Optimal Dividends in the Dual Model with Diffusion

We find the optimal dividend strategy in two related risk models under an affine penalty for ruin. The first risk model is the classical Cramer–Lundberg risk model, and the second is the so-called dual risk model. Under both models, for exponentially distributed jumps, we show that the optimal dividend strategy is a barrier strategy and obtain the barrier explicitly. Moreover, we prove that the optimal barrier increases with respect to the parameters of the affine penalty, while the penalized value function decreases with respect to the penalty. We also compute the expected time until ruin and show that the expected time of ruin increases with respect to the parameters of the affine penalty. Finally, we present some numerical examples to demonstrate the relationship between the results for the classical and dual risk models. Our main contributions are in comparing the classical and dual risk models side-by-side and in obtaining explicit expressions for the penalized value functions, the optimal barriers, and the expected times of ruin. Also, we contrast the free-boundary condition associated with the barrier strategies in the two models.

In the dual risk model, we consider the optimal dividend and capital injection problem, which involves a random time horizon and a ruin penalty. Both fixed and proportional costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends, and the penalized discounted both capital injections and ruin penalty during the horizon, which is described by the minimum of the time of ruin and an exponential random variable. The explicit solutions for optimal strategy and value function are obtained, when the income jumps follow a hyper-exponential distribution. Besides, some numerical examples are presented to illustrate our results.

Asymptotic Analysis for Optimal Dividends in a Dual Risk Model

In this paper, we investigate the optimal dividend problem under Parisian ruin with affine penalty payments at Parisian ruin time. The underlying risk process is assumed to be a spectrally negative Levy risk process. With the help of the dynamic programming principle, we prove that the value function associated to our optimal control problem is the smallest solution with certain characteristics to the corresponding Hamilton–Jacobi–Bellman (HJB) equation. In addition, the form of the performance function under barrier dividend strategy is expressed in terms of various extended scale functions. Then we identify a condition under which the performance function under certain barrier strategy is also a solution to the HJB equation, which in turn illustrates the optimalilty of such barrier dividend strategy among all admissible strategies. Various numerical examples are also given when the underlying risk process is compound Poisson process, Brownian motion with drift and jump-diffusion process.

On optimality of the barrier strategy for a general Lévy risk process

The paper studies the dual risk model with a barrier strategy under the concept of bankruptcy, in which one has a positive probability to continue business despite temporary negative surplus. Integrodifferential equations for the expectation of the discounted dividend payments and the probability of bankruptcy are derived. Moreover, when the gain size distribution is exponential, explicit solutions for the expected dividend payments and the bankruptcy probability are obtained for constant bankruptcy rate function. It also provided some numerical examples to illustrate the applications of the explicit solutions.

Optimal dividend strategy in dual risk model is well studied in the literatures. But to the best of our knowledge, all the previous works assumes deterministic interest rate. In this paper, we study the optimal dividends strategy in dual risk model, under a stochastic interest rate, assuming the discounting factor follows a geometric Brownian motion or exponential Lévy process. We will show that closed form solutions can be obtained.

Consider a spectrally negative risk process where, on ruin, the deficit is immediately paid, and the process restarts from 0. When the process reaches a threshold b, all the surplus above b is paid as dividend. Applying the theory of exit times for a spectrally negative Lévy process and its reflection at the maximum and at the minimum, we obtain recursive formulae for the following moments. (i) The moments of the discounted loss until the process reaches b. This is equivalent to the moments of the discounted dividends in the dual model under the barrier strategy. (ii) The moments of the discounted loss for models with and without a dividend barrier for the infinite horizon. (iii) The moments of the discounted dividends for the infinite horizon.

We consider the dual of the generalized Erlang(n) risk model with a barrier dividend strategy. We derive integro‐differential equations with boundary conditions satisfied by the expectation of the sum of discounted dividends until ruin and the moment‐generating function of the discounted dividend payments until ruin, respectively. The results are illustrated by several examples.

On the discounted penalty at ruin in a jump-diffusion and the perpetual put option

An anti-disturbance fault tolerant control (FTC) based on disturbance observer scheme is considered for a class of stochastic systems with faults and multiple disturbances in this paper. The disturbances are divided into two parts. One represents the stochastic disturbance with partial-known information which is formulated by an exogenous system. The other is independent Wiener process. A stochastic disturbance observer is designed to estimate exogenous disturbance using pole placement. And a composite fault diagnosis observer with disturbance observer is constructed to diagnose faults. Furthermore, a composite fault-tolerant controller is proposed to compensate modeled disturbance and accommodate faults. Finally, a simulation example is given to demonstrate the feasibility and effectiveness of the proposed scheme.

This paper proposes a disturbance observer-based anti-disturbance control strategy for a class of stochastic systems subject to two kinds of disturbances. One represents the stochastic disturbance with partially-known information. The other is an independent Wiener process. The stochastic disturbance observer can be constructed separately from the controller design to estimate the first kind of disturbance. Based on stochastic disturbance observer, a novel anti-disturbance control scheme is proposed to improve the ability of anti-disturbance for stochastic systems. A simulation example is given to demonstrate the effectiveness of the proposed method.

We establish the necessary and sufficient optimality conditions on a nondifferentiable minimax fractional programming problem. Subsequently, applying the optimality conditions, we constitute two dual models: Mond-Weir type and Wolfe type. On these duality types, we prove three duality theorems--weak duality theorem, strong duality theorem, and strict converse duality theorem.

In the present work, we discuss the thermoelastic interactions in a half-space in the context of the theory of dual phase-lag thermoelasticity due to stochastic conditions applied at the boundary. We consider a one-dimensional problem with traction-free boundary subjected to two types of time-dependent thermal distributions. The boundary conditions are assumed to be random in nature to make the problem more realistic and the noise added is considered to be white noise. The problem is solved using Laplace transform and short time approximation is used to find the approximate solution for field variables while taking the inverse transforms. The solution for the stochastic case has been obtained by using the concept of a Wiener process and stochastic simulation. Numerical analysis based on stochastic simulation is carried out using copper material along different sample paths and comparative analysis between the deterministic and the stochastic distributions of field variables is presented. Special attention is paid to highlight the effects of considering randomness added to the boundary conditions.

§1. Here we consider a two-dimensional stationary random process whose components ξ(t) and n(t) satisfy the stochastic differential equations where φ(t) and ψ(t) are two independent Wiener processes with