On optimal dividend payments and related problems

This work deals with an optimal dividend payment problem for a piecewise-deterministic compound Poisson insurance risk model. The objective is to maximize the expected discounted dividend payout up to the time of ruin. When the dividend payment rate is restricted, the value function is shown to be a solution of the corresponding Hamilton-Jacobi-Bellman equation, which in turn leads to a tractable methodology to find an optimal threshold dividend payment policy. For the case of unrestricted payment rate, the value function and an optimal barrier strategy are determined explicitly with exponential claim size distributions. A comparison of two examples is provided to illustrate the main results.

Assets are assumed to follow a diffusion process subject to some conditions. The owners can pay dividends at their discretion, but whenever assets reach zero, they have to reinvest money so that assets never go negative. With each dividend payment there is a fixed and a proportional cost, and so with reinvestments. The goal is to maximize the expected value of discounted net cash flow, i.e., dividends paid minus reinvestments. It is shown that there can be two different solutions depending on the model parameters and the costs as follows. (1) Whenever assets reach a barrier $y^*$ they are reduced to $y^*-\delta^*$ through a dividend payment, and whenever they reach 0 they are increased to $\gamma^*\leq y^*-\delta^*$ by a reinvestment. (2) There is no optimal policy, but the value function is approximated by policies of the form described in (1) for increasing barriers. We provide criteria to decide whether an optimal solution exists, and when it does not, we show how to calculate the value function. We discuss how the problem can be solved numerically and give numerical examples.

On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes

We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ( $i=1,2,\dots$ ) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature; that is, if the surplus process just before the ith arrival is at level u, then for $a>0$ the capital jumps up to the level $(1+a)u+C_i$ . The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arrival process of proportional gains. In the dividend calculations, we also consider a random perturbation of our basic risk process modeled by an independent Brownian motion with drift.

Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle

In this paper we study the optimal dividend payments for a company of limited liability whose cash reserves in the absence of dividends follow a Markov-modulated jump-diffusion process with positive drifts and negative exponential jumps, where parameters and discount rates are modulated by a finite-state irreducible Markov chain. The main aim is to maximize the expected cumulative discounted dividend payments until bankruptcy time when cash reserves are nonpositive for the first time. We extend the results of Jiang and Pistorius [15] to our setup by proving that it is optimal to adopt a modulated barrier strategy at certain positive regime-dependent levels and that the value function can be explicitly characterized as the fixed point of a contraction.

Optimal Dividend Policy When Cash Reserves Follow a Jump-Diffusion Process Under Markov-Regime Switching

In recent years, the Markov Regime-Switching model and the class of Archimedean\n\ncopulas have been widely applied to a variety of finance-related fields. The\n\nMarkov Regime-Switching model can reflect the reality that the underlying economy\n\nis changing over time. Archimedean copulas are one of the most popular\n\nclasses of copulas because they have closed form expressions and have great flexibility in modeling different kinds of dependencies.\n\nIn the thesis, we first consider a discrete-time risk process based on the compound\n\nbinomial model with regime-switching. Some general recursive formulas\n\nof the expected penalty function have been obtained. The orderings of ruin probabilities are investigated. In particular, we show that if there exists a stochastic\n\ndominance relationship between random claims at different regimes, then we can\n\norder ruin probabilities under different initial regimes.\n\nRegarding capital allocation problems, which are important areas in finance\n\nand risk management, this thesis studies the problems of optimal allocation of\n\npolicy limits and deductibles when the dependence structure among risks is modeled\n\nby an Archimedean copula. By employing the concept of arrangement\n\nincreasing and stochastic dominance, useful qualitative results of the optimal\n\nallocations are obtained.\n\nThen we turn our attention to a new family of risk measures satisfying a set\n\nof proposed axioms, which includes the class of distortion risk measures with\n\nconcave distortion functions. By minimizing the new risk measures, we consider\n\nthe optimal allocation of policy limits and deductibles problems based on the\n\nassumption that for each risk there exists an indicator random variable which\n\ndetermines whether the risk occurs or not. Several sufficient conditions to order\n\nthe optimal allocations are obtained using tools in stochastic dominance theory.

We consider the optimal stopping problem, based on the compound binomial model with capital injection controlled by optimal dividend strategy. First, the value function V(x) is established. Second, via the Bellman equation satisfied by the value function V(x) which maximizes the discounted value of define between dividend payment and the penalized discounted capital injection, we find the optimal stopping time τ*. Furthermore an optimal stopping model with capital injection in order to maximize the shareholders’ interests is obtained.

In this paper, some results on the dividend payments prior to ruin in the classical surplus process with stochastic interest are derived. An integro-differential equation with a boundary conditions satisfied by the expected present value of dividend payments is derived and solved. Furthermore, closed-form expressions for exponential claims are given.

This paper addresses the problem of finding an optimal dividend policy for a class of jump-diffusion processes. The jump component is a compound Poisson process with negative jumps, and the drift and diffusion components are assumed to satisfy some regularity and growth restrictions. Each dividend payment is changed by a fixed and a proportional cost, meaning that if ξ is paid out by the company, the shareholders receive kξ−K, where k and K are positive. The aim is to maximize expected discounted dividends until ruin. It is proved that when the jumps belong to a certain class of light-tailed distributions, the optimal policy is a simple lump sum policy, that is, when assets are equal to or larger than an upper barrier $\bar{u}^{*}$ , they are immediately reduced to a lower barrier $\underline{u}^{*}$ through a dividend payment. The case with K=0 is also investigated briefly, and the optimal policy is shown to be a reflecting barrier policy for the same light-tailed class. Methods to numerically verify whether a simple lump sum barrier strategy is optimal for any jump distribution are provided at the end of the paper, and some numerical examples are given.

De Finetti’s optimal dividend problem has recently been extended to the case when dividend payments can be made only at Poisson arrival times. In this paper we consider the version with bail-outs where the surplus must be nonnegative uniformly in time. For a general spectrally negative Lévy model, we show the optimality of a Parisian-classical reflection strategy that pays the excess above a given barrier at each Poisson arrival time and also reflects from below at 0 in the classical sense.

The finite-horizon optimal control problem with input constraints consists in controlling the state of a dynamical system over a finite time interval (possibly unknown) minimising a cost functional, while satisfying hard constraints on the input. In this framework, the minimum-time optimal control problem and some related problems are of interest for both theory and applications. For linear systems, the solution of the problem often relies upon the use of bang-bang control signals. For nonlinear systems, the “shape” of the optimal input is in general not known. The control input can be found solving a Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE): it typically consists of a combination of bang-bang controls and singular arcs. In this paper, a methodology to approximate the solution of the HJB PDE is proposed. This approximation yields a dynamic state feedback law. The theory is illustrated by means of two examples: the minimum-time optimal control problem for an industrial wastewater treatment plant and the Goddard problem, i.e. a maximum-range optimal control problem.

In this paper, a discrete time risk model with dividend payments is considered, in which the occurrences of the claims may be delayed. A system of differential equations with certain boundary conditions for the expected present value of dividend payments prior to ruin is derived and solved. Moreover, the closed form expressions are given when the claim size distributions belong to the rational family. The qualitative properties of ruin probabilities for this risk model are also obtained. Numerical results are provided to illustrate the applicability of the main result.

On the classical risk model with credit and debit interests under absolute ruin