Random recurrence equations and ruin in a Markov-dependent stochastic economic environment

Ruin probabilities as recurrence sequences in a discrete-time risk process

We apply the theory of linear recurrence sequences to find an expression for the ultimate ruin probability in a discrete-time risk process. We assume the claims follow an arbitrary distribution with support $\{0,1,\ldots,m\}$, for some integer $m\ge2$. The method requires to find the zeroes of an $m$ degree polynomial and to solve a system of $m$ linear equations. An approximation is derived from the exact ruin formula and several numerical results and plots are provided as examples.

The branching-ruin number and the critical parameter of once-reinforced random walk on trees

The motivation for this paper is the study of the phase transition for recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define a quantity, that we call the branching-ruin number of a tree, which provides (in the spirit of Furstenberg, 1970, and Lyons, 1990) a natural way to measure trees with polynomial growth. We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk. We define a sharp and effective (i.e. co…

The branching-ruin number as critical parameter of random processes on trees

The branching-ruin number of a tree, which describes its asymptotic growth and geometry, can be seen as a polynomial version of the branching number. This quantity was defined by Collevecchio, Kious and Sidoravicius (2018) in order to understand the phase transitions of the once-reinforced random walk (ORRW) on trees. Strikingly, this number was proved to be equal to the critical parameter of ORRW on trees. In this paper, we continue the investigation of the link between the branching-ruin number and the criticality of random processes on trees. First, we study random walks on random conductan…

随机环境下带投资的破产概率：平滑性

Ruin probabilities with investments in random environment: smoothness

本文研究了保险公司在依赖Markov过程的随机经济环境下投资风险资产的破产问题，证明了破产概率关于初始资本的平滑性，并推导出相应的积分微分方程。

Ladder Chains: A Variation of Random Walks

The authors propose a new variation of random walks called ladder chains $L(r,s,p)$. We extend concepts such as ruin probability, hitting time, transience and recurrence of random walks to ladder chain. Take $L(2,2,p)$ for instance, we find the linear difference equations that the ruin probability and the hitting time satisfy. We also prove the recurrence of a critical case $(p=\sqrt{2}-1)$. All approaches of these results can be generalized to solve similar problems for other ladder chains.

Asymptotic behaviour of ruin probabilities in a general discrete risk model using moment indices

We study the rough asymptotic behaviour of a general economic risk model in a discrete setting. Both financial and insurance risks are taken into account. Loss during the first $n$ years is modelled as a random variable $B_1+A_1B_2+\ldots+A_1\ldots A_{n-1}B_n$, where $A_i$ corresponds to the financial risk of the year $i$ and $B_i$ represents the insurance risk respectively. Risks of the same year $i$ are not assumed to be independent. The main result shows that ruin probabilities exhibit power law decay under general assumptions. Our objective is to give a complete characterisation of the rel…