# Optimal Control with Restrictions for a Diffusion Risk Model Under Constant Interest Force

Optimal Control with Restrictions for a Diffusion Risk Model Under Constant Interest Force In this paper, we study optimal dividend problems in a diffusion risk model for two different cases depending on whether reinsurance is incorporated. In either case, the dividend rate is bounded above by a constant, and the company earns investment income at a constant force of interest. Unlike existing approaches in the literature dealing with optimal problems with interest, we allow the force of interest to be greater than the discount factor, and we use a different method to solve the corresponding Hamilton–Jacobi–Bellman (HJB) equation instead of introducing a confluent hypergeometric function. We conclude that the optimal dividend policy is of a threshold type and show that the corresponding dividend barrier is nondecreasing in the dividend rate bound. In cases where there is no reinsurance, we construct an auxiliary reflecting control problem to find the nonzero dividend barrier. If proportional reinsurance is purchased, the optimal reinsurance strategy looks somewhat strange. The optimal retention level of risk first increases monotonically with risk reserve to some possible value (less than \$\$1\$\$ 1 ) and then stays at level \$\$1\$\$ 1 for a while or, if \$\$1\$\$ 1 has been reached, finally, it decreases to 0. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Optimal Control with Restrictions for a Diffusion Risk Model Under Constant Interest Force

, Volume 73 (1) – Feb 1, 2016
22 pages

/lp/springer_journal/optimal-control-with-restrictions-for-a-diffusion-risk-model-under-eeAxOFtTAF
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-015-9295-3
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we study optimal dividend problems in a diffusion risk model for two different cases depending on whether reinsurance is incorporated. In either case, the dividend rate is bounded above by a constant, and the company earns investment income at a constant force of interest. Unlike existing approaches in the literature dealing with optimal problems with interest, we allow the force of interest to be greater than the discount factor, and we use a different method to solve the corresponding Hamilton–Jacobi–Bellman (HJB) equation instead of introducing a confluent hypergeometric function. We conclude that the optimal dividend policy is of a threshold type and show that the corresponding dividend barrier is nondecreasing in the dividend rate bound. In cases where there is no reinsurance, we construct an auxiliary reflecting control problem to find the nonzero dividend barrier. If proportional reinsurance is purchased, the optimal reinsurance strategy looks somewhat strange. The optimal retention level of risk first increases monotonically with risk reserve to some possible value (less than \$\$1\$\$ 1 ) and then stays at level \$\$1\$\$ 1 for a while or, if \$\$1\$\$ 1 has been reached, finally, it decreases to 0.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Feb 1, 2016

### References

• Optimal dividends in the Brownian motion risk model with interest
Fang, Y; Wu, R

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