[Paper Review] Fluctuation identities for omega-killed Markov additive processes and dividend problem

Irmina Czarna, Adam Kaszubowski+2 more|arXiv (Cornell University)|Jun 21, 2018

Probability and Risk ModelsDecision Sciences15 references4 citations

TL;DR

This paper introduces generalized scale matrices, denoted $\mathcal{W}^{(\omega)}$ and $\mathcal{Z}^{(\omega)}$, for spectrally negative Markov additive processes (MAPs) exponentially killed with a bivariate intensity $\omega(x,i)$ depending on state and environment. It establishes fluctuation identities and resolvent equations for $\omega$-killed MAPs, extending classical scale matrix theory to Markov-modulated Lévy processes and enabling applications to dividend problems and the Omega model in risk theory.

ABSTRACT

In this paper we solve the exit problems for an one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $ω(\cdot,\cdot)$ dependent on the present level of the process and the present state of the environment. Moreover, we analyze respective resolvents. All identities are given in terms of new generalizations of classical scale matrices for the MAP. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the so-called Omega model, where bankruptcy occurs at rate $ω(\cdot,\cdot)$ when the surplus process becomes negative. Finally, we consider the Markov modulated Brownian motion (MMBM) and present the results for the particular choice of piecewise intensity function $ω(\cdot,\cdot)$.

Motivation & Objective

• To generalize classical scale matrix theory to Markov additive processes (MAPs) under $\omega$-killing, where killing intensity depends on both state and environment.
• To derive fluctuation identities and resolvent equations for $\omega$-killed MAPs, extending results from Lévy processes to the Markov-modulated framework.
• To provide a theoretical foundation for solving optimal dividend problems and ruin-related exit problems in controlled risk processes.
• To establish connections between the new $\omega$-scale matrices and the Omega model, where bankruptcy occurs at rate $\omega(x,i)$ when surplus is negative.
• To analyze the Markov-modulated Brownian motion (MMBM) case with piecewise constant $\omega(x,i)$, offering explicit solutions and numerical illustrations.

Proposed method

• Define $\omega$-killed MAPs as spectrally negative MAPs with exponential killing intensity $\omega(x,i)$ depending on level $x$ and environment state $i$.
• Introduce $\omega$-scale matrices $\mathcal{W}^{(\omega)}$ and $\mathcal{Z}^{(\omega)}$ as solutions to integral equations involving the killing intensity and classical scale matrices.
• Derive fluctuation identities for exit problems from intervals $[d,c]$, using potential measures and resolvent operators under $\omega$-killing.
• Establish boundary conditions and limiting behavior (as $d \to -\infty$, $c \to \infty$) to derive global identities and matrix inverses.
• Use the dominated convergence theorem and matrix inversion techniques to derive limiting forms of $\mathcal{W}^{(\omega)}$ and $\mathcal{Z}^{(\omega)}$.
• Apply the framework to the Omega model and MMBM by specifying piecewise constant $\omega(x,i)$, and derive explicit expressions for ruin and dividend functions.

Experimental results

Research questions

• RQ1 How can classical fluctuation identities for Lévy processes be extended to Markov additive processes under state- and environment-dependent killing?
• RQ2 What are the structural properties and integral equations satisfied by the generalized $\omega$-scale matrices $\mathcal{W}^{(\omega)}$ and $\mathcal{Z}^{(\omega)}$?
• RQ3 How do the $\omega$-scale matrices relate to the potential measures and resolvents of $\omega$-killed MAPs?
• RQ4 Can the new framework be applied to solve optimal dividend problems and ruin problems in the Omega model?
• RQ5 What are the explicit forms and numerical behaviors of $\omega$-scale matrices in the case of Markov-modulated Brownian motion with piecewise constant $\omega(x,i)$?

Key findings

• The $\omega$-scale matrices $\mathcal{W}^{(\omega)}$ and $\mathcal{Z}^{(\omega)}$ are defined as solutions to integral equations involving the killing intensity $\omega(x,i)$ and classical scale matrices.
• The fluctuation identities for $\omega$-killed MAPs are expressed in terms of $\mathcal{W}^{(\omega)}$ and $\mathcal{Z}^{(\omega)}$, generalizing classical results from Lévy processes.
• For the case of constant $\omega$, the $\omega$-scale matrices reduce to classical scale matrices, confirming consistency with prior literature.
• The limiting behavior of $\mathcal{W}^{(\omega)}(x,d)e^{-\mathbf{R}^{\beta}d}$ as $d \to -\infty$ is shown to be invertible, enabling the construction of $\mathcal{H}^{(\omega)}(x)$.
• The value function for dividends paid until ruin in the Omega model is derived using the $\omega$-scale matrices, providing a closed-form solution.
• For the Markov-modulated Brownian motion with piecewise constant $\omega(x,i)$, the paper provides explicit expressions and numerical computations for the $\omega$-scale matrices and associated ruin probabilities.