Subexponential asymptotics of hybrid fluid and ruin models

Second order corrections for the limits of normalized ruin times in the presence of heavy tails

In this paper we consider a compound Poisson risk model with regularly varying claim sizes. For this model in [1] an asymptotic formula for the finite time ruin probability is provided when the time is scaled by the mean excess function. In this paper we derive the rate of convergence for this finite time ruin probability when the claims have a finite second moment. [1] S. Asmussen and C. Klüppelberg. Large deviations results for subexponential tails, with applications to insurance risk. Stochastic Process. Appl., 64(1):103-125, 1996.

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。

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…

破产依赖的双变量随机流过程

Ruin-dependent bivariate stochastic fluid processes

本文提出一种新的双变量随机流过程模型，该模型引入了破产依赖的行为切换机制。通过数学框架分析其性质，并提供关键性能指标的闭式表达式，特别是破产时间的联合分布。

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…

A new class of large claim size distributions: Definition, properties, and ruin theory

We investigate a new natural class $\mathcal{J}$ of probability distributions modeling large claim sizes, motivated by the `principle of one big jump'. Though significantly more general than the (sub-)class of subexponential distributions $\mathcal{S}$, many important and desirable structural properties can still be derived. We establish relations to many other important large claim distribution classes (such as $\mathcal{D}$, $\mathcal{S}$, $\mathcal{L}$, $\mathcal {K}$, $\mathcal{OS}$ and $\mathcal{OL}$), discuss the stability of $\mathcal{J}$ under tail-equivalence, convolution, convolution…

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.

非标准风险模型的渐近行为：多变量亚指数索赔与恒定利息力

Asymptotics for a nonstandard risk model with multivariate subexponential claims and constant interest force

本文研究了多变量亚指数索赔和恒定利息力的非标准风险模型的渐近行为，推导了有限和无限时间 horizon 下的进入概率结果，并应用于资金链问题。

Asymptotics for the Time of Ruin in the War of Attrition

We consider two players, starting with $m$ and $n$ units, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probability $p(m,n)$ that the first player wins. When $m\sim Nx_{0}$, $n\sim N y_{0}$, we prove the fluid limit as $N\to \infty$. When $x_{0}=y_{0}$, then $z\to p(N,N+z\sqrt{N})$ converges to the standard normal CDF and the difference in fortunes scales diffusively. The exact limit of the time of ruin $τ_{N}$ is established as $(T-τ_N) \sim N^{-β}W^{\frac{1}β}$, …

Phase-type approximations perturbed by a heavy-tailed component for the Gerber-Shiu function of risk processes with two-sided jumps

We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with large probability $1-ε$ and heavy-tailed with small probability $ε$. We analyze the seminal Gerber-Shiu function coding the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. We derive its value as …

依赖时长的随机流体过程与太阳能收益建模

Duration-dependent stochastic fluid processes and solar energy revenue modeling

本文通过引入依赖时长的马尔可夫到达过程扩展经典随机流体过程，用于灵活建模太阳能收益，并基于均匀化方法和n-桥概念给出收益过程首次返回及项目价值的拉普拉斯变换与高效计算算法。

Two-node queueing network with a heavy-tailed random input: the strong stability case

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional queue-length process. The tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, presence of heavy tails totally changes the asymptotics. Here we focus on the case of strong stability where both nodes release fluid with sufficiently high speeds to minimise their mutual influence. We s…

On the asymptotic behavior of weakly lacunary series

Let $f$ be a measurable function satisfying $$f(x+1)=f(x), \qquad \int_0^1 f(x) dx=0, \qquad \textrm{Var} ~f < + \infty,$$ and let $(n_k)_{k\ge 1}$ be a sequence of integers satisfying $n_{k+1}/n_k \ge q >1$ $(k=1, 2, \ldots)$. By the classical theory of lacunary series, under suitable Diophantine conditions on $n_k$, $(f(n_kx))_{k\ge 1}$ satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences $(n_k)_{k\ge 1}$ as well, but as Fukuyama (2009) showed, the behavior of $f(n_kx)$ is generally not permutation-i…