 It can be simplified due to the homogeneity (see Proposition 1). We present also the equations for the distribution functions of the laws of It and I∞ , 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∞ and integrable jumps the equations coincide with the known ones from [15].

We introduce, for fixed t>0 0$]]>, a time reversal process Y=(Ys)0≤s≤t with Ys=Xt−X(t−s)−. 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 It defined by (1) and the process Y=(Ys)0≤s≤t .

In the following lemma we claim that Y is PII and we precise its semimartingale triplet. For that we introduce the functions b¯=(b¯u)0≤u≤t , c¯=(c¯u)0≤u≤t and K¯=(K¯u)0≤u≤t putting (6) b¯u=1{t}(u)(bt−b0)+bt−u, (7) c¯u=1{t}(u)(ct−c0)+ct−u, (8) K¯u(A)=1{t}(u)(Kt(A)−K0(A))+Kt−u(A), where 1{·} is the indicator function and A∈B(R) . It means, for instance for b¯=(b¯u)0≤u≤t , that b¯u=bt−uif0≤u<t,btifu=t. So, the function b¯ can have a discontinuity at t, since in general b0≠bt . 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=(Vs)0≤s≤t and J=(Js)0≤s≤t , with (10) Vs=e−YsJs,Js=∫0seYudu. 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, It=Vt (P-a.s.), and then they have the same laws. As we will see, the process V=(Vs)0≤s≤t is a Markov process with respect to the natural filtration FY=(FsY)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 It .

We define the set of functions C={f∈Cb2(R+)|supy∈R+|f′(y)y|<∞,supy∈R+|f″(y)y2|<∞} such that f(0)=f′(0)=0 . Here we use the usual notation Cb2(R+) for the set of twice continuously differentiable functions with bounded derivatives on R+ .

For 0≤s≤t we put (11) a‾s=−b‾s+12c‾s+∫R(e−x−1+x)K¯s(dx). We notice that the assumptions (3) and (4) imply that (λ-a.s.) ∫R|e−x−1+x|K¯s(dx)<+∞, so that a‾s is (λ-a.s.) well defined. We introduce also for f∈C the generator (AsV)0≤s<t of the process V via (12) AsV(f)(y)=(1+ya‾s)f′(y)+12c‾sf″(y)y2+∫Rf(ye−x)−f(y)−f′(y)y(e−x−1)K¯s(dx).

The proof of Theorem 1 will be divided in four parts. In Lemma 4 we give a semimartingale decomposition of the process (e−Ys)0≤s≤t , then in Lemma 5 we prove a semimartingale decomposition of (f(Vs))0≤s≤t with f belonging to 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). Lemma 4.

For 0≤s≤t (16) e−Ys=e−Y0+As+Ns where A=(As)0≤s≤t is a process with locally bounded variation and N=(Ns)0≤s≤t is a local martingale.