The Discounted Penalty Function with Multi-Layer Dividend Strategy in the Phase-Type Risk Model

The Gerber–Shiu discounted penalty function in a delayed renewal risk model with multi-layer dividend strategy

On a perturbed Sparre Andersen risk model with multi-layer dividend strategy

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

This chapter complements the material of Chap. 4 in the sense that explicit identification of requisite quantities is possible with stronger parametric assumptions about either the (marginal) distribution of the claim sizes or that of the interclaim times. Convenient parametric assumptions involve Erlang (or Coxian) distributions, due to the generality and mathematical tractability inherent in the use of such models. Sect. 5.1 directly generalizes the classical Poisson model of Chap. 3 by essentially assuming a (marginal) Erlang form for the interclaim times. A Lagrange polynomial approach together with Laplace transform arguments is used to explicitly identify all components of the Gerber-Shiu function. The independent model with an arbitrary interclaim time distribution and exponential claim sizes is then considered in Sect. 5.2. The conditioning argument of Sect. 5.2 is then extended in Sect. 5.3 to a model where the (marginal) claim size distribution is assumed to be of Coxian form. In particular, identification of the Gerber-Shiu function is shown to be possible for special choices of the penalty function, and more generally for an arbitrary penalty function of the deficit. Finally, the Coxian assumption of Sect. 5.3 is replaced by a (possibly infinite) mixture of Erlang distributions, resulting in tractable results for ruin probabilities, the deficit at ruin, and related quantities.

In the classical risk theory ruin probabilities were studied with the interclaim time d.f. K(t) = l-e−t. In Sparre Andersen (1957) the theory was extended to other interclaim-time distributions but doing so one has to make difference between an ordinary Sparre Andersen model and a stationary Sparre Andersen model.

In this paper, a compound Poisson risk model with time-dependent claims is studied under a multi-layer dividend strategy. A piecewise integro-differential equation for the Gerber-Shiu function is derived and solved. Asymptotic formulas of the ruin probability are obtained when the claim size distributions are heavy-tailed.

The Gerber–Shiu discounted penalty function in the delayed renewal risk process with random income

Viscoelastic thin-walled structures such as plates, panels and shells, with mounted objects in the form of additional masses are widely used in modern technology. The role of such additional masses is often played by longitudinal and transverse ribs, tie-plates and fixtures. When designing such structures, it is relevant to study their dynamic behavior depending on the mass distribution, viscoelastic and inhomogeneous properties of the material, etc. In this paper, the dynamic stability of a viscoelastic shell carrying concentrated masses is considered, taking into account the nonlinear and inhomogeneous properties of the material. A mathematical model of the problem is described by a system of integro-differential equations in partial derivatives. With the Bubnov-Galerkin method, the problem is reduced to solving a system of ordinary nonlinear integro-differential equations. To solve the resulting system with the Koltunov-Rzhanitsyn singular kernel, a numerical method based on the use of quadrature formulas is applied. The effect of the viscoelastic and inhomogeneous properties of the shell material, location, and the amount of concentrated masses on stability is studied.

A similar surplus analysis as that of previous chapters, but with important variations on the dependent Sparre Andersen model, is considered in this chapter. In Sect. 7.1, the joint distribution of the first interclaim time and claim size distributions is allowed to assume a different form in order to accomodate the possibility that a claim does not occur at time 0. A special case of this model involving the equilibrium distribution of subsequent (marginal) interclaim times, motivated by stationarity arguments, is also discussed. A discrete model is considered in Sect. 7.2, where special emphasis is placed on analysis using a discrete Coxian type assumption for the (marginal) interclaim times. This model generalizes the compound binomial model, a discrete analog of the classical Poisson model often used in financial applications, which is discussed here as well.

Convergence of the collocation method and the mechanical quadrature method for systems of singular integro-differential equations in Lebesgue spaces

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a Kn distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.

We consider the Sparre Andersen risk process in the presence of a constant dividend barrier, and propose a new expected discounted penalty function which is different from that of Gerber and Shiu. We find that iteration mothed can be used to compute the values of expected discounted dividends until ruin and the new penalty function. Applying the new function and the recursion method proposed in Section 5, we obtain the arbitrary moments of discounted dividend payments until ruin.

In this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the survival probability is well given when the claim amount distribution is Erlang distribution or mixed Erlang distribution. The expressions for moments of the time to ruin with the model above are given.

In this work, a modified residual power series method is implemented for providing efficient analytical and approximate solutions for a class of coupled system of nonlinear fractional integrodifferential equations. The proposed algorithm is based on the concept of residual error functions and generalized power series formula. The fractional derivative is described under the Caputo concept. To illustrate the potential, accuracy, and efficiency of the proposed method, two numerical applications of the coupled system of nonlinear fractional integrodifferential equations are tested. The numerical results confirm the theoretical predictions and depict that the suggested scheme is highly convenient, is quite effective, and practically simplifies computational time. Consequently, the proposed method is simple, accurate, and convenient in handling different types of fractional models arising in the engineering and physical systems.

In this paper, fuzzy neural network (FNN) can be trained with crisp and fuzzy data. The work of this paper is an expansion of the research of fuzzy linear Fredholm integro-differential equations. In this work, we interpret a fuzzy integro-differential equations. In this paper, a novel hybrid method based on FNN and Newton–Cotes methods with positive coefficient for the solution of system of fuzzy linear Fredholm integro-differential equations of the second kind with fuzzy initial values is presented. The problem formulation of the proposed UA is quite straightforward. To obtain the “best-approximated” solution of system of fuzzy linear Fredholm integro-differential equations, the adjustable parameters of PFNN are systematically adjusted by using the learning algorithm. Within this paper, the fuzzy neural network model is used to obtain an estimate for the fuzzy parameters in a statistical sense. Based on the extension principle, a simple algorithm from the cost function of the fuzzy neural network is proposed, in order to find the approximate parameters. We propose a learning algorithm from the cost function for adjusting of fuzzy weights. Here neural network is considered as a part of a larger field called neural computing or soft computing. Finally, we illustrate our approach by some numerical examples.