A fully nonlinear free boundary problem arising from optimal dividend and risk control model

Guan, Chonghu; Yi, Fahuai; Chen, Xiaoshan

doi:10.3934/mcrf.2019020

Cited by 8 publications

(14 citation statements)

References 13 publications

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“…A usual approach to study this kind of problem is to transform it into a variational inequality problem for its gradient. Then the gradient constraint becomes value constraint and the new variational inequality becomes a well-studied obstacle problem; see [8,9,14,15].…”

Section: Hjb Equationmentioning

confidence: 99%

“…We first prove the existence of a solution to Problem (3.13), and then construct a solution to Problem (3.1), or equivalently (3.8), from it. As mentioned earlier, we adopt the standard penalty approximation method; see [9,14,30].…”

Section: Rewrite (B7) Asmentioning

confidence: 99%

“…We consider a controlled diffusion surplus process, which is a good approximation of the classical Cramér-Lundberg process as well-justified by Grandell [12]. A closest model to this paper is considered by Guan, Yi and Chen [14], where the risk control model is relatively simple and the type of reinsurance policy is constrained to proportional ones. It turns out that the reinsurance scheme is restrictive.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic optimal reinsurance and dividend-payout in finite time horizon

Guan,

Xu,

Zhou

2020

Preprint

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This paper studies a dynamic optimal reinsurance and dividend-payout problem for an insurer in a finite time horizon. The goal of the insurer is to maximize its expected cumulative discounted dividend payouts until bankruptcy or maturity which comes earlier. The insurer is allowed to dynamically choose reinsurance contracts over the whole time horizon. This is a singular control problem and the corresponding Hamilton-Jacobi-Bellman equation is a variational inequality with fully nonlinear operator and with gradient constraint. A comparison principle and C 2,1 smoothness for the solution are established by penalty approximation method. We find that the surplus-time space can be divided into three non-overlapping regions by a ceded risk and time dependent reinsurance barrier and a time dependent dividend-payout barrier. The insurer should be exposed to higher risk as surplus increases; exposed to all risk once surplus upward crosses the reinsurance barrier; and pay out all reserves in excess of the dividend-payout barrier. The localities of these regions are explicitly estimated.

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Section: Hjb Equationmentioning

confidence: 99%

Section: Rewrite (B7) Asmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamic optimal reinsurance and dividend-payout in finite time horizon

Guan,

Xu,

Zhou

2020

Preprint

Self Cite

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show abstract

“…We consider a controlled diffusion surplus process, which is a good approximation of the clas-sical Cramér-Lundberg process as well-justified by Grandell [18]. A closest model to this paper is considered by Guan, Yi and Chen [16], where the risk control model is relatively simple and the type of reinsurance policy is constrained to the proportional one. It turns out that the reinsurance scheme is restrictive.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Optimal Reinsurance and Dividend Payout in Finite Time Horizon

Guan

Zhou

2023

Mathematics of OR

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This paper studies a dynamic optimal reinsurance and dividend-payout problem for an insurance company in a finite time horizon. The goal of the company is to maximize the expected cumulative discounted dividend payouts until bankruptcy or maturity, whichever comes earlier. The company is allowed to buy reinsurance contracts dynamically over the whole time horizon to cede its risk exposure with other reinsurance companies. This is a mixed singular–classical stochastic control problem, and the corresponding Hamilton–Jacobi–Bellman equation is a variational inequality with a fully nonlinear operator and subject to a gradient constraint. We obtain the [Formula: see text] smoothness of the value function and a comparison principle for its gradient function by the penalty approximation method so that one can establish an efficient numerical scheme to compute the value function. We find that the surplus-time space can be divided into three nonoverlapping regions by a risk-magnitude and time-dependent reinsurance barrier and a time-dependent dividend-payout barrier. The insurance company should be exposed to a higher risk as its surplus increases, be exposed to the entire risk once its surplus upward crosses the reinsurance barrier, and pay out all its reserves exceeding the dividend-payout barrier. The estimated localities of these regions are also provided.

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“…Some previous arguments in Bahman et al (2019) based on constant free boundary points cannot be applied in our framework with time dependence. Instead, we resort to some PDE techniques proposed in , , Dai and Yi (2009), Guan et al (2019) and modify some arguments to work for our particular stochastic control problem over a finite time horizon. Thanks to the homogeneity of the power utility, we are able to first reduce the dimension of the problem and focus on a one-dimensional nonlinear parabolic problem.…”

Section: Introductionmentioning

confidence: 99%

Optimal consumption under a drawdown constraint over a finite horizon

Chen¹,

Li²,

Yi³

2022

Preprint

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This paper studies a finite horizon utility maximization problem on excessive consumption under a drawdown constraint up to the bankruptcy time. Our control problem is an extension of the one considered in Bahman et al. (2019) to the model with a finite horizon and also an extension of the one considered in Jeon and Oh (2022) to the model with zero interest rate. Contrary to Bahman et al. ( 2019), we encounter a parabolic nonlinear HJB variational inequality with a gradient constraint, in which some time-dependent free boundaries complicate the analysis significantly. Meanwhile, our methodology is built on technical PDE arguments, which differs substantially from the martingale approach in Jeon and Oh (2022). Using dual transform and considering some auxiliary parabolic variational inequalities with both gradient and function constraints, we establish the existence and uniqueness of the classical solution to the HJB variational inequality and characterize all associated free boundaries in analytical form. Consequently, the piecewise optimal feedback controls and some time-dependent thresholds of the wealth variable for different control expressions can be obtained.

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A fully nonlinear free boundary problem arising from optimal dividend and risk control model

Cited by 8 publications

References 13 publications

Dynamic optimal reinsurance and dividend-payout in finite time horizon

Dynamic optimal reinsurance and dividend-payout in finite time horizon

Dynamic Optimal Reinsurance and Dividend Payout in Finite Time Horizon

Optimal consumption under a drawdown constraint over a finite horizon

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