scholarly journals Optimal Dividend Payment in De Finetti Models: Survey and New Results and Strategies

Risks ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 96
Author(s):  
Christian Hipp
Keyword(s):  
Iteration Scheme ◽  
Dividend Payment ◽  
Monotone Iteration ◽  
Dividend Payments ◽  
Optimal Dividend ◽  
Hamilton Jacobi Bellman Equation ◽  
De Finetti ◽  
Hamilton Jacobi Bellman ◽  
Monotone Iteration Scheme ◽  
The Value Function

We consider optimal dividend payment under the constraint that the with-dividend ruin probability does not exceed a given value α. This is done in most simple discrete De Finetti models. We characterize the value function V(s,α) for initial surplus s of this problem, characterize the corresponding optimal dividend strategies, and present an algorithm for its computation. In an earlier solution to this problem, a Hamilton-Jacobi-Bellman equation for V(s,α) can be found which leads to its representation as the limit of a monotone iteration scheme. However, this scheme is too complex for numerical computations. Here, we introduce the class of two-barrier dividend strategies with the following property: when dividends are paid above a barrier B, i.e., a dividend of size 1 is paid when reaching B+1 from B, then we repeat this dividend payment until reaching a limit L for some 0≤L≤B. For these strategies we obtain explicit formulas for ruin probabilities and present values of dividend payments, as well as simplifications of the above iteration scheme. The results of numerical experiments show that the values V(s,α) obtained in earlier work can be improved, they are suboptimal.

scholarly journals Optimal dividend strategies for two collaborating insurance companies

2017 ◽  
Vol 49 (2) ◽  
pp. 515-548 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Pablo Azcue ◽  
Nora Muler
Keyword(s):  
Value Function ◽  
Insurance Companies ◽  
Two Dimensional ◽  
Dividend Payments ◽  
Optimal Dividend ◽  
Optimal Value ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman ◽  
The One ◽  
The Value Function

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.

scholarly journals Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs

2007 ◽  
Vol 39 (3) ◽  
pp. 669-689 ◽  
Author(s):  
Jostein Paulsen
Keyword(s):  
Optimal Policy ◽  
Value Function ◽  
Diffusion Processes ◽  
Fixed Cost ◽  
Model Parameters ◽  
Dividend Payment ◽  
Optimal Dividends ◽  
Dividend Payments ◽  
Optimal Dividend ◽  
The Value Function

The problem of optimal dividends paid until absorbtion at zero is considered for a rather general diffusion model. With each dividend payment there is a proportional cost and a fixed cost. It is shown that there can be essentially three different solutions depending on the model parameters and the costs. (i) Whenever assets reach a barrier y*, they are reduced to y* - δ* through a dividend payment, and the process continues. (ii) Whenever assets reach a barrier y*, everything is paid out as dividends and the process terminates. (iii) There is no optimal policy, but the value function is approximated by policies of one of the two above forms for increasing barriers. A method to numerically find the optimal policy (if it exists) is presented and numerical examples are given.

scholarly journals Optimal Expected Utility of Dividend Payments with Proportional Reinsurance under VaR Constraints and Stochastic Interest Rate

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Yuzhen Wen ◽  
Chuancun Yin
Keyword(s):  
Optimal Strategy ◽  
Bellman Equation ◽  
Insurance Company ◽  
Time Value ◽  
Dividend Payments ◽  
Discounted Utility ◽  
Stochastic Interest ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman ◽  
The Value Function

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.

scholarly journals Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs

2007 ◽  
Vol 39 (03) ◽  
pp. 669-689 ◽  
Author(s):  
Jostein Paulsen
Keyword(s):  
Optimal Policy ◽  
Value Function ◽  
Diffusion Processes ◽  
Fixed Cost ◽  
Model Parameters ◽  
Dividend Payment ◽  
Optimal Dividends ◽  
Dividend Payments ◽  
Optimal Dividend ◽  
The Value Function

The problem of optimal dividends paid until absorbtion at zero is considered for a rather general diffusion model. With each dividend payment there is a proportional cost and a fixed cost. It is shown that there can be essentially three different solutions depending on the model parameters and the costs. (i) Whenever assets reach a barrier y *, they are reduced to y * - δ* through a dividend payment, and the process continues. (ii) Whenever assets reach a barrier y *, everything is paid out as dividends and the process terminates. (iii) There is no optimal policy, but the value function is approximated by policies of one of the two above forms for increasing barriers. A method to numerically find the optimal policy (if it exists) is presented and numerical examples are given.

Solving the Hamilton-Jacobi-Bellman equation of stochastic control by a semigroup perturbation method

1984 ◽  
Vol 16 (1) ◽  
pp. 16-16
Author(s):  
Domokos Vermes
Keyword(s):  
Bellman Equation ◽  
Value Function ◽  
Sufficient Optimality Condition ◽  
Hamilton Jacobi Bellman Equation ◽  
Deterministic Processes ◽  
Random Jumps ◽  
Hamilton Jacobi Bellman ◽  
Necessary And Sufficient ◽  
The Value Function ◽  
Deterministic Trajectories

We consider the optimal control of deterministic processes with countably many (non-accumulating) random iumps. A necessary and sufficient optimality condition can be given in the form of a Hamilton-jacobi-Bellman equation which is a functionaldifferential equation with boundary conditions in the case considered. Its solution, the value function, is continuously differentiable along the deterministic trajectories if. the random jumps only are controllable and it can be represented as a supremum of smooth subsolutions in the general case, i.e. when both the deterministic motion and the random jumps are controlled (cf. the survey by M. H. A. Davis (p.14)).

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