Delayed Capital Injections for a Risk Process with Markovian Arrivals

A. S. Dibu (), M. J. Jacob (), Apostolos D. Papaioannou () and Lewis Ramsden ()
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A. S. Dibu: National Institute of Technology Calicut
M. J. Jacob: National Institute of Technology Calicut
Apostolos D. Papaioannou: University of Liverpool
Lewis Ramsden: The York Management School University of York

Methodology and Computing in Applied Probability, 2021, vol. 23, issue 3, 1057-1076

Abstract: Abstract In this paper we propose a generalisation to the Markov Arrival Process (MAP) risk model, by allowing for a delayed receipt of required capital injections whenever the surplus of an insurance firm is negative. Delayed capital injections often appear in practice due to the time taken for administrative and processing purposes of the funds from a third party or the shareholders of a firm. We introduce a MAP risk model that allows for capital injections to be received instantaneously, or with a random delay, depending on the amount of deficit experienced by the firm. For this model, we derive a system of Fredholm integral equations of the second kind for the Gerber-Shiu function and obtain an explicit expression (in matrix form) in terms of the Gerber-Shiu function of the MAP risk model without capital injections. In addition, we show that the expected discounted accumulated capital injections and the expected discounted overall time in red, up to the time of ruin, satisfy a similar integral equation, which can also be solved explicitly. Finally, to illustrate the applicability of our results, numerical examples are given.

Keywords: Delayed capital injections; Markov Arrival Process; Systems of Fredholm integral equations; Gerber-Shiu function; Expected discounted-accumulated capital injections until ruin; 45B05; 60J25; 91B05 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s11009-020-09796-9

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