Geometric Stable processes and related fractional differential equations

We are interested in the differential equations satisfied by the density of the Geometric Stable processes $\mathcal{G}_{\alpha}^{\beta}=\left\{\mathcal{G}_{\alpha}^{\beta}(t);t\geq 0\right\} $, with stability \ index $% \alpha \in (0,2]$ and asymmetry parameter $\beta \in \lbrack -1,1]$, both in the univariate and in the multivariate cases. We resort to their representation as compositions of stable processes with an independent Gamma subordinator. As a preliminary result, we prove that the latter is governed by a differential equation expressed by means of the shift operator. As a consequence, we obtain the space-fractional equation satisfied by the density of $\mathcal{G}_{\alpha}^{\beta}.$ For some particular values of $% \alpha $ and $\beta ,$ we get some interesting results linked to well-known processes, such as the Variance Gamma process and the first passage time of the Brownian motion.

The main features of the GS laws are the heavy tails and the unboundedness at zero. These two characteristics, together with their stability properties (with respect to geometric summation) and domains of attraction, make them attractive in modelling financial data, as shown, for example, in [15]. As particular cases, when the symmetry parameter β is equal to 1, the support of the GS r.v. is limited to R + and its law coincides, for 0 < α ≤ 1, with the Mittag-Leffler distribution, as shown in [9] and [12]. Moreover the GS distribution is sometimes referred to as "asymmetric Linnik distribution", since it can be considered a generalization of the latter (to which it reduces for β = µ = 0, see [13], [7]). The Linnik distribution exhibits fat tails, finite mean for 1 < α ≤ 2 and also finite variance only for α = 2 (when it takes the name of Laplace distribution) and is applied in particular to model temporal changes in stock prices (see [2]).
The univariate GS process will be denoted as G β α (t), t ≥ 0 and defined by having the onedimensional distribution coinciding with G β α and characteristic exponent equal to (see [21], [6]). Moreover the following representation holds where S β α (t) is, for any t, a stable law with parameters µ = 0, β ∈ [−1, 1], σ = t 1/α and {Γ(t), t ≥ 0} is an independent Gamma subordinator. We will use the following notation, for a generic process We note that, for β = 0, the process G β α reduces to a symmetric GS process (that we will denote simply as G α ), while, for β = 1, it is called GS subordinator (since it is increasing and Lévy); we will denote it as G ′ α . The space-fractional differential equation that we obtain here, as governing equations of G β α , are expressed in terms of Riesz and Riesz-Feller derivatives. We recall that the Riesz fractional derivative R D α x is defined through its Fourier transform, which reads, for α > 0 and for an infinitely differentiable function u, where the Fourier transform is defined as F {u(x); θ} := +∞ −∞ e iθx u(x)dx (see [16] and [10], p.131). Alternatively it can be explicitly represented as follows, for α ∈ (0, 2], (see [19]). The more general Riesz-Feller definition is given by where ψ α β (θ) := −|θ| α e i γπ 2 signθ , |γ| ≤ min{α, 2 − α} (see [10], p.359 and [16]) and ψ α β (θ) coincides with the characteristic exponent of the stable random variable S β α , in the Feller parametrization, for γ = 2 π arctan −β tan πα 2 . Indeed (2) can be rewritten (for µ = 0) as We recall now the following result on stable processes proved in [16] (in the special case c = 1), which will be used later: let p β α (x, t), x ∈ R, t ≥ 0, be the transition density of the stable process S β α , then p β α satisfies the following space-fractional differential equation, for α ∈ (0, 2], x ∈ R, t ≥ 0: and the additional condition ∂ ∂t p β α (x, t) Our main result concerns the space-fractional equation satisfied by the density g β α (x, t), x ∈ R, t ≥ 0, of the GS process G β α . As a preliminary step we derive the partial differential equation satisfied by the density f Γ (x, t), x, t ≥ 0, of the Gamma subordinator Γ and then we resort to the representation (3) of the GS process. Indeed we prove that f Γ (x, t) satisfies where b is the rate parameter of Γ (see (15) below) and e −∂t is a particular case (for k = 1) of the shift operator, defined as for any analytical function f : In the n-dimensional case, we prove that the governing equation of the GS vector process in R n is analogous to (12), but the Riesz-Feller fractional derivative is substituted, in this case, by the fractional derivative operator ∇ α M defined by where S n := {s ∈ R n : ||s|| = 1} and M is the spectral measure (see [17], with a change of sign due to the different definition of Fourier transform). The multivariate GS law has been first introduced in [1] (in the isotropic case) and called multivariate Linnik distribution. As special cases of the previous results the governing equations of some well-known processes are obtained: indeed, in the symmetric case and for α = 2, the GS process reduces to the Variance Gamma process, while, for α = 1, it coincides with a Cauchy process subordinated to a Gamma subordinator. On the other hand, in the positively asymmetric case, G β α reduces to a GS subordinator, which is used in particular as random time argument of the subordinated Brownian motion, via successive iterations (see [6], [21]) Moreover, for α = 1/2, we can obtain, as a corollary, the fractional equation satisfied by the density g ′ 1/2 (x, t) of the first-passage time of a standard Brownian motion B through a Gamma distributed random barrier, i.e.

Preliminary results
We start by deriving the differential equation satisfied by the density of the Gamma subordinator, since it will be applied in the study of the equation governing the GS process (thanks to the representation (3)).
The one-dimensional distribution of the Gamma subordinator {Γ a,b (t), t ≥ 0} , of parameters a, b > 0 is given by (see, for example, [3], p.52). Hereafter we will consider, for the sake of simplicity, the case a = 1 and denote Γ 1,b := Γ. The Fourier transform of (15) is given by Lemma 1 The density (15) of the Gamma subordinator satisfies (for a = 1), the following equation where e −∂t is the partial derivative version of the shift operator defined in (11), for k = 1. The initial and boundary conditions are the following Proof. The first condition in (18) can be checked easily by considering (16) and the definition of the Dirac delta function, i.e. δ(x) := 1 2π R e −iθx dθ. The second one is immediately satisfied by (15). As far as equation (17) is concerned, the Fourier transform of its left-hand side, with respect to x, is given by = [by (18) For the right-hand side of (17) we have that which coincides with (19).
An alternative result on the differential equation satisfied by f Γ can be obtained by considering the following differential operator: for any given infinitely differentiable function f (x), We could use for (20) the formalism A k,x f (x) = log(1 + D x /k).
If moreover D j x f (x) |x|=∞ = 0, for any j ≥ 0, the Fourier transform of (20) can be written as follows: Lemma 2 The following differential equation is satisfied by the density of the Gamma subordinator: with the conditions Proof. The conditions (23) are immediately verified by (15). Moreover, by taking the Fourier transform of the r.h.s. of (22), we get From the previous Lemma we can conclude that the infinitesimal generator of the Gamma process can be written as A x = − log(1 + D x ).

Univariate GS process
By resorting to the representation (3) and applying the previous results, we can obtain the differential equation satisfied by the density of the univariate GS process G β α . This can be done, for t > 1, by considering Lemma 1 together with the result (9) on S β α , as follows: by (3), we can write We consider hereafter the simple case b = 1. We then apply (17), for b = 1, and we get In the last step we have applied the first equation in (9) and we have considered that, for t > 1, f Γ (0, t) = 1. In the next theorem we prove the same result in an alternative way, which can be applied for any t ≥ 0.

Proposition 3
The density g β α of the GS process G β α satisfies the following equation, for any x, t ≥ 0 and α ∈ (0, 2], where c > 0 is the spreading rate of dispersion defined in (8).
Proof. By (24) and (8) we can write the characteristic function of G β α as where ψ α β (θ) is defined in (7); thus the Fourier transform of the space-fractional differential equation (25) can be written as On the other hand we get which coincides with (28). The conditions (26) are clearly satisfied since and lim |x|→∞ g β α (x, t) = 0 (by (9) and (24)).

Symmetric GS process
In the special case of a symmetric GS process G α we can easily derive from Proposition 3 the following result, which is expressed in terms of the Riesz derivative R D α x , defined in (4). In its regularized form, for α ∈ (0, 2], the derivative R D α x can be explicitly represented as (see [16]).

Corollary 4
The density g α of the symmetric GS process G α satisfies the following equation, for any x, t ≥ 0 and α ∈ (0, 2], where c = σ α and with conditions Remark 5 We consider now some interesting special cases of the previous results. For α = 1, we show, from the previous corollary, that the density g 1 (x, t) of a Cauchy process C subordinated to an independent Gamma subordinator (i.e. the process defined as {C(Γ(t)), t ≥ 0}) satisfies the following equation, for any x, t ≥ 0: with conditions (31) and ∂/∂|x| := R D 1 x . For α = 2, we derive the governing equation of the density g 2 (x, t) of the Variance Gamma process, since the latter can be represented as a standard Brownian motion B subordinated to an independent Gamma subordinator, i.e. as {B(Γ(t)), t ≥ 0} . Indeed we get that g 2 (x, t) satisfies, for any x, t ≥ 0, the second order differential equation where c = σ 2 and with conditions (31).
We derive now another equation satisfied by the density of the symmetric GS process, which, unlike (30), involves a standard time derivative and a space fractional differential operator which generalizes (20). Let us define the fractional version of A k,x , for any α > 0, as where R D ν x is the Riesz derivative of order ν > 0. We note that in the non-symmetric case (i.e. for β = 0) we can not define the analogue to (32) since the Riesz-Feller derivative is not defined for a fractional order greater than 2.

Proposition 6
The density g α of the symmetric GS process G α satisfies the following equation, for any x, t ≥ 0 and α ∈ (0, 2], where c = σ α and with conditions Proof. The Fourier transform of (32) is given by Therefore we get = log 1 1 + c|θ| α The expression (36) clearly coincides with the Fourier transform of the left-hand side of (33).
The previous result agrees with the expression of the infinitesimal generator A x of the GS process, which is given by A x = − log 1 + − d 2 dx 2 α/2 (see [8]).

GS subordinator
In the positively asymmetric case, i.e. for β = 1, the process G β α reduces to a GS subordinator (we will denote it as G ′ α ).

Remark 8
We now consider the special case α = 1/2 of the previous result. It is well-known that the stable law with parameters α = 1/2, µ = 0, β = 1, σ > 0 coincides with the Lévy density. Moreover if we define as the first-passage time of a standard Brownian motion B, we have that since T z is equal in distribution to a stable subordinator S ′ 1/2 of index 1/2 and variance σ = z 2 (whose density is denoted as p ′ 1/2 (x, z)). Therefore, from the previous corollary, we can derive that the density of the time-changed process T Γ(t) , t ≥ 0 , given by satisfies the following equation for any x, t ≥ 0: with conditions (38) and ∂ 1/2 /∂|x| 1/2 := RF D 1/2 x,1 . The constant in (39) can be derived by considering that, in this case, c = √ σ (cos(π/4)) −1 and we assume that σ = 1. The process T Γ(t) can be interpreted as the first-passage time of a Brownian motion through a random barrier, represented by a Gamma process. Thus we can conclude that