[Paper Review] Fractional Gamma process and fractional Gamma-subordinated processes

Luisa Beghin|arXiv (Cornell University)|May 8, 2013

Fractional Differential Equations Solutions13 references3 citations

TL;DR

This paper introduces fractional versions of the Gamma subordinator and its subordinated processes by time-changing via stable subordinators or their inverses, deriving fractional differential equations involving novel fractional shift operators. The key contribution is the derivation of fractional Fokker-Planck-type equations for the distributions of Variance Gamma, Geometric Stable, and Negative Binomial processes, with distinct equations for ν < 1 and ν > 1 using Caputo and Riemann-Liouville fractional derivatives.

ABSTRACT

We define and study fractional versions of the well-known Gamma subordinator $Γ:=\{Γ(t),$ $t\geq 0\},$ which are obtained by time-changing $% Γ$ by means of an independent stable subordinator or its inverse. Their densities are proved to satisfy differential equations expressed in terms of fractional versions of the shift operator (with fractional parameter greater or less than one, in the two cases). As a consequence, the fractional generalization of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding fractional differential equations are obtained.

Motivation & Objective

• To develop fractional generalizations of the Gamma subordinator and its associated processes using time-changed Lévy processes.
• To define new fractional shift operators based on Caputo and Riemann-Liouville fractional derivatives for ν < 1 and ν > 1, respectively.
• To derive fractional differential equations governing the one-dimensional distributions of the fractional Gamma processes.
• To extend the framework to subordinated processes, including Variance Gamma, Geometric Stable, and Negative Binomial processes, and derive their fractional dynamics.
• To establish the Lévy symbols of the resulting fractional processes, confirming their Lévy process structure for ν > 1.

Proposed method

• Define two fractional Gamma processes: Γν(t) = Γ(ℒν(t)) for ν < 1 and Ḡν(t) = Γ(𝒜1/ν(t)) for ν > 1, using inverse and stable subordinators.
• Introduce the fractional shift operator 𝒪c,xν for ν < 1 using the Caputo fractional derivative, and the operator 𝒪̄c,xν for ν > 1 using the right-sided Riemann-Liouville derivative.
• Derive fractional differential equations for the densities of Γν and Ḡν by applying the fractional shift operators to their transition densities.
• Apply the same framework to subordinated processes by composing Brownian motion or compound Poisson processes with the fractional Gamma processes.
• Use the subordination principle to derive the characteristic functions and Lévy symbols of the resulting fractional processes.
• Establish the connection between the fractional differential equations and the infinitesimal generators of the processes via the fractional shift operators.

Experimental results

Research questions

• RQ1 How can the standard Gamma subordinator be generalized to a fractional version using time-changed Lévy processes?
• RQ2 What fractional differential equations govern the transition densities of the fractional Gamma processes for ν < 1 and ν > 1?
• RQ3 How do the fractional shift operators based on Caputo and Riemann-Liouville derivatives relate to the dynamics of the fractional Gamma processes?
• RQ4 What are the fractional Fokker-Planck-type equations satisfied by the distributions of Variance Gamma, Geometric Stable, and Negative Binomial processes under the fractional time change?
• RQ5 Under what conditions does the fractional subordinated process remain a Lévy process, and what is its Lévy symbol?

Key findings

• The density of the fractional Gamma process Γν(t) satisfies the fractional differential equation 𝒪−1,tνfΓν(z,t) = −1/p fΓν(z,t) + 1/p fΓν(z−1,t) for ν < 1.
• The density of the fractional Gamma process Ḡν(t) satisfies the fractional differential equation 𝒪̄−1,tνfḠν(z,t) = −1/p fḠν(z,t) + 1/p fḠν(z−1,t) for ν > 1.
• The fractional Variance Gamma process Xν(t) = B(Γν(t)) satisfies the fractional differential equation 𝒪−1,tνqkν(t) = 1/2 qkν(t) − 1/2 qk−1ν(t) for ν < 1.
• The fractional Negative Binomial process Mν(t) = M(Ḡν(t)) satisfies the fractional differential equation 𝒪̄−1,tνq̄kν(t) = 1/p q̄kν(t) − (1−p)/p q̄k−1ν(t) for ν > 1.
• For ν > 1, the fractional Variance Gamma, Geometric Stable, and Negative Binomial processes are Lévy processes with Lévy symbols given by η(t) = −[log(1 + u²/2)]^{1/ν}, η(t) = −[log(1 + (−iu)^α)]^{1/ν}, and η(t) = −[log((1−(1−p)e^{iu})/p)]^{1/ν}, respectively.
• The fractional shift operators 𝒪c,xν and 𝒪̄c,xν generalize the classical shift operator e^{cDx} to fractional order, enabling the derivation of fractional Fokker-Planck equations for the transition densities.