Delayed Capital Injections for a Risk Process with Markovian Arrivals

A. S. Dibu, M. J. Jacob, Apostolos D. Papaioannou, Lewis Ramsden
2020 Methodology and Computing in Applied Probability  
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
more » ... ously, 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. Methodology and Computing in Applied Probability where u 0 is the insurer's initial capital, c > 0 is the continuously received constant premium rate, {N t } t 0 is a counting process denoting the number of claims received up to time t 0 with a sequence of random variables {σ i } i∈N denoting the i-th claim epoch and {X k } k∈N + forms a sequence of i.i.d. random variables representing the amount of the kth claim, having (common) cumulative distribution function (c.d.f.) F X (·), density function f X (·), and finite mean μ = E(X) < ∞. In this paper, we consider a fairly general setting for the above risk model by assuming that the claim counting process, {N t } t 0 , with {σ i } i∈N claim arrival epochs, is governed by a Markovian Arrival Process (MAP), which includes the classical and (most) Sparre Andersen risk models, as well as models with correlated inter-claim times as special cases. A MAP with representation MAP m α , D 0 , D 1 , of order m 1, is a two-dimensional Markov process, denoted by {(N t , J t )} t 0 , on the state space N 0 ×{1, . . . , m}, where {J t } t 0 denotes the state of an underlying Continuous-Time Markov Chain (CTMC), defined on the state space E. For E = {1, . . . , m}, the underlying CTMC exhibits two different categories of state transition: (a) transitions between states i and j = i ∈ E, occurring without an accompanying claim, which are given by the (i, j )-th element of D 0 , namely D 0 (i, j ) ≥ 0, and (b) transitions from state i to state j (possibly with j = i) in E with an accompanying claim, which are given by the (i, j )-th element of D 1 , namely D 1 (i, j ) ≥ 0. The diagonal elements of D 0 are assumed to be negative such that each row of the matrix D 0 + D 1 is zero. Moreover, the initial state distribution of the underlying CTMC is given by the probability row vector α = (α 1 , . . . , α m ), with α i = P(J (0) = i), for i ∈ E, where denotes the transpose vector/matrix. For more details of MAP's see Neuts (1979), Ramaswami (1999), and Badescu et al. (2005a) and the references therein. Due to the matrix form of the MAP set up, it has been natural throughout the literature to assume that claim amounts, {X k } k∈N + , follow a phase-type (PH) distribution, which allows for analysis of the overall risk model through the extensive theory of matrix analytic methods, see for e.g. Badescu et al. (2005a, b), Ahn and Badescu (2007) , among others. In addition, given that a claim occurs during a transition from state i to j ∈ E (with rate D 1 (i, j )), the versatility of the MAP allows for the claim size distribution to be dependent on the environmental states prior to and following the claim arrival epoch, i.e. states i, j ∈ E. This flexibility provides the framework for modelling claim sizes that depend on external environmental factors, such as automobile liabilities and weather conditions or claims arising from natural disasters. Under this general formulation, any claim that occurs during a transition from state i to j ∈ E is assumed (without loss of generality, see Ramaswami (2006) pp. 501 and Appendix 2 of the same paper for details) to have a phase-type distribution of order n 1, with representation P H n ( γ ij , H ij ), which is equivalent in distribution to the absorption time of an (n + 1)-dimensional CTMC where the transitions between its n transient states are given by the transition rate matrix H ij , transitions into the absorb-Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article'
doi:10.1007/s11009-020-09796-9 fatcat:qatqleiakffznn7uor3gknrtsa