Banach Contraction Principle and ruin probabilities in regime-switching models

Note on stability of the ruin time density in a Sparre Andersen risk model with exponential claim sizes

“…Proposition 2.1 and Theorem 2.2 below give upper bounds for d(p τ , p τ ). The relevant results on stability (continuity) of the infinite horizon ruin probability can be found, for instance, in [13,17] for the Cramer-Lundberg risk process, and for more general risk model in [2,3,9,10]. On the other hand, we are not aware of any continuity inequalities for densities of ruin time.…”

Finite-horizon general insolvency risk measures in a regime-switching Sparre Andersen model

“…Lower and upper estimates for the vector of finite-horizon as well as ultimate deficit distributions at ruin has been proven in Gajek and Rudź (2018c). The generality of the regime-switching Sparre Andersen model allows an immediate application of the above mentioned methods in several continuous and discrete time risk models listed in Gajek and Rudź (2018a).…”

“…Other references can be found, e.g., in Silvestrov (2014Silvestrov ( , 2015. For a discussion of recent developments of Markov switching models and the operator approach in ruin theory, we also refer the reader to Gajek and Rudź (2018a). In particular, a research methodology based on Banach Contraction Principle is used there to approximate a vector of ultimate ruin probabilities in a regime-switching Sparre Andersen model.…”

General methods for bounding multidimensional ruin probabilities in regime-switching models

“…Regime-switching models have received considerable attention recently, see, e.g. Q. Liu et al [20], Jacka and Ocejo [15], Momeya [22], Xu et al [28], R.H. Liu [19], G. Wang et al [27], Landriault et al [18], Chen et al [4], Guillou et al [13], Lu [21], Asmussen [1], Gajek and Rudź [9][10][11][12] and the references therein for an overview of selected developments and applications of Markov-modulated models and related problems.…”

“…hold for any n ∈ N (see Theorem 3.6 for details). Moreover, for any i ∈ S, the sequences {D i n } n∈N and {U i n } n∈N converge monotonically, as n → ∞, to i with the exponential rate of convergence: A distinct mathematical methodology for obtaining the unique fixed point of the risk operator, based on Banach Contraction Principle, can be found in Gajek and Rudź [10]. Some stochastic developments of Banach-type fixed point theorems are discussed in Saipara et al [25].…”