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Journal articles on the topic 'Optimal dividend control problem'

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Published: 11 March 2023

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1

Ekström, Erik, and Bing Lu. "The Optimal Dividend Problem in the Dual Model." Advances in Applied Probability 46, no. 3 (September 2014): 746–65. http://dx.doi.org/10.1239/aap/1409319558.

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We study de Finetti's optimal dividend problem, also known as the optimal harvesting problem, in the dual model. In this model, the firm value is affected both by continuous fluctuations and by upward directed jumps. We use a fixed point method to show that the solution of the optimal dividend problem with jumps can be obtained as the limit of a sequence of stochastic control problems for a diffusion. In each problem, the optimal dividend strategy is of barrier type, and the rate of convergence of the barrier and the corresponding value function is exponential.

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2

Ekström, Erik, and Bing Lu. "The Optimal Dividend Problem in the Dual Model." Advances in Applied Probability 46, no. 03 (September 2014): 746–65. http://dx.doi.org/10.1017/s0001867800007357.

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We study de Finetti's optimal dividend problem, also known as the optimal harvesting problem, in the dual model. In this model, the firm value is affected both by continuous fluctuations and by upward directed jumps. We use a fixed point method to show that the solution of the optimal dividend problem with jumps can be obtained as the limit of a sequence of stochastic control problems for a diffusion. In each problem, the optimal dividend strategy is of barrier type, and the rate of convergence of the barrier and the corresponding value function is exponential.

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3

Pérez, José-Luis, Kazutoshi Yamazaki, and Xiang Yu. "On the Bail-Out Optimal Dividend Problem." Journal of Optimization Theory and Applications 179, no. 2 (June 23, 2018): 553–68. http://dx.doi.org/10.1007/s10957-018-1340-3.

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4

Zhu, Jinxia. "OPTIMAL FINANCING AND DIVIDEND DISTRIBUTION WITH TRANSACTION COSTS IN THE CASE OF RESTRICTED DIVIDEND RATES." ASTIN Bulletin 47, no. 1 (October 5, 2016): 239–68. http://dx.doi.org/10.1017/asb.2016.29.

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AbstractWe consider the optimal financing (capital injections) and dividend payment problem for a Brownian motion model in the case of restricted dividend rates. The company has no obligation to inject capitals and therefore, the bankruptcy risk is present. Capital injections, if any, will incur both fixed and proportional transaction costs and dividend payments incur proportional transaction costs. The aim is to find the optimal strategy to maximize the expected present value of dividend payments minus the total cost of capital injections up to the time of bankruptcy. The problem is formulated as a mixed impulse-regular control problem. We address the problem via studying three cases of two auxiliary functions. We derive important analytical properties of the auxiliary functions and use them to study the value function and then identify the optimal control strategy. We show that the optimal dividend control is of threshold type and the optimal financing strategy prescribes to either never inject capitals or inject capitals only when the surplus reaches 0 with a fixed lump sum amount.

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5

Lindensjö, Kristoffer, and Filip Lindskog. "Optimal dividends and capital injection under dividend restrictions." Mathematical Methods of Operations Research 92, no. 3 (July 16, 2020): 461–87. http://dx.doi.org/10.1007/s00186-020-00720-y.

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AbstractWe study a singular stochastic control problem faced by the owner of an insurance company that dynamically pays dividends and raises capital in the presence of the restriction that the surplus process must be above a given dividend payout barrier in order for dividend payments to be allowed. Bankruptcy occurs if the surplus process becomes negative and there are proportional costs for capital injection. We show that one of the following strategies is optimal: (i) Pay dividends and inject capital in order to reflect the surplus process at an upper barrier and at 0, implying bankruptcy never occurs. (ii) Pay dividends in order to reflect the surplus process at an upper barrier and never inject capital—corresponding to absorption at 0—implying bankruptcy occurs the first time the surplus reaches zero. We show that if the costs of capital injection are low, then a sufficiently high dividend payout barrier will change the optimal strategy from type (i) (without bankruptcy) to type (ii) (with bankruptcy). Moreover, if the costs are high, then the optimal strategy is of type (ii) regardless of the dividend payout barrier. We also consider the possibility for the owner to choose a stopping time at which the insurance company is liquidated and the owner obtains a liquidation value. The uncontrolled surplus process is a Wiener process with drift.

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6

Sun, Shi Liang, Xiao Qian Huang, and Lu Lian. "Control Strategy of Proportional Reinsurance with Dividend Process." Applied Mechanics and Materials 488-489 (January 2014): 1301–5. http://dx.doi.org/10.4028/www.scientific.net/amm.488-489.1301.

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This paper is concerned with the problem of proportional reinsurance control strategy with dividend process in insurance company. The existence of optimal control strategy is proved by variational inequality,and the specific construct of optimal control strategy and the explicit structure of optimal return function are derived.

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7

Chevalier, Etienne, Vathana Ly Vath, and Simone Scotti. "An Optimal Dividend and Investment Control Problem under Debt Constraints." SIAM Journal on Financial Mathematics 4, no. 1 (January 2013): 297–326. http://dx.doi.org/10.1137/120866816.

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8

Chen, Mi, Xiaofan Peng, and Junyi Guo. "Optimal dividend problem with a nonlinear regular-singular stochastic control." Insurance: Mathematics and Economics 52, no. 3 (May 2013): 448–56. http://dx.doi.org/10.1016/j.insmatheco.2013.02.010.

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9

De Angelis, Tiziano. "Optimal dividends with partial information and stopping of a degenerate reflecting diffusion." Finance and Stochastics 24, no. 1 (October 18, 2019): 71–123. http://dx.doi.org/10.1007/s00780-019-00407-1.

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Abstract We study the optimal dividend problem for a firm’s manager who has partial information on the profitability of the firm. The problem is formulated as one of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a two-dimensional degenerate diffusion whose first component is singularly controlled. Moreover, the process is absorbed when its first component hits zero. The free boundary problem (FBP) associated to the value function of the control problem is challenging from the analytical point of view due to the interplay of degeneracy and absorption. We find a probabilistic way to show that the value function of the dividend problem is a smooth solution of the FBP and to construct an optimal dividend strategy. Our approach establishes a new link between multidimensional singular stochastic control problems with absorption and problems of optimal stopping with ‘creation’. One key feature of the stopping problem is that creation occurs at a state-dependent rate of the ‘local time’ of an auxiliary two-dimensional reflecting diffusion.

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10

Albrecher, Hansjörg, Pablo Azcue, and Nora Muler. "Optimal dividend strategies for two collaborating insurance companies." Advances in Applied Probability 49, no. 2 (June 2017): 515–48. http://dx.doi.org/10.1017/apr.2017.11.

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Abstract We consider a two-dimensional optimal dividend problem in the context of two insurance companies with compound Poisson surplus processes, who collaborate by paying each other's deficit when possible. We study the stochastic control problem of maximizing the weighted sum of expected discounted dividend payments (among all admissible dividend strategies) until ruin of both companies, by extending results of univariate optimal control theory. In the case that the dividends paid by the two companies are equally weighted, the value function of this problem compares favorably with the one of merging the two companies completely. We identify the optimal value function as the smallest viscosity supersolution of the respective Hamilton–Jacobi–Bellman equation and provide an iterative approach to approximate it numerically. Curve strategies are identified as the natural analogue of barrier strategies in this two-dimensional context. A numerical example is given for which such a curve strategy is indeed optimal among all admissible dividend strategies, and for which this collaboration mechanism also outperforms the suitably weighted optimal dividend strategies of the two stand-alone companies.

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11

Meng, Hui, Ming Zhou, and Tak Kuen Siu. "OPTIMAL DIVIDEND–REINSURANCE WITH TWO TYPES OF PREMIUM PRINCIPLES." Probability in the Engineering and Informational Sciences 30, no. 2 (December 9, 2015): 224–43. http://dx.doi.org/10.1017/s0269964815000352.

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A combined optimal dividend/reinsurance problem with two types of insurance claims, namely the expected premium principle and the variance premium principle, is discussed. Dividend payments are considered with both fixed and proportional transaction costs. The objective of an insurer is to determine an optimal dividend–reinsurance policy so as to maximize the expected total value of discounted dividend payments to shareholders up to ruin time. The problem is formulated as an optimal regular-impulse control problem. Closed-form solutions for the value function and optimal dividend–reinsurance strategy are obtained in some particular cases. Finally, some numerical analysis is given to illustrate the effects of safety loading on optimal reinsurance strategy.

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12

LI, WEIPING. "OPTIMAL DIVIDEND POLICY AND STOCK PRICES." International Journal of Theoretical and Applied Finance 23, no. 04 (June 2020): 2050023. http://dx.doi.org/10.1142/s0219024920500235.

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We model a corporation dividend as an exchange option on stochastic cash flow and capital budge. Then we solve optimal dividend policy problem completely based on the dividend model under the assumption that the cash reservoir of a corporation follows a mean reverting process from empirical evidence and economic arguments. Our optimal dividend controls depend on explicitly with the cash flow and the capital budget of the corporation, and maximizes the HARA utility performance. We specify the unique optimal dividend control for the cash flow and the capital budge. Multiplicity or absence of optimal dividend policies are given. The stock price of the corporation is studied in terms of our stochastic dividend model. We find an explicit relation among the volatility of the stock price, the volatility of the cash flow and the volatility of the capital budget. The ex-dividend stock price is positively proportional to the stochastic cash flow and the probability of the dividend delta with respect to the cash flow, and negatively proportional to the capital budget and the probability of the dividend delta with respect to the capital budget. Hence, our approach provides another passage through which countercyclical volatility of the stock price can arise from the countercyclical cash flow and capital budget directly.

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13

Cheng, Xiang, Zhuo Jin, and Hailiang Yang. "OPTIMAL INSURANCE STRATEGIES: A HYBRID DEEP LEARNING MARKOV CHAIN APPROXIMATION APPROACH." ASTIN Bulletin 50, no. 2 (May 2020): 449–77. http://dx.doi.org/10.1017/asb.2020.9.

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AbstractThis paper studies deep learning approaches to find optimal reinsurance and dividend strategies for insurance companies. Due to the randomness of the financial ruin time to terminate the control processes, a Markov chain approximation-based iterative deep learning algorithm is developed to study this type of infinite-horizon optimal control problems. The optimal controls are approximated as deep neural networks in both cases of regular and singular types of dividend strategies. The framework of Markov chain approximation plays a key role in building the iterative equations and initialization of the algorithm. We implement our method to classic dividend and reinsurance problems and compare the learning results with existing analytical solutions. The feasibility of our method for complicated problems has been demonstrated by applying to an optimal dividend, reinsurance and investment problem under a high-dimensional diffusive model with jumps and regime switching.

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14

Tian, Linlin, Lihua Bai, and Junyi Guo. "Optimal Singular Dividend Problem Under the Sparre Andersen Model." Journal of Optimization Theory and Applications 184, no. 2 (October 31, 2019): 603–26. http://dx.doi.org/10.1007/s10957-019-01600-0.

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15

Chen, Mi, and Kam Chuen Yuen. "Optimal dividend and reinsurance in the presence of two reinsurers." Journal of Applied Probability 53, no. 2 (June 2016): 554–71. http://dx.doi.org/10.1017/jpr.2016.20.

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Abstract In this paper the optimal dividend (subject to transaction costs) and reinsurance (with two reinsurers) problem is studied in the limit diffusion setting. It is assumed that transaction costs and taxes are required when dividends occur, and that the premiums charged by two reinsurers are calculated according to the exponential premium principle with different parameters, which makes the stochastic control problem nonlinear. The objective of the insurer is to determine the optimal reinsurance and dividend policy so as to maximize the expected discounted dividends until ruin. The problem is formulated as a mixed classical-impulse stochastic control problem. Explicit expressions for the value function and the corresponding optimal strategy are obtained. Finally, a numerical example is presented to illustrate the impact of the parameters associated with the two reinsurers' premium principle on the optimal reinsurance strategy.

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16

Zhu, Jinxia. "DIVIDEND OPTIMIZATION FOR A REGIME-SWITCHING DIFFUSION MODEL WITH RESTRICTED DIVIDEND RATES." ASTIN Bulletin 44, no. 2 (February 13, 2014): 459–94. http://dx.doi.org/10.1017/asb.2014.2.

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AbstractWe consider the optimal dividend control problem to find an optimal strategy under the constraint that dividend rates is restricted such that the expected total discounted dividends are maximized for an insurance company. The evolution of the reserve is modeled by a diffusion process with drift and volatility coefficients modulated by an observable Markov chain. We consider the regime-switching threshold strategy which pays out dividends at the maximal possible rate when the current reserve is above some critical level dependent on the regime of the Markov chain at the time, and pays nothing when the reserve is below that level. We give sufficient conditions under which such type of strategy is optimal for the regime-switching model.

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17

YAMAZAKI, AKIRA. "EQUILIBRIUM EQUITY PRICE WITH OPTIMAL DIVIDEND POLICY." International Journal of Theoretical and Applied Finance 20, no. 02 (March 2017): 1750012. http://dx.doi.org/10.1142/s0219024917500121.

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This paper proposes an equilibrium model for evaluating equity with optimal dividend policy in a jump-diffusion market. In this model, a representative investor having power utility over an aggregate consumption process evaluates the equity as the expected value of the discounted dividends with his stochastic discount factor, while a firm paying the dividends from its own cash reserves manages to maximize the equity price. This situation is formulated as a singular stochastic control problem of jump-diffusion processes. We solve this problem and give the equilibrium equity price and the optimal dividend policy. Numerical examples show that the aggregate consumption process and the investor’s risk aversion have a significant impact on the equity price and the dividend policy. This model provides a structural explanation of equity risk premiums that is consistent with the standard theory of asset pricing.

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18

Christensen, Sören, and Kristoffer Lindensjö. "Moment-constrained optimal dividends: precommitment and consistent planning." Advances in Applied Probability 54, no. 2 (June 2022): 404–32. http://dx.doi.org/10.1017/apr.2021.38.

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AbstractA moment constraint that limits the number of dividends in an optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz’s consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame-perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion.

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19

Christensen, Sören, and Kristoffer Lindensjö. "Moment-constrained optimal dividends: precommitment and consistent planning." Advances in Applied Probability 54, no. 2 (June 2022): 404–32. http://dx.doi.org/10.1017/apr.2021.38.

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AbstractA moment constraint that limits the number of dividends in an optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz’s consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame-perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion.

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20

Keppo, Jussi, A. Max Reppen, and H. Mete Soner. "Discrete Dividend Payments in Continuous Time." Mathematics of Operations Research 46, no. 3 (August 2021): 895–911. http://dx.doi.org/10.1287/moor.2020.1081.

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We propose a model in which dividend payments occur at regular, deterministic intervals in an otherwise continuous model. This contrasts traditional models where either the payment of continuous dividends is controlled or the dynamics are given by discrete time processes. Moreover, between two dividend payments, the structure allows for other types of control; we consider the possibility of equity issuance at any point in time. The value is characterized as the fixed point of an optimal control problem with periodic initial and terminal conditions. We prove the regularity and uniqueness of the corresponding dynamic programming equation and the convergence of an efficient numerical algorithm that we use to study the problem. The model enables us to find the loss caused by infrequent dividend payments. We show that under realistic parameter values, this loss varies from around 1%–24% depending on the state of the system and that using the optimal policy from the continuous problem further increases the loss.

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21

Junca, Mauricio, Harold A. Moreno-Franco, José Luis Pérez, and Kazutoshi Yamazaki. "Optimality of refraction strategies for a constrained dividend problem." Advances in Applied Probability 51, no. 03 (September 2019): 633–66. http://dx.doi.org/10.1017/apr.2019.32.

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AbstractWe consider de Finetti’s problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.

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22

Ferrari, Giorgio, and Patrick Schuhmann. "An Optimal Dividend Problem with Capital Injections over a Finite Horizon." SIAM Journal on Control and Optimization 57, no. 4 (January 2019): 2686–719. http://dx.doi.org/10.1137/18m1184588.

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23

Zhu, Dan, and Chuancun Yin. "Stochastic Optimal Control of Investment and Dividend Payment Model under Debt Control with Time-Inconsistency." Mathematical Problems in Engineering 2018 (July 9, 2018): 1–8. http://dx.doi.org/10.1155/2018/7928953.

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This paper considers the optimal debt ratio, investment, and dividend payment policies for insurers with time-inconsistency. The surplus process of an insurance company is determined by the change of asset value and the change of liabilities. The asset can be invested in financial market which contains a risky asset and a risk-free asset, and when the insurer incurs a liability, he/she earns some premium. The objective is to maximize the expected nonconstant discounted utility of dividend payment until a determinate time. This is a time-inconsistent control problem. We obtain the modified HJB equation and the closed-form expressions for the optimal debt ratio, investment, and dividend payment policies under logarithmic utility.

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24

Eisenberg, Julia, Stefan Kremsner, and Alexander Steinicke. "Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate." Mathematics 9, no. 18 (September 14, 2021): 2257. http://dx.doi.org/10.3390/math9182257.

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We investigate a dividend maximization problem under stochastic interest rates with Ornstein-Uhlenbeck dynamics. This setup also takes negative rates into account. First a deterministic time is considered, where an explicit separating curve α(t) can be found to determine the optimal strategy at time t. In a second setting, we introduce a strategy-independent stopping time. The properties and behavior of these optimal control problems in both settings are analyzed in an analytical HJB-driven approach, and we also use backward stochastic differential equations.

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25

Guan, Chonghu, Fahuai Yi, and Xiaoshan Chen. "A fully nonlinear free boundary problem arising from optimal dividend and risk control model." Mathematical Control & Related Fields 9, no. 3 (2019): 425–52. http://dx.doi.org/10.3934/mcrf.2019020.

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26

Yao, Dingjun, Hailiang Yang, and Rongming Wang. "OPTIMAL DIVIDEND AND REINSURANCE STRATEGIES WITH FINANCING AND LIQUIDATION VALUE." ASTIN Bulletin 46, no. 2 (January 25, 2016): 365–99. http://dx.doi.org/10.1017/10.1017/asb.2015.28.

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AbstractThis study investigates a combined optimal financing, reinsurance and dividend distribution problem for a big insurance portfolio. A manager can control the surplus by buying proportional reinsurance, paying dividends and raising money dynamically. The transaction costs and liquidation values at bankruptcy are included in the risk model. Under the objective of maximising the insurance company's value, we identify the insurer's joint optimal strategies using stochastic control methods. The results reveal that managers should consider financing if and only if the terminal value and the transaction costs are not too high, less reinsurance is bought when the surplus increases or dividends are always distributed using the barrier strategy.

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27

Yang, Xixi, Jiyang Tan, Hanjun Zhang, and Ziqiang Li. "An Optimal Control Problem in a Risk Model with Stochastic Premiums and Periodic Dividend Payments." Asia-Pacific Journal of Operational Research 34, no. 03 (June 2017): 1740013. http://dx.doi.org/10.1142/s0217595917400139.

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In this paper, a discrete-time risk model is considered. We assume that the premium received in each time interval is a positive real-valued random variable, and the sequence of premiums is a Markov chain. In any time interval the probability of a claim occurrence is related to the premium received in the corresponding period. We discuss control strategies for dividends paid periodically to the shareholders under two cases: absence and presence of ceiling restriction for dividend rates. We provide algorithms and some properties for the optimal control strategies by transforming the value function.

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28

Strietzel, Philipp Lukas, and Henriette Elisabeth Heinrich. "Optimal Dividends for a Two-Dimensional Risk Model with Simultaneous Ruin of Both Branches." Risks 10, no. 6 (June 2, 2022): 116. http://dx.doi.org/10.3390/risks10060116.

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We consider the optimal dividend problem in the so-called degenerate bivariate risk model under the assumption that the surplus of one branch may become negative. More specific, we solve the stochastic control problem of maximizing discounted dividends until simultaneous ruin of both branches of an insurance company by showing that the optimal value function satisfies a certain Hamilton–Jacobi–Bellman (HJB) equation. Further, we prove that the optimal value function is the smallest viscosity solution of said HJB equation, satisfying certain growth conditions. Under some additional assumptions, we show that the optimal strategy lies within a certain subclass of all admissible strategies and reduce the two-dimensional control problem to a one-dimensional one. The results are illustrated by a numerical example and Monte Carlo simulated value functions.

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29

Guan, Chonghu, and Fahuai Yi. "A Free Boundary Problem Arising from a Stochastic Optimal Control Model with Bounded Dividend Rate." Stochastic Analysis and Applications 32, no. 5 (September 2, 2014): 742–60. http://dx.doi.org/10.1080/07362994.2014.922778.

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30

Yao, Dingjun, Hailiang Yang, and Rongming Wang. "Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle." Economic Modelling 37 (February 2014): 53–64. http://dx.doi.org/10.1016/j.econmod.2013.10.026.

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31

Chen, Peimin, and Bo Li. "Classical and Impulse Stochastic Control on the Optimization of Dividends with Residual Capital at Bankruptcy." Discrete Dynamics in Nature and Society 2017 (2017): 1–14. http://dx.doi.org/10.1155/2017/2693568.

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In this paper, we consider the optimization problem of dividends for the terminal bankruptcy model, in which some money would be returned to shareholders at the state of terminal bankruptcy, while accounting for the tax rate and transaction cost for dividend payout. Maximization of both expected total discounted dividends before bankruptcy and expected discounted returned money at the state of terminal bankruptcy becomes a mixed classical-impulse stochastic control problem. In order to solve this problem, we reduce it to quasi-variational inequalities with a nonzero boundary condition. We explicitly construct and verify solutions of these inequalities and present the value function together with the optimal policy.

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32

Chen, Xiaoshan, Chonghu Guan, and Fahuai Yi. "A Free Boundary Problem of Liquidity Management for Optimal Dividend and Insurance in Finite Horizon." SIAM Journal on Control and Optimization 59, no. 4 (January 2021): 2524–45. http://dx.doi.org/10.1137/20m1329949.

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33

Guo, Xin. "Some risk management problems for firms with internal competition and debt." Journal of Applied Probability 39, no. 1 (March 2002): 55–69. http://dx.doi.org/10.1239/jap/1019737987.

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Consider an optimization problem for a company with the following parameters: a constant liability payment rate (δ), an average return (μ) and a risk (σ) proportional to the size of the business unit, and an internal competition factor (α) between different units. The goal is to maximize the expected present value of the total dividend distributions, via controls (Ut, Zt), where Ut is the size of the business unit and Zt is the total dividend payoff up to time t. We formulate this as a stochastic control problem for a diffusion process Xt and derive an explicit solution by solving the corresponding Hamilton-Jacobi-Bellman equation. The resulting optimal control policy involves a mixture of a nonlinear control for Ut and a singular control for Zt. The optimal strategies are different for the cases δ < 0 and δ = 0. When δ > 0, it is optimal to play bold: the initial optimal investment size should be proportional to the debt rate δ. Under this optimal rule, however, the probability of bankruptcy in finite time is 1. When δ = 0, i.e. when the company is free of debt, the probability of going broke in finite time reduces to 0. Moreover, when δ = 0, the value function is singular at X0 = 0. Our analytical result shows considerable consistency with daily business practices. For instance, it shows that ‘too many people is counter-productive’. In fact, the maximal optimal size of the business unit should be inversely proportional to α. This eliminates the redundant and simplistic technical assumption of a known uniform upper bound on the size of the firm.

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34

Guo, Xin. "Some risk management problems for firms with internal competition and debt." Journal of Applied Probability 39, no. 01 (March 2002): 55–69. http://dx.doi.org/10.1017/s0021900200021501.

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Consider an optimization problem for a company with the following parameters: a constant liability payment rate (δ), an average return (μ) and a risk (σ) proportional to the size of the business unit, and an internal competition factor (α) between different units. The goal is to maximize the expected present value of the total dividend distributions, via controls (U t , Z t ), where U t is the size of the business unit and Z t is the total dividend payoff up to time t. We formulate this as a stochastic control problem for a diffusion process X t and derive an explicit solution by solving the corresponding Hamilton-Jacobi-Bellman equation. The resulting optimal control policy involves a mixture of a nonlinear control for U t and a singular control for Z t . The optimal strategies are different for the cases δ < 0 and δ = 0. When δ > 0, it is optimal to play bold: the initial optimal investment size should be proportional to the debt rate δ. Under this optimal rule, however, the probability of bankruptcy in finite time is 1. When δ = 0, i.e. when the company is free of debt, the probability of going broke in finite time reduces to 0. Moreover, when δ = 0, the value function is singular at X 0 = 0. Our analytical result shows considerable consistency with daily business practices. For instance, it shows that ‘too many people is counter-productive’. In fact, the maximal optimal size of the business unit should be inversely proportional to α. This eliminates the redundant and simplistic technical assumption of a known uniform upper bound on the size of the firm.

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35

Wen, Yuzhen, and Chuancun Yin. "Optimal Expected Utility of Dividend Payments with Proportional Reinsurance under VaR Constraints and Stochastic Interest Rate." Journal of Function Spaces 2020 (August 11, 2020): 1–13. http://dx.doi.org/10.1155/2020/4051969.

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In this paper, we consider the problem of maximizing the expected discounted utility of dividend payments for an insurance company taking into account the time value of ruin. We assume the preference of the insurer is of the CRRA form. The discounting factor is modeled as a geometric Brownian motion. We introduce the VaR control levels for the insurer to control its loss in reinsurance strategies. By solving the corresponding Hamilton-Jacobi-Bellman equation, we obtain the value function and the corresponding optimal strategy. Finally, we provide some numerical examples to illustrate the results and analyze the VaR control levels on the optimal strategy.

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36

Kyprianou, A. E., and Z. Palmowski. "Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process." Journal of Applied Probability 44, no. 2 (June 2007): 428–43. http://dx.doi.org/10.1239/jap/1183667412.

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We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

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Kyprianou, A. E., and Z. Palmowski. "Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process." Journal of Applied Probability 44, no. 02 (June 2007): 428–43. http://dx.doi.org/10.1017/s0021900200117930.

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Abstract:

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

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38

Kyprianou, A. E., and Z. Palmowski. "Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process." Journal of Applied Probability 44, no. 02 (June 2007): 428–43. http://dx.doi.org/10.1017/s0021900200003077.

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Abstract:

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

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39

Yan, Qingyou, Le Yang, Tomas Baležentis, Dalia Streimikiene, and Chao Qin. "Optimal Dividend and Capital Injection Problem with Transaction Cost and Salvage Value: The Case of Excess-of-Loss Reinsurance Based on the Symmetry of Risk Information." Symmetry 10, no. 7 (July 12, 2018): 276. http://dx.doi.org/10.3390/sym10070276.

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This paper considers the optimal dividend and capital injection problem for an insurance company, which controls the risk exposure by both the excess-of-loss reinsurance and capital injection based on the symmetry of risk information. Besides the proportional transaction cost, we also incorporate the fixed transaction cost incurred by capital injection and the salvage value of a company at the ruin time in order to make the surplus process more realistic. The main goal is to maximize the expected sum of the discounted salvage value and the discounted cumulative dividends except for the discounted cost of capital injection until the ruin time. By considering whether there is capital injection in the surplus process, we construct two instances of suboptimal models and then solve for the corresponding solution in each model. Lastly, we consider the optimal control strategy for the general model without any restriction on the capital injection or the surplus process.

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40

Fornasier, Massimo, Benedetto Piccoli, and Francesco Rossi. "Mean-field sparse optimal control." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, no. 2028 (November 13, 2014): 20130400. http://dx.doi.org/10.1098/rsta.2013.0400.

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We introduce the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints, modelling parsimonious interventions on the dynamics of a moving population divided into leaders and followers, to an infinite dimensional optimal control problem with a constraint given by a system of ODE for the leaders coupled with a PDE of Vlasov-type, governing the dynamics of the probability distribution of the followers. In the classical mean-field theory, one studies the behaviour of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect. In this paper, we address instead the situation where the leaders are actually influenced also by an external policy maker , and we propagate its effect for the number N of followers going to infinity. The technical derivation of the sparse mean-field optimal control is realized by the simultaneous development of the mean-field limit of the equations governing the followers dynamics together with the Γ -limit of the finite dimensional sparse optimal control problems.

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41

Hipp, Christian. "Company Value with Ruin Constraint in Lundberg Models." Risks 6, no. 3 (July 20, 2018): 73. http://dx.doi.org/10.3390/risks6030073.

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In this note we study the problem of company values with a ruin constraint in classical continuous time Lundberg models. For this, we adapt the methods and results for discrete de Finetti models to time and state continuous Lundberg models. The policy improvement method works also in continuous models, but it is slow and needs discretization. Better results can be obtained faster using the barrier method for discrete models which can be adjusted for Lundberg models. In this method, dividend strategies are considered which are based on barrier sequences. In our continuous state model, optimal barriers can be computed with the Lagrange method leading to a backward recursion scheme. The resulting dividend strategies will not always be optimal: in the case without ruin constraint, there are examples in which band strategies are superior. We also develop equations for optimal control of dynamic reinsurance to maximize the company value under a ruin constraint. These identify the optimal reinsurance strategy in no action regions and allow for an interactive computation of the value function. We apply the methods in a numerical example with exponential claims.

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Xu, Lin, Hao Wang, and Dingjun Yao. "Optimal Investment and Consumption for an Insurer with High-Watermark Performance Fee." Mathematical Problems in Engineering 2015 (2015): 1–14. http://dx.doi.org/10.1155/2015/413072.

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The optimal investment and consumption problem is investigated for an insurance company, which is subject to the payment of high-watermark fee from profit. The objective of insurance company is to maximize the expected cumulated discount utility up to ruin time. The consumption behavior considered in this paper can be viewed as dividend payment of the insurance company. It turns out that the value function of the proposed problem is the viscosity solution to the associated HJB equation. The regularity of the viscosity is discussed and some asymptotic results are provided. With the help of the smooth properties of viscosity solutions, we complete the verification theorem of the optimal control policies and the potential applications of the main result are discussed.

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Gardashova, Latafat A. "Application of DEO Method to Solving Fuzzy Multiobjective Optimal Control Problem." Applied Computational Intelligence and Soft Computing 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/971894.

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In the present paper a problem of optimal control for a single-product dynamical macroeconomic model is considered. In this model gross domestic product is divided into productive consumption, gross investment, and nonproductive consumption. The model is described by a fuzzy differential equation (FDE) to take into account imprecision inherent in the dynamics that may be naturally conditioned by influence of various external factors, unforeseen contingencies of future, and so forth. The considered problems are characterized by four criteria and by several important aspects. On one hand, the problem is complicated by the presence of fuzzy uncertainty as a result of a natural imprecision inherent in information about dynamics of real-world systems. On the other hand, the number of the criteria is not small and most of them are integral criteria. Due to the above mentioned aspects, solving the considered problem by using convolution of criteria into one criterion would lead to loss of information and also would be counterintuitive and complex. We applied DEO (differential evolution optimization) method to solve the considered problem.

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Zhang, Xin, Jie Xiong, and Shuaiqi Zhang. "Optimal reinsurance-investment and dividends problem with fixed transaction costs." Journal of Industrial & Management Optimization 13, no. 5 (2017): 0. http://dx.doi.org/10.3934/jimo.2020008.

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Yin, Chuancun, and Kam Chuen Yuen. "Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs." Journal of Industrial & Management Optimization 11, no. 4 (2015): 1247–62. http://dx.doi.org/10.3934/jimo.2015.11.1247.

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Ilmayasinta, Nur, and Heri Purnawan. "Optimal Control in a Mathematical Model of Smoking." Journal of Mathematical and Fundamental Sciences 53, no. 3 (December 3, 2021): 380–94. http://dx.doi.org/10.5614/j.math.fund.sci.2021.53.3.4.

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This paper presents a dynamic model of smoking with optimal control. The mathematical model is divided into 5 sub-classes, namely, non-smokers, occasional smokers, active smokers, individuals who have temporarily stopped smoking, and individuals who have stopped smoking permanently. Four optimal controls, i.e., anti-smoking education campaign, anti-smoking gum, anti-nicotine drug, and government prohibition of smoking in public spaces are considered in the model. The existence of the controls is also presented. The Pontryagin maximum principle (PMP) was used to solve the optimal control problem. The fourth-order Runge-Kutta was employed to gain the numerical solutions.

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Kremsner, Stefan, Alexander Steinicke, and Michaela Szölgyenyi. "A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics." Risks 8, no. 4 (December 9, 2020): 136. http://dx.doi.org/10.3390/risks8040136.

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In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks.

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Ko, Dongnam, and Enrique Zuazua. "Model predictive control with random batch methods for a guiding problem." Mathematical Models and Methods in Applied Sciences 31, no. 08 (July 2021): 1569–92. http://dx.doi.org/10.1142/s0218202521500329.

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We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space. The numerical simulation of such models quickly becomes unfeasible for a large number of interacting agents, as the number of interactions grows [Formula: see text] for [Formula: see text] agents. For reducing the computational cost to [Formula: see text], we use the Random Batch Method (RBM), which provides a computationally feasible approximation of the dynamics. First, the considered time interval is divided into a number of subintervals. In each subinterval, the RBM randomly divides the set of particles into small subsets (batches), considering only the interactions inside each batch. Due to the averaging effect, the RBM approximation converges to the exact dynamics in the [Formula: see text]-expectation norm as the length of subintervals goes to zero. For this approximated dynamics, the corresponding optimal control can be computed efficiently using a classical gradient descent. The resulting control is not optimal for the original system, but for a reduced RBM model. We therefore adopt a Model Predictive Control (MPC) strategy to handle the error in the dynamics. This leads to a semi-feedback control strategy, where the control is applied only for a short time interval to the original system, and then compute the optimal control for the next time interval with the state of the (controlled) original dynamics. Through numerical experiments we show that the combination of RBM and MPC leads to a significant reduction of the computational cost, preserving the capacity of controlling the overall dynamics.

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Buckley, I. R. C., and R. Korn. "Optimal Index Tracking Under Transaction Costs and Impulse Control." International Journal of Theoretical and Applied Finance 01, no. 03 (July 1998): 315–30. http://dx.doi.org/10.1142/s0219024998000187.

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We apply impulse control techniques to a cash management problem within a mean-variance framework. We consider the strategy of an investor who is trying to minimise both fixed and proportional transaction costs, whilst minimising the tracking error with respect to an index portfolio. The cash weight is constantly fluctuating due to the stochastic inflow and outflow of dividends and liabilities. We show the existence of an optimal strategy and compute it numerically.

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Rapaport, Alain, Terence Bayen, Matthieu Sebbah, Andres Donoso-Bravo, and Alfredo Torrico. "Dynamical modeling and optimal control of landfills." Mathematical Models and Methods in Applied Sciences 26, no. 05 (February 25, 2016): 901–29. http://dx.doi.org/10.1142/s0218202516500214.

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We propose a simple model of landfill and study a minimal time control problem where the re-circulation leachate is the manipulated variable. We propose a scheme to construct the optimal strategy by dividing the state space into three subsets [Formula: see text], [Formula: see text] and the complementary. On [Formula: see text] and [Formula: see text], the optimal control is constant until reaching target, while it can exhibit a singular arc outside these two subsets. Moreover, the singular arc could have a barrier. In this case, we prove the existence of a switching curve that passes through a point of prior saturation under the assumption that the set [Formula: see text] intersects the singular arc. Numerical computations allow then to determine the switching curve and depict the optimal synthesis.

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