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

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.

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.

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.

A model in which a foraging animal can reproduce if its energy reserves reach a critical level is presented. The animal's reserves are modelled as a finite-state Markov chain. The animal has the choice between foraging options which may have the same mean net gain but differ in their variances. In addition to starvation, animals are subject to death because of predation, bad weather, and so on. We focus on the case where the rate of mortality due to these sources is the same under all options. We investigate policies that maximise the expected lifetime reproductive success. It is found that the optimal value function is concave at low reserves (low-variance action region) and convex at high reserves (high-variance action region). The value function under the low-variance action has also the same shape and the same inflexion point. This result allows us to compute optimal policies just by looking at the low-variance value function. The result is also used to show that increasing the mortality rate increases the high-variance region under the optimal policy. The pattern of risk-sensitive behaviour predicted by this model is in contrast to that predicted by a similar model in which no reproduction occurs and the optimality criterion is to minimise the probability of death (McNamara (1990)).

In this work, we study a dynamic portfolio optimization problem related to pairs trading, which is an investment strategy that matches a long position in one security with a short position in another security with similar characteristics. The relationship between pairs, called a spread, is modeled by a Gaussian mean-reverting process whose drift rate is modulated by an unobservable continuous-time, finite-state Markov chain. Using the classical stochastic filtering theory, we reduce this problem with partial information to an equivalent one with full information and solve it for the logarithmic utility function, where the terminal wealth is penalized by the riskiness of the portfolio according to the realized volatility of the wealth process. We characterize optimal dollar-neutral strategies as well as optimal value functions under full and partial information and show that the certainty equivalence principle holds for the optimal portfolio strategy. Finally, we provide a numerical analysis for a toy example with a two-state Markov chain.

In this paper, having been inspired by the work of Kunita and Seko, we study the pricing of δ‐penalty game call options on a stock with a dividend payment. For the perpetual case, our result reveals that the optimal stopping region for the option seller depends crucially on the dividend rate d. More precisely, we show that when the penalty δ is small, there are two critical dividends 0 < d1 < d2 < ∞ such that the optimal stopping region for the option seller takes one of the following forms: (1) an interval if d < d1; (2) a singleton if d∈ [d1, d2]; or (3) an empty set if d > d2. When d∈ [d1, d2], the value function is not continuously differentiable at the optimal stopping boundary for the option seller, therefore our result in the perpetual case cannot be established by the free boundary approach with smooth‐fit conditions imposed on both free boundaries. For the finite time horizon case, the dependence of the optimal stopping region for the option seller on the time to maturity is exhibited; more precisely, when both δ and d are small, we show that there are two critical times 0 < T1 < T2 < T, such that the optimal stopping region for the option seller takes one of the following forms: (1) an interval if t < T1; (2) a singleton if t∈ [T1, T2]; or (3) an empty set if t > T2. In summary, for both the perpetual and the finite horizon cases, we characterize in terms of model parameters how the optimal stopping region for the option seller shrinks when the dividend rate d increases and the time to maturity decreases; these results complete the original work of Emmerling for the perpetual case and Kunita and Seko for the finite maturity case. In addition, for the finite time horizon case, we also extend the probabilistic method for the establishment of existence and regularity results of the classical American option pricing problem to the game option setting. Finally, we characterize the pair of optimal stopping boundaries for both the seller and the buyer as the unique pair of solutions to a couple of integral equations and provide numerical illustrations.

In this study, we are currently investigating the controllability of nonlinear fractional differential control systems with delays in the state function. The solution representations of fractional delay differential equations have been established by using the delayed Mittag-Leffler function. Firstly we obtain the result of the controllability of a linear fractional control system with delay. Then, for the controllability criteria of nonlinear fractional delay system, we establish the set of sufficient conditions of nonlinear fractional differential systems with delay in their state function by using Schauder’s fixed point theorem. In the end, a numerical example is constructed to support the results.

Optimal feedback production planning in a stochastic N-machine flowshop

In this paper we derive some results on the dividend payments prior to ruin in a Markovmodulated risk process in which the rate for the Poisson claim arrival process and the distribution of the claim sizes vary in time depending on the state of an underlying (external) Markov jump process {J(t); t ≥ 0}. The main feature of the model is the flexibility in modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate, and that the states of {J(t); t ≥ 0} could describe, for example, epidemic types in health insurance or weather conditions in car insurance. A system of integro-differential equations with boundary conditions satisfied by the nth moment of the present value of the total dividends prior to ruin, given the initial environment state, is derived and solved. We show that the probabilities that the surplus process attains a dividend barrier from the initial surplus without first falling below zero and the Laplace transforms of the time that the surplus process first hits a barrier without ruin occurring can be expressed in terms of the solution of the above-mentioned system of integro-differential equations. In the two-state model, explicit results are obtained when both claim amounts are exponentially distributed.

Value function in maximum hands-off control for linear systems

Nearly Optimal Control Scheme Using Adaptive Dynamic Programming Based on Generalized Fuzzy Hyperbolic Model

The authors study the performance of a non-blocking space-division packet switch, which experiences nonuniform traffic intensities that also change as a function of time. A finite-state Markov chain is used as an underlying process to govern the time variation of traffic for the entire switch. The packet arrivals at each input form an independent Bernoulli process modulated by the underlying Markov chain. Also, the output address of each packet is independently and randomly assigned with probability distributions, which are gain modulated by the Markov chain. Provided that the traffic at each output is not dominated by individual inputs, it is shown that the service time of each input queue for sufficiently large switches can be characterized by an independent Markov-modulated phase-type process. A matrix geometric solution for the resultant quasi-birth-death-type queuing process is presented. The maximum throughput is obtained at the system saturation. The performance of the switch is numerically examined under various traffic conditions. A contention priority scheme is also proposed to improve the switch performance. >

Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates

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.