Uniform estimates for ruin probabilities in the renewal risk model with upper-tail independent claims and premiums
Yinghua Dong Yuebao Wang
Journal of Industrial & Management Optimization 2011, 7(4): 849-874 doi: 10.3934/jimo.2011.7.849
In this paper, we discuss a nonstandard renewal risk model, where the price process of the investment portfolio is modelled as a geometric Lévy process, the claim sizes and premium sizes form sequences of identically distributed and upper-tail independent random variables, respectively, the claim size and its corresponding inter-claim time satisfy a certain dependence structure described via a conditional tail probability of the claim size given the inter-claim time before the claim occurs, and there is a similar dependence structure between the premium size and the inter-arrival time before the premium is paid. When the claim-size distribution belongs to the extended-regular-varying class, we obtain a uniform tail asymptotics for stochastically discounted aggregate claims. Furthermore, assuming that the tail of the premium-size distribution is lighter than that of the claim-size distribution, the uniform estimates for the finite- and infinite-time ruin probabilities are presented respectively.
keywords: upper-tail independence extended-regular variation dependence Asymptotics uniformity. Lévy process ruin probabilities
Stochastic dynamics: Markov chains and random transformations
Felix X.-F. Ye Yue Wang Hong Qian
Discrete & Continuous Dynamical Systems - B 2016, 21(7): 2337-2361 doi: 10.3934/dcdsb.2016050
This article outlines an attempt to lay the groundwork for understanding stochastic dynamical descriptions of biological processes in terms of a discrete-state space, discrete-time random dynamical system (RDS), or random transformation approach. Such mathematics is not new for continuous systems, but the discrete state space formulation significantly reduces the technical requirements for its introduction to a much broader audiences. In particular, we establish some elementary contradistinctions between Markov chain (MC) and RDS descriptions of a stochastic dynamics. It is shown that a given MC is compatible with many possible RDS, and we study in particular the corresponding RDS with maximum metric entropy. Specifically, we show an emergent behavior of an MC with a unique absorbing and aperiodic communicating class, after all the trajectories of the RDS synchronizes. In biological modeling, it is now widely acknowledged that stochastic dynamics is a more complete description of biological reality than deterministic equations; here we further suggest that the RDS description could be a more refined description of stochastic dynamics than a Markov process. Possible applications of discrete-state RDS are systems with fluctuating law of motion, or environment, rather than inherent stochastic movements of individuals.
keywords: Markov chain entropy. random dynamical system Stochastic process
Random time ruin probability for the renewal risk model with heavy-tailed claims
Yuebao Wang Qingwu Gao Kaiyong Wang Xijun Liu
Journal of Industrial & Management Optimization 2009, 5(4): 719-736 doi: 10.3934/jimo.2009.5.719
In this paper, we investigate the asymptotic behavior of the random time ruin probability for the renewal risk model with heavy-tailed claim sizes. Under the assumption that the claim sizes are independent and long-tailed, we give the equivalent conditions on asymptotic behavior for the random time ruin probability, where the independent or dependent structure among the inter-arrival times is not needed. While, under the assumption that the claim sizes are of some negative dependence structure and consistently varying tails, we obtain the sufficient condition of asymptotic behavior for the random time ruin probability which will require some negative dependence structure among the inter-arrival times.
keywords: Renewal risk model random time ruin probability heavy-tailed claims negative dependence.
Minimality of p-adic rational maps with good reduction
Aihua Fan Shilei Fan Lingmin Liao Yuefei Wang
Discrete & Continuous Dynamical Systems - A 2017, 37(6): 3161-3182 doi: 10.3934/dcds.2017135

A rational map with good reduction in the field $\mathbb{Q}_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $\mathbb{P}^1(\mathbb{Q}_p)$ over $\mathbb{Q}_p$. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, $\mathbb{P}^1(\mathbb{Q}_p)$ is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of periodic orbits and minimal subsystems. For any prime $p$, a criterion of minimality for rational maps with good reduction is obtained. When $p=2$, a condition in terms of the coefficients of the rational map is proved to be necessary for the map being minimal and having good reduction, and sufficient for the map being minimal and $1$-Lipschitz. It is also proved that a rational map having good reduction of degrees $2$, $3$ and $4$ can never be minimal on the whole space $\mathbb{P}^1(\mathbb{Q}_2)$.

keywords: p-adic dynamical system minimal decomposition projective line good reduction rational map
Parameter identification techniques applied to an environmental pollution model
Yuepeng Wang Yue Cheng I. Michael Navon Yuanhong Guan
Journal of Industrial & Management Optimization 2018, 14(2): 817-831 doi: 10.3934/jimo.2017077

The retrieval of parameters related to an environmental model is explored. We address computational challenges occurring due to a significant numerical difference of up to two orders of magnitude between the two model parameters we aim to retrieve. First, the corresponding optimization problem is poorly scaled, causing minimization algorithms to perform poorly (see Gill et al., practical optimization, AP, 1981,401pp). This issue is addressed by proper rescaling. Difficulties also arise from the presence of strong nonlinearity and ill-posedness which means that the parameters do not converge to a single deterministic set of values, but rather there exists a range of parameter combinations that produce the same model behavior. We address these computational issues by the addition of a regularization term in the cost function. All these computational approaches are addressed in the framework of variational adjoint data assimilation. The used observational data are derived from numerical simulation results located at only two spatial points. The effect of different initial guess values of parameters on retrieval results is also considered. As indicated by results of numerical experiments, the method presented in this paper achieves a near perfect parameter identification, and overcomes the indefiniteness that may occur in inversion process even in the case of noisy input data.

keywords: Conjugate gradient method variational adjoint method data assimilation regularization

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