The Gerber-Shiu penalty functions for two classes of renewal risk processes

In this paper, we study the Gerber-Shiu functions for a risk model with two independent classes of risks. We suppose that both of the two claim number processes are renewal processes with phase-type inter-claim times. By re-composing and analyzing the Markov chains associated with two given phase-type distributions, we obtain systems of integro-differential equations for two types of Gerber-Shiu functions. Explicit expressions for the Laplace transforms of the two types of Gerber-Shiu functions are established, respectively. And explicit results for the Gerber-Shiu functions are derived when the initial surplus is zero and when the two claim amount distributions are both from the rational family. Finally, an example is considered to illustrate the applicability of our main results.     

Time consistent dynamic risk processes

A crucial property for dynamic risk measures is the time consistency. In this paper, a characterization of time consistency in terms of a “cocycle condition” for the minimal penalty function is proved for general dynamic risk measures continuous from above. Then the question of the regularity of paths is addressed. It is shown that, for a time consistent dynamic risk measure normalized and non-degenerate, the process associated with any bounded random variable has a càdlàg modification, under a mild condition always satisfied in the case of continuity from below. When normalization is not assumed, a right continuity condition on the penalty has to be added.     

On the Gerber-Shiu function for a risk model with multi-layer dividend strategy

In this paper, we present a new approach to the study of the Gerber-Shiu discounted function for the risk model with multi-layer dividend strategy. The formulae for the Gerber-Shiu discounted function and ruin probability were obtained and the special case where the claim size distribution is a combination of exponentials is considered in detail.     

On the approximation of functions satisfying defective renewal equations

Functions satisfying a defective renewal equation arise commonly in applied probability models. Usually these functions do not admit an explicit expression. In this work, we consider their approximation by means of a gamma-type operator given in terms of the Laplace transform of the initial function. We investigate which conditions on the initial parameters of the renewal equation give the optimal order of uniform convergence of the approximation. We apply our results to ruin probabilities in the classical risk model, paying special attention to mixtures of gamma claim amounts.     

On some classes of distance distribution function-valued submeasures

In this paper we continue the study of a submeasure notion introduced in Hutník and Mesiar (2009) [1] involving a class of operations which provides a generalization ofτ_T-submeasures. We construct pseudo-metrics and metrics generated by such probabilistic submeasures. Two possible generalizations of our submeasure notion are discussed.     

On the DFR property of the compound geometric distribution with applications in risk theory

In 1988, Shanthikumar proved that the sum of a geometrically distributed number of i.i.d. DFR random variables is also DFR. In this paper, motivated by the inverse problem, we study monotonicity properties related to defective renewal equations, and obtain that if a compound geometric distribution is DFR, then the random variables of the sums are NWU (a class that contains DFR). Furthermore, we investigate some applications of risk theory and give a characterization of the exponential distribution.     

Stochastic optimization algorithms for barrier dividend strategies

This work focuses on finding optimal barrier policy for an insurance risk model when the dividends are paid to the share holders according to a barrier strategy. A new approach based on stochastic optimization methods is developed. Compared with the existing results in the literature, more general surplus processes are considered. Precise models of the surplus need not be known; only noise-corrupted observations of the dividends are used. Using barrier-type strategies, a class of stochastic optimization algorithms are developed. Convergence of the algorithm is analyzed; rate of convergence is also provided. Numerical results are reported to demonstrate the performance of the algorithm.     

On convergence determining and separating classes of functions

Herein, we generalize and extend some standard results on the separation and convergence of probability measures. We use homeomorphism-based methods and work on incomplete metric spaces, Skorokhod spaces, Lusin spaces or general topological spaces. Our contributions are twofold: we dramatically simplify the proofs of several basic results in weak convergence theory and, concurrently, extend these results to apply more immediately in a number of settings, including on Lusin spaces.     

Sharp estimates for the CDF of quadratic forms of MPE random vectors

In this paper we develop an efficient analytical expansion of the cumulative distribution function (cdf) XBX^t where X=(X₁,…,X_n+1) with n≥2, follows a multivariate power exponential distribution (MPE). Our approach provides a sharp estimate of the cumulative distribution function of a quadratic form of MPE, together with explicit error estimates.     

Optimal control of the insurance company with proportional reinsurance policy under solvency constraints

This paper considers the optimal control problem of the insurance company with proportional reinsurance policy under solvency constraints. The management of the company controls the reinsurance rate and dividends payout processes to maximize the expected present value of the dividend until the time of bankruptcy. This is a mixed singular-regular control problem. However, the optimal dividend payout barrier may be too low to be acceptable. The company may be prohibited to pay dividend according to external reasons because this low dividend payout barrier will result in bankruptcy soon. Therefore, some constraints on the insurance company’s dividend policy will be imposed. One reasonable and normal constraint is that if b is the minimum dividend barrier, then the bankrupt probability should not be larger than some predetermined ε within the time horizon T. This paper is to work out the optimal control policy of the insurance company under the solvency constraints.     

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

The main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz-Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved.     

Mathematical investigation of the Gerber-Shiu function in the case of dependent inter-claim time and claim size

In this paper we investigate the well-known Gerber-Shiu expected discounted penalty function in the case of dependence between the inter-claim times and the claim amounts. We set up an integral equation for it and we prove the existence and uniqueness of its solution in the set of bounded functions. We show that if δ>0, the limit property of the solution is not a regularity condition, but the characteristic of the solution even in the case when the net profit condition is not fulfilled. It is the consequence of the choice of the penalty function for a given density function. We present an example when the Gerber-Shiu function is not bounded, consequently, it does not tend to zero. Using an operator technique we also prove exponential boundedness.     

Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model

Consider an insurer who is allowed to make risk-free and risky investments. The price process of the investment portfolio is described as a geometric Lévy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of Pareto type, we obtain a simple asymptotic formula which holds uniformly for all time horizons. The same asymptotic formula holds for the finite-time and infinite-time ruin probabilities. Restricting our attention to the so-called constant investment strategy, we show how the insurer adjusts his investment portfolio to maximize the expected terminal wealth subject to a constraint on the ruin probability.     

Optimality of the threshold dividend strategy for the compound Poisson model

In this paper, we consider the optimal dividend problem for the compound Poisson risk model. We assume that dividends are paid to the shareholders according to an admissible strategy with dividend rate bounded by a constant. Our objective is to find a dividend policy so as to maximize the expected discounted value of dividends until ruin. We give sufficient conditions under which the optimal strategy is of threshold type.     

The expected discounted penalty function for two classes of risk processes perturbed by diffusion with multiple thresholds

In this paper, we consider a perturbed risk model with two independent classes of risks under multiple thresholds in which both of the two inter-claim times have phase-type distributions. We obtain the integro-differential equations with boundary conditions for the expected discounted penalty function. Explicit expressions are derived if the two classes claim amount distributions both belong to the rational family.     

Moderate deviations for a risk model based on the customer-arrival process

In this paper, we investigate the moderate deviations for a customer-arrival-based insurance risk model, in which customer’s actual claim sizes are described as independent and identically distributed heavy-tailed random variables multiplying a shot function, and the model can be treated as a Poisson shot noise process.     

Optimal proportional reinsurance and investment under partial information

In this paper, we study the optimal proportional reinsurance and investment strategy for an insurer that only has partial information at its disposal, under the criterion of maximizing the expected utility of the terminal wealth. We assume that the surplus of the insurer is governed by a jump diffusion process, and that reinsurance is used by the insurer to reduce risk. In addition, the insurer can invest in financial markets. We give a characterization for the optimal strategy within a non-Markovian setting. Malliavin calculus for Lévy processes is used for the analysis.     

On the expected discounted penalty function in a Markov-dependent risk model with constant dividend barrier

This article considers a Markov-dependent risk model with a constant dividend barrier. A system of integro-differential equations with boundary conditions satisfied by the expected discounted penalty function, with given initial environment state, is derived and solved. Explicit formulas for the discounted penalty function are obtained when the initial surplus is zero or when all the claim amount distributions are from rational family. In two state model, numerical illustrations with exponential claim amounts are given.     

Numerical solutions of quantile hedging for guaranteed minimum death benefits under a regime-switching jump-diffusion formulation

This work develops numerical approximation methods for quantile hedging involving mortality components for contingent claims in incomplete markets, in which guaranteed minimum death benefits (GMDBs) could not be perfectly hedged. A regime-switching jump-diffusion model is used to delineate the dynamic system and the hedging function for GMDBs, where the switching is represented by a continuous-time Markov chain. Using Markov chain approximation techniques, a discrete-time controlled Markov chain with two component is constructed. Under simple conditions, the convergence of the approximation to the value function is established. Examples of quantile hedging model for guaranteed minimum death benefits under linear jumps and general jumps are also presented.     

Information,no-arbitrage and completeness for asset price models with a change point

We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time τ$\tau$. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions.