Julie Thøgersen modtager 1. pris for foredrag

06.06.2016 | Lars Madsen

Julie Thøgersen har modtaget 1. prisen blandt foredragsholderne ved 9th Conference on Actuarial Science & Finance on Samos afholdt 18-22 maj på Samos.

Julies vindende foredrag havde titlen: Heavy tails and Markov dependence, with insurance applications


Let \( (\xi_n, Y_n)_{n \in \mathbb{N}} \) be a Markov renewal process characterized by state space \( \mathcal{E} \) and semi-Markov kernel \( \textbf{F} \) with the following measures as elements \begin{align*} F_{ij} (t) = \mathbb{P}(\xi_{n+1} = j, T_n \leq t \; \vert \; \xi_n = i) \end{align*} where \( T_n= Y_n-Y_{n-1} \) are the interarrival times. Hence, the transition matrix of \( (\xi_n) \) has entries \( p_{ij}= \lVert F_{ij} \rVert \) for \( i,j \in \mathcal{E} \) and the interarrival times has conditional distribution \( \mathbb{P} (T_n \leq t \; \vert\; \xi_n = i, \xi_{n+1} =j) =p_{ij} F_{ij}(t) \). Now consider the Markov renewal equation \begin{align*} Z_i(x) = z_i(x) +\sum_{j \in \mathcal{E}} \int_0^x Z_j(x-y)F_{ij}(dy). \end{align*} This has a variety of applications, e.g. ruin probability subject to regime shifting in insurance.

More specifically, we consider subexponential conditional interarrival distributions and a terminating Markov chain which therefore has a substochastic transition matrix. The main objective is to obtain the asymptotics of the Markov renewal equation for this specific case.

In order to do so the subexponentiality is needed to be defined locally, which is a relatively recent concept introduced by Asmussen, Foss and Korshunov (2003) [1]. The asymptotic result is split into three cases depending on the relative heavyness between the conditional distribution of the interarrival times and the \( z_i \)'s. Further, a version Kesten's Lemma including multiple distribution functions is also needed and prooved, which is an additional contribution of this paper.

Keywords: Markov renewal equations, local subexponentiality, sums of independent variables, substochastic.

[1] Søren Asmussen, Serguei Foss, and Dimitry Korshunov (2003), "Asymptotics for sums of random variables with local subexponential behaviour". Journal of Theoretical Probability, vol. 2(iss), 489-518.