Q-scale function, Banach contraction principle, and ultimate ruin probability in a Markov-modulated jump–diffusion risk model

We consider an insurer who has a fixed amount of funds allocated as the initial surplus for a risk portfolio, so that the probability of ultimate ruin for this portfolio is at a known level. We consider the question of whether the insurer can reduce this ultimate ruin probability by allocating part of the initial funds to the purchase of a reinsurance contract. This reinsurance contract would restore the insurer's surplus to a positive level k every time the surplus fell between 0 and k. The insurer's objective is to choose the level k that minimizes the ultimate ruin probability. Using different examples of reinsurance premium calculation and claim size distribution we show that this objective can be achieved, often with a substantial reduction in the ultimate ruin probability from the situation when there is no reinsurance. We also show that by purchasing reinsurance the insurer can release funds for other purposes without altering its ultimate ruin probability.

Ruin probabilities and penalty functions with stochastic rates of interest

Schauder's fixed point theorem and the Banach contraction principle are used to study the polynomial-like iterative functional equation with variable coefficients. We give sufficient conditions for the existence, uniqueness, and stability of the strongly convex and strongly concave solutions. Finally, some examples were considered by our results.

In this paper, we present a threshold proportional reinsurance strategy and we analyze the effect on some solvency measures: ruin probability and time of ruin. This dynamic reinsurance strategy assumes a retention level that is not constant and depends on the level of the surplus. In a model with inter-occurrence times being generalized Erlang(n)-distributed, we obtain the integro-differential equation for the Gerber–Shiu function. Then, we present the solution for inter-occurrence times exponentially distributed and claim amount phase-type(N). Some examples for exponential and phase-type(2) claim amount are presented. Finally, we show some comparisons between threshold reinsurance and proportional reinsurance.

Banach contraction principle is a fundamental result in Metric Fixed Point Theory. It is a very popular and powerful tool in solving the existence problems in pure and applied sciences. In this chapter, Banach contraction principle and its converse are presented. Moreover, various applications of this famous principle, including mixed Volterra–Fredholm-type integral equations and systems of nonlinear matrix equations, are provided. Some results of this chapter appeared in [3, 5, 13, 19].

ABSTRACTIn this paper, we consider a class of neutral stochastic differential systems with variable delays and give sufficient conditions to ensure that the zero solution is asymptotically stable in the mean square by means of a generalisation of Banach's contraction principle. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. The results are shown to improve the previous globally stable results derived in the literature.

In this article, continuous‐time risk models are introduced; the time of ruin, and the surplus immediately before and at ruin caused by a claim are defined. Then the probability of ruin, the (discounted) moment of the severity of ruin caused by a claim, the joint moment of the severity of ruin and the time of ruin caused by a claim, and the (discounted) defective joint/marginal distribution and probability functions of the surplus immediately before and at ruin are studied.

The purpose of this paper is to study the positive almost periodic solutions to the nonlinear integral equation. Using Hilbert projective metric techniques and contraction mapping principle, we establish existence and uniqueness results for the integral equation. Some examples are given.

According to the characteristics of risk model with perturbed by diffusion, a new kind of stochastic process— F-B process is proposed following from Fuzzy Markov Skeleton process. Then the ruin probabilities (caused by oscillation or by a claim) are discussed by F-B process. We derive a distribution of the ruin probability and a distribution of the surplus immediately before the time of ruin. Then continuous model and discrete model, risk model with compound Poisson process and risk model with double Poisson process, are used to illustrate the implementation of the model. Next limit theorems of two continuous risk models perturbed by diffusion are discussed. Finally simulation experiment is introduced.

Fixed point theory is one of the most powerful tools in nonlinear analysis. The Banach contraction principle is the simplest and most versatile elementary result in fixed point theory. The principle has many applications and was extended by several authors. In this paper, we introduce a concept of α–ψ-contractive type mappings and establish fixed point theorems for such mappings in complete fuzzy metric spaces. Starting from the Banach contraction principle, the presented theorems are the extension, generalization, and improvement of many existing results in the literature. Some example and application to ordinary differential equations are given to illustrate the usability of obtained results.

The main objective of the paper is to study the existence theorem of the newly defined generalized fuzzy ψ-variational inequality problem, generalized fuzzy ψ-complementarity problem. Uniqueness of the problems are also studied. Existence of these problems are established with the help of unique point condition of η. The existence of the solution of generalized fuzzy ψ-variational inequality problems is studied with the help of Banach contraction principle. The existence of the solution of generalized fuzzy ψ-complementarity problem is established in η-invex space.Conclusion: Equality of the solution sets of generalized fuzzy ψ-variational inequality problem and generalized fuzzy ψ-complementarity problem is shown using unique point condition of the bifunction η. Unique of the solution of generalized fuzzy ψ-variational inequality problem and generalized fuzzy ψ-complementarity problem is studied using strictly monotonicity condition. Existence theorems of the solution of generalized fuzzy ψ-variational inequality problem are shown using Banach contraction principle in the presence of Lipschitz continuous, strongly monotone mappings T_ψ and weakly monotonicity of η. Existence theorems of the solution of generalized fuzzy ψ-complementarity problem in the presence of condition C_0.

We extend the application of nearly contraction mapping principle introduced by Sahu (2005) for existence of fixed points of demicontinuous mappings to certain hemicontinuous nearly Lipschitzian nonlinear mappings in Banach spaces. We have applied certain results due to Sahu (2005) to obtain conditions for existence—and to introduce an asymptotic iterative process for construction—of fixed points of these hemicontractions with respect to a new auxiliary operator.

This paper explores the applications of Fixed-point theory in metric spaces, focusing on the role of real numbers in matrix spaces and iterative methods. Using Banach's Contraction Principle, the study provides mathematical proofs to establish the existence and uniqueness of fixed points, while also analyzing the convergence and stability of iterative methods. In a complete metric space (𝑋, 𝑑), a contraction mapping 𝑓: 𝑋 → 𝑋 has a unique fixed point 𝑥∗ such that 𝑓 (𝑥∗) = 𝑥∗ and 𝑑 (𝑓 (𝑥), 𝑓 (𝑦)) ≤ 𝛼 𝑑 (𝑥, 𝑦), where 0 ≤ 𝛼 < 1. The Intermediate Value Theorem ensures that for a continuous function 𝑓on [𝑎, 𝑏] with 𝑓(𝑎) ≤ 𝐿 ≤ 𝑓(𝑏), there is a point 𝑐 ∈ [𝑎, 𝑏] such that 𝑓(𝑐) = 𝐿. The Mean Value Theorem states that for a differentiable function 𝑓 on (𝑎,), there exists a point 𝑐 ∈ (𝑎, 𝑏) such that 𝑓′(𝑐)=𝑓(𝑏) – 𝑓(𝑎) / 𝑏−𝑎. The Cauchy-Schwarz inequality for vectors 𝑢and 𝑣 gives ∣⟨𝑢, 𝑣⟩∣ ≤ ∥𝑢∥ ∥𝑣∥. For the function (𝑥) = ½ (𝑥+3), the fixed point is 𝑥∗ = 3. In matrix space, the operator (𝐴) = ½ 𝐴 has a fixed point 𝐴 = 0. For 𝑥 = cos (𝑥), the successive approximations method converges to 𝑥∗ ≈ 0.7391, and the Newton-Raphson method achieves faster convergence to the same fixed point.This research highlights the practical applications of fixed-point theory, supported by real numbers, as a foundation for accurate mathematical models. It enhances outcome prediction in optimization, stability analysis, and system modeling, emphasizing the critical role of real numbers in both theoretical and applied mathematics.

of thesis entitled STUDY ON INSURANCE RISK MODELS WITH SUBEXPONENTIAL TAILS AND DEPENDENCE STRUCTURES Submitted by CHEN YIQING for the degree of Doctor of Philosophy at The University of Hong Kong in August 2009 In recent years, insurance risk studies in the presence of dependence structure and heavy-tailed claims are popular topics in actuarial science. In this thesis, we consider the renewal risk model with certain dependence structures, namely negative dependence and quasi-asymptotic independence. We also assume that claim sizes follow a heavy-tailed distribution, in particular, a subexponential distribution. We focus on studying the impact of heavy tails and dependence structures on ruin probabilities and the tail probabilities of aggregate claims. The asymptotic behavior of the ultimate ruin probability in the renewal risk model was studied in the presence of negatively dependent claim sizes and a constant force of interest. A purely probabilistic approach was employed to establish a simple asymptotic formula for the ruin probability. This result coincides with some known results in the recent literature, and discovers a phenomenon that heavy tails can offset the impact of the dependence structure of claim sizes on the ruin probability. A two-dimensional renewal risk model was considered in which the insurance company consists of two dependent classes of business. The dependence between the two classes comes from the fact that their claims arrive according to a common renewal counting process. For this model, several ruin probabilities were defined and simple asymptotic formulae for these ruin probabilities were derived. In particular, the obtained asymptotic formulae hold uniformly for all time horizons. This uniformity greatly enhances the theoretical value of the results. The next problem was to study precise estimates of the large deviation probabilities of aggregate claims with sizes which are negatively dependent. Specially, the problem for the so-called quasi renewal risk model was investigated in which inter-arrival times are also negatively dependent. A new dependence structure was considered and was termed “quasi-asymptotic independence” structure, which is more general and verifiable than some commonlyused dependence structures. The proposed dependence structure covers a wide range of positive and negative dependence. For the case of Pareto-like tails, the tail probability of the sum was proved to be asymptotically equivalent to the sum of tail probabilities. This confirms again that heavy tails offset the impact of dependence between summands. This result was then extended to randomly weighted sums and random sums. All the obtained results have obvious applications to concrete actuarial problems. In the last part of the thesis, each primary random variable representing the net loss during a period was associated with a random weight or a stochastic discount factor. Therefore, the weighted sum represents the stochastic present value of aggregate losses. Suppose that the primary random variables are long tailed, the weights are positive and bounded, and the primary random variables and their weights are independent. The tail probability of the maximum of the weighted sums was shown to be asymptotically equivalent to that of the last weighted sum. STUDY ON INSURANCE RISK MODELS WITH SUBEXPONENTIAL TAILS AND DEPENDENCE STRUCTURES

Weak contraction mapping principle is a generalization of the Banach contraction mapping principle. Weakly contractive mappings are intermediate to contraction mappings and nonexpansive mappings. They have been studied in several contexts. Metric fixed point theory in partially ordered spaces have rapidly developed in recent times. In this paper we extend the concept of weak contraction to subset of a partially ordered generalized metric space which are chains by themselves. It is noted that this weak contraction is different from weak contraction on the whole space. We prove here that under certain assumptions the weakly contractive mapping on certain chains will have a fixed point. Two illustrative examples are given.