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Keywords = expected discounted capital injection function

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25 pages, 730 KB  
Article
Optimal Surplus-Dependent Reinsurance under Regime-Switching in a Brownian Risk Model
by Julia Eisenberg, Lukas Fabrykowski and Maren Diane Schmeck
Risks 2021, 9(4), 73; https://doi.org/10.3390/risks9040073 - 13 Apr 2021
Cited by 4 | Viewed by 3421
Abstract
In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company’s surplus process is assumed to follow a Brownian motion with drift, and [...] Read more.
In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company’s surplus process is assumed to follow a Brownian motion with drift, and the reinsurance price is modelled by a continuous-time Markov chain with two states. The presence of regime-switching substantially complicates the optimal reinsurance problem, as the surplus-independent strategies turn out to be suboptimal. We develop a recursive approach that allows to represent a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation and the corresponding reinsurance strategy as the unique limits of the sequence of solutions to ordinary differential equations and their first- and second-order derivatives. Via Ito’s formula, we prove the constructed function to be the value function. Two examples illustrate the recursive procedure along with a numerical approach yielding the direct solution to the HJB equation. Full article
(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)
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21 pages, 2189 KB  
Article
On a Periodic Capital Injection and Barrier Dividend Strategy in the Compound Poisson Risk Model
by Wenguang Yu, Peng Guo, Qi Wang, Guofeng Guan, Qing Yang, Yujuan Huang, Xinliang Yu, Boyi Jin and Chaoran Cui
Mathematics 2020, 8(4), 511; https://doi.org/10.3390/math8040511 - 2 Apr 2020
Cited by 29 | Viewed by 4403
Abstract
In this paper, we assume that the reserve level of an insurance company can only be observed at discrete time points, then a new risk model is proposed by introducing a periodic capital injection strategy and a barrier dividend strategy into the classical [...] Read more.
In this paper, we assume that the reserve level of an insurance company can only be observed at discrete time points, then a new risk model is proposed by introducing a periodic capital injection strategy and a barrier dividend strategy into the classical risk model. We derive the equations and the boundary conditions satisfied by the Gerber-Shiu function, the expected discounted capital injection function and the expected discounted dividend function by assuming that the observation interval and claim amount are exponentially distributed, respectively. Numerical examples are also given to further analyze the influence of relevant parameters on the actuarial function of the risk model. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications)
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