Potential measures for spectrally negative Markov additive processes with applications in ruin theory

Abstract

The Markov additive process (MAP) has become an increasingly popular modeling tool in the applied probability literature. In many applications, quantities of interest are represented as functionals of MAPs and the potential measure, also known as the resolvent measure, have played a key role in the representations of explicit solutions to these functionals. In this paper, closed-form solutions to potential measures for spectrally negative MAPs are found using a novel approach based on algebraic operations of matrix operators. This approach also provides a connection between results from fluctuation theoretic techniques and those from classical differential equation techniques. In the end, the paper presents a number of applications to ruin-related quantities as well as verification of known results concerning exit problems.

Phase-type approximations perturbed by a heavy-tailed component for the Gerber-Shiu function of risk processes with two-sided jumps

“…where f (q) 11 = det T (+) + sI . Recall now that we only need the (1, 1) element of the matrix r (q) ǫ (u, z).…”

Potential Measures of One-Sided Markov Additive Processes with Reflecting and Terminating Barriers

“…where f ≥ 0 is a measurable function, which is equal to 0 for the cemetery state of the environment. Various interesting applications of this equation applied to insurance risk are presented in [14].…”

“…Moreover, some results for the two-sided reflection process appeared in [15]. The potential measure of a spectrally negative MAP (under certain assumptions) in the case of a single lower terminating barrier was obtained in [14], using a very different approach based on algebraic operations of matrix operators. Finally, some very interesting applications of potential measures to insurance risk can be found in [10], [14], and [26].…”

The Markov Additive Risk Process Under an Erlangized Dividend Barrier Strategy

“…For example, Cheung and Landriault (2009, Section 4) studied a dividend barrier strategy in which the barrier is allowed to depend on J; whereas Zhang et al (2011) investigated the absolute ruin problem under debit interest. Moreover, Salah and Morales (2012) studied the Gerber-Shiu expected discounted penalty function (Gerber and Shiu (1998)) in a more general spectrally negative MAP risk process; whereas generalizations of the Gerber-Shiu function were analyzed by Cheung and Landriault (2010), Cheung and Feng (2013), and Feng and Shimizu (2014). While the afore-mentioned papers involve analytic derivations of the quantities of interest, we remark that MAP risk processes may also be studied using a more probabilistic approach via connection to Markov-modulated fluid flow (MMFF) processes (e.g.…”

“…22) whereM δ (u) andg δ (x) are defined in(3.16) and(3.17), respectively. The matrix ∞ 0g δ (x)dx is known to be strictly substochastic (see Feng and Shimizu (2014, Appendix D)), and therefore (3.22) can be regarded as a matrix version of defective renewal equation.…”