Optimal reinsurance for Gerber–Shiu functions in the Cramér–Lundberg model

Complementing existing results on minimal ruin probabilities, we minimize expected discounted penalty functions (or Gerber–Shiu functions) in a Cramér–Lundberg model by choosing optimal reinsurance. Reinsurance strategies are modeled as time dependent control functions, which lead to a setting from the theory of optimal stochastic control and ultimately to the problem’s Hamilton–Jacobi–Bellman equation. We show existence and uniqueness of the solution found by this method and provide numerical examples involving light and heavy tailed claims and also give a remark on the asymptotics.     

Pseudo-Taylor expansions and the Carathéodory–Fejér problem

We give a new solvability criterion for the boundary Carathéodory–Fejér problem: given a point $x\in \mathbb{R}$ and, a finite set of target values $a^0,a^1,\dots ,a^n\in \mathbb{C}$, to construct a function f in the Pick class such that the limit of $f^{(k)}(z)/k!$ as $z\to x$ nontangentially in the upper half-plane is $a^k$ for $k=0,1,\dots ,n$. The criterion is in terms of positivity of an associated Hankel matrix. The proof is based on a reduction method due to Julia and Nevanlinna.     

A Cramér–Rao inequality for non-differentiable models

We compute a variance lower bound for unbiased estimators in statistical models. The construction of the bound is related to the original Cramér–Rao bound, although it does not require the differentiability of the model. Moreover, we show our efficiency bound to be always greater than the Cramér–Rao bound in smooth models, thus providing a sharper result.     

Cramér transform and t-entropy

t-entropy is the convex conjugate of the logarithm of the spectral radius of a weighted composition operator (WCO). Let \(X\) be a nonnegative random variable. We show how the Cramér transform with respect to the spectral radius of WCO is expressed by the t-entropy and the Cramér transform of the given random variable \(X\) .     

An optimal control problem inL ∞1

We consider an optimal control problem inL , where the cost functional has a penalty term which involves the structure of the control law. This type of penalization allows us to restrict our attention only to piecewise constant controls, with assigned switching times, i.e., the controls belong to finite-dimensional control spaces. This fact and our assumptions on the dynamics give as a consequence the compactness of the minimizing sequences inC([0, 1], ) ×L ([0, 1], ). The existence of a minimum of the cost functional is then obtained by a direct method. This result allows us to avoid the usual convexity assumption on the cost functionalC and on the multivalued vector field associated to the dynamics when we have to consider the controls in all ofL .This research was partially supported by Research Project M.P.I. 40%, Teoria del Controllo dei Sistemi Dinamici.     

On the Cramér–von Mises test with parametric hypothesis for poisson processes

The problem of the goodness of-fit testing for inhomogeneous Poisson process with parametric basic hypothesis is considered. A test statistic of the Cramér–von Mises type with parameter replaced by the maximum likelihood estimator is proposed and its asymptotic behavior is studied. It is shown that in the case of shift parameter, the limit distribution of the test statistics (under hypothesis) does not depend on the true value of this parameter.     

A remark on the series of H. Cramér

One establishes some asymptotic representations for an analogue of H. Cramér's series, with the aid of which one describes the asymptotics of the probabilities of large deviations of the norm of the sum of independent, identically distributed random variables in a Hilbert space (L. V. Osipov, Teor. Veroyatn. Primen.,23, 510–526).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 181–185, 1980.     

Optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk process under the Heston model

In this paper, we study the optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk model. The insurer is allowed to purchase reinsurance and invest in one risk-free asset and one risky asset whose price process satisfies the Heston model. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. By applying stochastic optimal control approach, we obtain the optimal strategy and value function explicitly. In addition, a verification theorem is provided and the properties of the optimal strategy are discussed. Finally, we present a numerical example to illustrate the effects of model parameters on the optimal investment–reinsurance strategy and the optimal value function.     

Elliptic solutions in the Hénon–Heiles model

Equations of motion corresponding to the Hénon–Heiles Hamiltonian are considered. A method enabling one to find all elliptic solutions of an autonomous ordinary differential equation or a system of autonomous ordinary differential equations is described. New families of elliptic solutions of a fourth-order equation related to the Hénon–Heiles system are obtained. A classification of elliptic solutions up to the sixth order inclusively is presented.     

A mixed integer nonlinear programming model for the optimal repair–replacement in the firm

Optimal repair–replacement problem is an important aspect of economic decision making at the firm and aggregate levels. In this paper, we extend the continuous time optimal replacement model in the firm under technological progress by considering the possibility of repairing/replacing the machines during their lifetime period. In our model, two possible decisions can be recognized by the managers in which the machines are repaired under the efficiency condition or replaced under the availability of technological progress in the firm. As a special case, we restrict the model to the more real case in which all the growth, purchase price and repair cost functions are assumed to be in the exponential form. The solvability of the model in this case is also discussed.     

Time-consistent mean–variance proportional reinsurance and investment problem in a defaultable market

In this paper, we consider an optimal time-consistent reinsurance-investment problem incorporating a defaultable security for a mean–variance insurer under a constant elasticity of variance (CEV) model. In our model, the insurer’s surplus process is described by a jump-diffusion risk model, the insurer can purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset, a defaultable bond and a risky asset whose price process is assumed to follow a CEV model. Using a game theoretic approach, we establish the extended Hamilton–Jacobi–Bellman system for the post-default case and the pre-default case, respectively. Furthermore, we obtain the closed-from expressions for the time-consistent reinsurance-investment strategy and the corresponding value function in both cases. Finally, we provide numerical examples to illustrate the impacts of model parameters on the optimal time-consistent strategy.     

Remark on the Cramér Type Condition for ω2-Statistic

In 1984, Bentkus,^{(2, 3)} and independently,Nagaev and Chebotarev,⁽¹¹⁾ introduced somemodifications of the well-known one-dimensional Cramér conditionlim sup_|t| |E e ^it| < 1 for the case of Hilbert space (denote it by H). Although many H-valued random variables satisfy these conditions, there remained open the question if these conditionshold for the random variable X ₀ = I _{{ < x007D;}–x ( is uniformly distributed on [0, 1]), whichtakes its values in H = L ₂[0,1], and in a sense generates the so-called²-statistic. It is shown in the present paper thatX ₀ does not satisfy the condition of Nagaev andChebotarev,⁽¹¹⁾ and also, at least partly, thecondition of Bentkus.^{(2, 3)} One more versionof the Cramér condition in the case of Hilbert space is suggestedin the paper not for individual summands, but for the whole sum. Ageneralization of the estimate of the characteristic function of squarednorm of the sum of independent truncations is obtained in terms ofquantities, which define this modification. It is shown thatX ₀ satisfies this version of the Cramércondition.     

An impulsively controlled predator–pest model with disease in the pest

In this paper, we consider an integrated pest management model with disease in the pest and a stage structure for its natural predator, which is subject to impulsive and periodic controls. A nonlinear incidence rate expressed in an abstract form, is used to describe the propagation of the disease, which is spread through the periodic release of infective pests, the functional response of the mature predator also being given in an abstract, unspecified form. Sufficient conditions for the local and global stability of the susceptible pest-eradication periodic solution are found by means of Floquet theory and comparison methods, the permanence of the system also being discussed. These stability conditions are shown to be biologically significant, being reformulated as balance conditions for the susceptible pest class.     

An optimal Hardy–Morrey inequality

In this work we improve the sharp Hardy inequality in the case p?>?n by adding an optimal weighted H?lder seminorm. To achieve this we first obtain a local improvement. We also obtain a refinement of both the Sobolev inequality for p?>?n and the Hardy inequality, the latter having the best constant.