An application of fractional differential equations to risk theory

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...

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Bibliographic Details
Published in:Finance and stochastics 2019-10, Vol.23 (4), p.1001-1024
Main Authors: Constantinescu, Corina D., Ramirez, Jorge M., Zhu, Wei R.
Format: Article
Language:English
Subjects:
Density
Differential equations
Economic Theory/Quantitative Economics/Mathematical Methods
Economics
Finance
Insurance
Laplace transforms
Management
Markov analysis
Mathematical models
Mathematics
Mathematics and Statistics
Operators
Operators (mathematics)
Probability Theory and Stochastic Processes
Quantitative Finance
Random variables
Renewal
Risk
Statistics for Business
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Summary: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.
ISSN:0949-2984
1432-1122
DOI:10.1007/s00780-019-00400-8