Abstract
This paper defines a new class of fractional differential operators alongside a family of random variables whose density functions solve fractional differential equations equipped with these operators. These equations can be further used to construct fractional integro-differential equations for the ruin probabilities in collective renewal risk models, with inter-arrival time distributions from the aforementioned family. Gamma-time risk models and fractional Poisson risk models are two specific cases among them, whose ruin probabilities have explicit solutions when claim size distributions exhibit rational Laplace transforms.
Highlights
The concept of first passage time is widely used in financial mathematics and actuarial science
C.Constantinescu@liverpool.ac.uk 1 Institute for Financial and Actuarial Mathematics, University of Liverpool, L69 7ZL, Liverpool, UK 2 Universidad Nacional de Colombia, Sede Medellin, Cra 65 59A-110 Medellin, Colombia is aimed at solving equations for the probability of ruin expressed as a function of the initial capital of the risk process
Motivated by risk theory applications, we consider a new class of risk processes, while extending those from Li and Garrido [23], Albrecher et al [2] and Biard and Saussereau [6] into a fractional derivative framework
Summary
The concept of first passage time is widely used in financial mathematics and actuarial science. The present paper derives explicit ruin probabilities in risk models with claim sizes whose distributions have rational Laplace transforms, and with inter-arrival time densities solving fractional differential equations. If the claim size distributions have rational Laplace transforms, these integrodifferential equations can be further reduced to linear boundary value problems Their symbolic computation approach permits extensions to models with premia dependent on reserves ( discussed in Djehiche [15] regarding the upper and lower bounds of finite-time ruin probabilities), the associated boundary problems involving linear ordinary differential equations with variable coefficients; see Albrecher et al [1]. We consider the case of claim sizes described by sums of heterogeneous gamma random variables and show that the corresponding ruin probabilities solve fractional differential equations with constant coefficients.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
