Applications of a change of measures technique for compound mixed renewal processes to the ruin problem

Ruin and dividend measures in the renewal dual risk model

In this manuscript we consider the dual risk model with financial application, where the random gains occur under a renewal process. We particularly work the Erlang(n) case for common distribution of the inter-arrival times, from there it is easy to understand that our method or procedures can be generalised to other cases under the matrix-exponential family case. We work several and different problems involving future dividends and ruin. We also show that our results are valid even if the usual income condition is not satisfied. In most known works under the dual model, the main target under …

Uniform Markov Renewal Theory and Ruin Probabilities in Markov Random Walks

Let {X_n,n\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability π. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that {(X_n,S_n),n\geq0} is a Markov random walk. In this paper, we prove a uniform Markov renewal theorem with an estimate on the rate of convergence. This result is applied to boundary crossing problems for {(X_n,S_n),n\geq0}. To be more precise, for given b\geq0, define the stopping time τ=τ(b)=inf{n:S_n>b}. When a drift μof the random walk S_n is 0, we derive a one-term Ed…

On ruin probabilities in the presence of risky investments and random switching

We study the asymptotic behavior of ruin probabilities, as the initial reserve goes to infinity, for a reserve process model where claims arrive according to a renewal process, while between the claim times the process has the dynamics of geometric Brownian motion-type Itô processes with time-dependent random coefficients. These coefficients are ``reset" after each claim time, switching to new values independent of the past history of the process. We use the implicit renewal theory to obtain power-function bounds for the eventual ruin probability. In the special case when the random drift and …

The probability of exceeding a piecewise deterministic barrier by the heavy-tailed renewal compound process

We analyze the asymptotics of crossing a high piecewise linear barriers by a renewal compound process with the subexponential jumps. The study is motivated by ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions when the initial reserves of both companies tend to infinity.

分布方程与带有投资的Sparre Andersen模型的破产问题

Distributional equations and the ruin problem for the Sparre Andersen model with investments

本文基于Sparre Andersen模型的投资问题，利用隐含再生理论方法补充了Eberlein等人关于破产概率渐进行为的研究结果。

On Approximating Ruin Probability of Double Stochastic Compound Poisson Processes

Consider a surplus process which both of collected premium and payed claim size are two independent compound Poisson processes. This article derives two approximated formulas for the ruin probability of such surplus process, say double stochastic compound poisson process. More precisely, it provides two mixture exponential approximations for ruin probability of such double stochastic compound poisson process. Applications to long_term Bonus_Malus systems and a heavy-tiled claim size distribution have been given. Improvement of our findings compared to the Cramer- Lundberg upper bound has been …

Ruin Probabilities for Risk Processes with Non-Stationary Arrivals and Subexponential Claims

In this paper, we obtain the finite-horizon and infinite-horizon ruin probability asymptotics for risk processes with claims of subexponential tails for non-stationary arrival processes that satisfy a large deviation principle. As a result, the arrival process can be dependent, non-stationary and non-renewal. We give three examples of non-stationary and non-renewal point processes: Hawkes process, Cox process with shot noise intensity and self-correcting point process. We also show some aggregate claims results for these three examples.

改进的破产概率：由复合混合泊松过程驱动的Cramér-Lundberg模型

Modified ruin probability for a Cramér-Lundberg model driven by a compound mixed Poisson process

研究改进的破产概率，探讨在重尾和轻尾情况下，改进与经典破产概率的等价性及端点原子结果和密度渐近性。

Ruin probabilities with dependence on the number of claims within a fixed time window

We analyse the ruin probabilities for a renewal insurance risk process with inter-arrival time distributions depending on the claims that arrived within a fixed (past) time window. This dependence could be explained through a regenerative structure. The main inspiration of the model comes from the Bonus-Malus feature. We discuss first asymptotic results of ruin probabilities for different regimes of claim distributions. For numerical results, we recognise an embedded Markov additive process. Via an appropriate change of measure, ruin probabilities could be computed to a closed form formulae. A…

On subexponential tails for the maxima of negatively driven compound renewal and Lévy processes

We study subexponential tail asymptotics for the distribution of the maximum $M_t:=\sup_{u\in[0,t]}X_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and Lévy processes. For both we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes particularly includes Cramér-Lundberg risk process.

Large deviations principle for terminating multidimensional compound renewal processes with application to polymer pinning models

Large deviations principle is obtained for terminating multidimensional compound renewal processes. We also obtained the asymptotic of large deviations for the case when a Gibbs change of the original probability measure takes place. The random processes mentioned in the paper are widely used in polymer pinning models.

Approximation of the fixed-probability level for a compound renewal process

Dealing with compound renewal process with generally distributed jump sizes and inter-renewal intervals, we focus on the approximation for the fixed-probability level, which is the core of inverse level crossing problem. We are developing an analytical technique presented in [15]-[17] and based on Kendall's identity; this yields (see [18]) inverse Gaussian approximation in the direct level crossing problem. These issues are of great importance in risk theory.

复合马尔可夫二项风险模型的有限时间破产

Finite-Time Ruin for the Compound Markov Binomial Risk Model

本文研究由有限状态遍历马尔可夫链调制的复合二项风险模型的有限时间破产概率，通过推广 Ballot 定理处理零初始盈余情形，并利用多元 Lagrangian 反演给出任意初始盈余下的 Seal 型破产公式。

重尾复合更新过程与Lévy过程在随机时间区间上最大值的渐近性

Maxima over random time intervals for heavy-tailed compound renewal and Lévy processes

本文研究了带负漂移的重尾复合更新过程和Lévy过程在随机时间区间上最大值的次指数尾部渐近性，结果对不依赖未来增量的随机时间一致成立，并与随机游走理论建立联系。

Escape probabilities of compound renewal processes with drift

We consider the problem of determining escape probabilities from an interval of a general compound renewal process with drift. This problem is reduced to the solution of a certain integral equation. In an actuarial situation where only negative jumps arise we give a general solution for escape and survival probabilities under Erlang$(n)$ and hypo-exponential arrivals. These ideas are generalized to the class of arrival distributions having rational Laplace transforms. In a general situation with two-sided jumps we also identify important families of solvable cases. A parallelism with the "scal…

A characterization of equivalent martingale measures in a renewal risk model with applications to premium calculation principles

Generalizing earlier work of Delbaen and Haezendonck for given compound renewal process $S$ under a probability measure $P$ we characterize all probability measures $Q$ on the domain of $P$ such that $Q$ and $P$ are progressively equivalent and $S$ remains a compound renewal process under $Q$. As a consequence, we prove that any compound renewal process can be converted into a compound Poisson process through a change of measures and we show how this approach is related to premium calculation principles.

Ruin Probabilities and Overshoots for General Levy Insurance Risk Processes

We formulate the insurance risk process in a general Levy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to -\infty a.s. and the positive tail of the Levy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64 (1996) 103-125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207-226] for ruin probabilities and the overshoot in random walk and compou…

Large deviations for fractional Poisson processes

We prove large deviation principles for two versions of fractional Poisson processes. Firstly we consider the main version which is a renewal process; we also present large deviation estimates for the ruin probabilities of an insurance model with constant premium rate, i.i.d. light tail claim sizes, and a fractional Poisson claim number process. We conclude with the alternative version where all the random variables are weighted Poisson distributed. Keywords: Mittag Leffler function; renewal process; random time cha

Local theorems for arithmetic compound renewal processes, when Cramer's condition holds

We continue the study of the compound renewal processes (c.r.p.), where the moment Cramer's condition holds (see [1]-[10], where the study of c.r.p. was started). In the paper arithmetic c.r.p. Z(n) are studied. In such processes random vector ξ = (τ,ζ) has the arithmetic distribution, where τ > 0 defines the distance between jumps, ζ defines the values of jumps. For this processes the fine asymptotics in the local limit theorem for probabilities P(Z(n) = x) has been obtained in Cramer's deviation region of x $\in$ Z. In [6]-[10] the similar problem has been solved for non-lattice c.r.p., when…

混合更新风险模型中等价鞅概率测度的刻画及其在风险理论中的应用

A characterization of equivalent martingale probability measures in a mixed renewal risk model with applications in Risk Theory

本文刻画了使复合混合更新过程在逐步等价测度下变为复合混合 Poisson 过程的所有概率测度，并应用于破产问题和保费计算原则。

Ruin probabilities for risk processes in a bipartite network

This paper studies risk balancing features in an insurance market by evaluating ruin probabilities for single and multiple components of a multivariate compound Poisson risk process. The dependence of the components of the process is induced by a random bipartite network. In analogy with the non-network scenario, a network ruin parameter is introduced. This random parameter, which depends on the bipartite network, is crucial for the ruin probabilities. Under certain conditions on the network and for light-tailed claim size distributions we obtain Lundberg bounds and, for exponential claim size…

A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

Generalizing earlier works of Delbaen & Haezendonck [5] as well as of [18] and [16] for given compound mixed renewal process S under a probability measure P, we characterize all those probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound mixed renewal process under Q with improved properties. As a consequence, we prove that any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures. Applications related to the ruin problem and to the computation of premium calculation princip…

Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the init…

Efficient simulation of ruin probabilities when claims are mixtures of heavy and light tails

We consider the classical Cramér-Lundberg risk model with claim sizes that are mixtures of phase-type and subexponential variables. Exploiting a specific geometric compound representation, we propose control variate techniques to efficiently simulate the ruin probability in this situation. The resulting estimators perform well for both small and large initial capital. We quantify the variance reduction as well as the efficiency gain of our method over another fast standard technique based on the classical Pollaczek-Khinchine formula. We provide a numerical example to illustrate the performance…

Strong memoryless times and rare events in Markov renewal point processes

Let W be the number of points in (0,t] of a stationary finite-state Markov renewal point process. We derive a bound for the total variation distance between the distribution of W and a compound Poisson distribution. For any nonnegative random variable ζ, we construct a ``strong memoryless time'' \hat ζsuch that ζ-t is exponentially distributed conditional on {\hat ζ\leq t, ζ>t}, for each t. This is used to embed the Markov renewal point process into another such process whose state space contains a frequently observed state which represents loss of memory in the original process. We then write…

带标记Hawkes过程与复合带标记Hawkes过程的泛函大数定律

Functional Laws of Large Numbers for Marked Hawkes Processes and Compound Marked Hawkes Processes

该研究推导了一般标记空间下带标记Hawkes过程及复合带标记Hawkes过程的泛函大数定律，补充了已有文献结论，并给出了其在保险破产过程时限分析中的应用示例。

Merging of bivariate compound Binomial processes with shocks

The paper investigates a discrete time Binomial risk model with different types of polices and shock events may influence some of the claim sizes. It is shown that this model can be considered as a particular case of the classical compound Binomial model. As far as we work with parallel Binomial counting processes in infinite time, if we consider them as independent, the probability of the event they to have at least once simultaneous jumps would be equal to one. We overcome this problem by using thinning instead of convolution operation. The bivariate claim counting processes are expressed in…

带相位型跳变的阈值依赖破产模型与排队模型的瞬态行为

Transient Behavior of Threshold-Dependent Ruin and Queueing Models with Phase-Type Jumps

该研究针对带相位型跳变的阈值依赖Cramér–Lundberg风险模型与M/G/1型排队模型，利用谱正Lévy过程波动理论，刻画指数时刻前的破产概率与工作量拉普拉斯-斯蒂尔杰斯变换，给出相位型分布下辅助量的显式表示。

社区保险中的线性风险分担：在复利泊松模型中减少破产概率

Linear Risk Sharing in Community-Based Insurance: Ruin Reduction in the Compound Poisson Model

本文研究社区保险中索赔发生时的比例风险分担，通过比较单独运营与参与池化的破产概率，证明在共同尺度家族、完全分配和精算公平条件下，池化可降低个体破产概率，且通过指数和对数正态示例验证了比例分担规则对偿付能力的影响。

Bivariate Compound Poisson Risk Processes with Shocks

Contemporary insurance theory is concentrated on models with different types of polices and shock events may influence the payments on some of them. Jordanova (2018) considered a model where a shock event contributes to the total claim amount with one and the same value of the claim sizes to different types of polices. Jordanova and Veleva (2021) went a step closer to real-life situations and allowed a shock event to cause different claim sizes to different types of polices. In that paper, the counting process is assumed to be Multinomial. Here it is replaced with different independent homogen…

A ruin model with a resampled environment

This paper considers a Cramér-Lundberg risk setting, where the components of the underlying model change over time. These components could be thought of as the claim arrival rate, the claim-size distribution, and the premium rate, but we allow the more general setting of the cumulative claim process being modelled as a spectrally positive Lévy process. We provide an intuitively appealing mechanism to create such parameter uncertainty: at Poisson epochs we resample the model components from a finite number of $d$ settings. It results in a setup that is particularly suited to describe situations…

复合多元Hawkes过程：大偏差与稀有事件模拟

Compound Multivariate Hawkes Processes: Large Deviations and Rare Event Simulation

本文为随机标记多元Hawkes过程诱导的多元复合过程建立大偏差原理，证明极限累积量的本质光滑性，并应用其推导多元风险过程的边际破产概率渐近结果，同时提出基于重要性抽样的稀有事件模拟方法并证明其渐近有效性。

累积过程的渐近偏差界

Asymptotic deviation bounds for cumulative processes

本文借助大偏差原理推导累积过程的渐近偏差界，放宽了累积随机变量与更新过程的依赖限制，拓展了已有研究，且可应用于带抑制的Hawkes过程。

若干复合过程的积分

Integrals of some compound processes

该研究基于复合泊松过程理论构建通用框架，证明伯恩斯坦从属的时间变换性质，推导复合过程积分的显式表示等结果，为相关随机过程研究提供理论支撑。

更新理论中的淬灭大偏差

Quenched large deviations in renewal theory

本文研究随机环境下取值于 Banach 空间的更新-报酬过程，推导出淬灭大偏差原理并用校正子变分公式确定速率函数，通过复合 Poisson 过程等三个例子验证理论。

Optimal dividend problem for a generalized compound Poisson risk model

In this note we study the optimal dividend problem for a company whose surplus process, in the absence of dividend payments, evolves as a generalized compound Poisson model in which the counting process is a generalized Poisson process. This model including the classical risk model and the Polya-Aeppli risk model as special cases. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. We show that under some conditions the optimal dividend strategy is formed by a barrier strategy.

Optimal scheduling of critically loaded multiclass GI/M/n+M queues in an alternating renewal environment

In this paper, we study optimal control problems for multiclass GI/M/n+M queues in an alternating renewal (up-down) random environment in the Halfin-Whitt regime. Assuming that the downtimes are asymptotically negligible and only the service processes are affected, we show that the limits of the diffusion-scaled state processes under non-anticipative, preemptive, work-conserving scheduling policies, are controlled jump diffusions driven by a compound Poisson jump process. We establish the asymptotic optimality of the infinite-horizon discounted and long-run average (ergodic) problems for the q…

Integro-differential equations linked to compound birth processes with infinitely divisible addends

Stochastic modelling of fatigue (and other material's deterioration), as well as of cumulative damage in risk theory, are often based on compound sums of independent random variables, where the number of addends is represented by an independent counting process. We consider here a cumulative model where, instead of a renewal process (as in the Poisson case), a linear birth (or Yule) process is used. This corresponds to the assumption that the frequency of \textquotedblleft damage" increments accelerates according to the increasing number of \textquotedblleft damages". We start from the partial…

A fractional generalization of the Poisson processes

It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time is exponentially distributed we have a Poisson process, which is Markovian. However, other waiting time distributions are also relevant in applications, in particular such ones with a fat tail caused by a…

Approximations to ultimate ruin probabilities with a Wienner process perturbation

In this paper, we adapt the classic Cramér-Lundberg collective risk theory model to a perturbed model by adding a Wiener process to the compound Poisson process, which can be used to incorporate premium income uncertainty, interest rate fluctuations and changes in the number of policyholders. Our study is part of a Master dissertation, our aim is to make a short overview and present additionally some new approximation methods for the infinite time ruin probabilities for the perturbed risk model. We present four different approximation methods for the perturbed risk model. The first method is b…

A parsimonious model for intraday European option pricing

A stochastic model for pure-jump diffusion (the compound renewal process) can be used as a zero-order approximation and as a phenomenological description of tick-by-tick price fluctuations. This leads to an exact and explicit general formula for the martingale price of a European call option. A complete derivation of this result is presented by means of elementary probabilistic tools.