scholarly journals Optimal Control of Capital Injections by Reinsurance with a Constant Rate of Interest

2011 ◽  
Vol 48 (03) ◽  
pp. 733-748
Author(s):  
Julia Eisenberg ◽  
Hanspeter Schmidli
Keyword(s):  
Diffusion Approximation ◽  
Value Function ◽  
Risk Model ◽  
Classical Risk Model ◽  
Surplus Process ◽  
The Individual ◽  
Reinsurance Treaty ◽  
Riskless Asset ◽  
Capital Injections ◽  
The Value Function

We consider a classical risk model and its diffusion approximation, where the individual claims are reinsured by a reinsurance treaty with deductible b ∈ [0, b̃]. Here b = b̃ means ‘no reinsurance’ and b= 0 means ‘full reinsurance’. In addition, the insurer is allowed to invest in a riskless asset with some constant interest rate m > 0. The cedent can choose an adapted reinsurance strategy {b t } t≥0, i.e. the parameter can be changed continuously. If the surplus process becomes negative, the cedent has to inject additional capital. Our aim is to minimise the expected discounted capital injections over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton-Jacobi-Bellman approach in the case of a diffusion approximation. In the case of the classical risk model, we show the existence of a ‘weak’ solution and calculate the value function numerically.

scholarly journals Optimal Control of Capital Injections by Reinsurance with a Constant Rate of Interest

2011 ◽  
Vol 48 (3) ◽  
pp. 733-748 ◽  
Author(s):  
Julia Eisenberg ◽  
Hanspeter Schmidli
Keyword(s):  
Diffusion Approximation ◽  
Value Function ◽  
Risk Model ◽  
Classical Risk Model ◽  
Surplus Process ◽  
The Individual ◽  
Reinsurance Treaty ◽  
Riskless Asset ◽  
Capital Injections ◽  
The Value Function

We consider a classical risk model and its diffusion approximation, where the individual claims are reinsured by a reinsurance treaty with deductible b ∈ [0, b̃]. Here b = b̃ means ‘no reinsurance’ and b= 0 means ‘full reinsurance’. In addition, the insurer is allowed to invest in a riskless asset with some constant interest rate m > 0. The cedent can choose an adapted reinsurance strategy {bt}t≥0, i.e. the parameter can be changed continuously. If the surplus process becomes negative, the cedent has to inject additional capital. Our aim is to minimise the expected discounted capital injections over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton-Jacobi-Bellman approach in the case of a diffusion approximation. In the case of the classical risk model, we show the existence of a ‘weak’ solution and calculate the value function numerically.

scholarly journals Optimal capital injections and dividends with tax in a risk model in discrete time

2020 ◽  
Vol 10 (1) ◽  
pp. 235-259
Author(s):  
Katharina Bata ◽  
Hanspeter Schmidli
Keyword(s):  
Discrete Time ◽  
Optimal Strategy ◽  
Bellman Equation ◽  
Value Function ◽  
Risk Model ◽  
Barrier Type ◽  
Optimal Capital ◽  
Capital Injections ◽  
The Value Function ◽  
Special Case

AbstractWe consider a risk model in discrete time with dividends and capital injections. The goal is to maximise the value of a dividend strategy. We show that the optimal strategy is of barrier type. That is, all capital above a certain threshold is paid as dividend. A second problem adds tax to the dividends but an injection leads to an exemption from tax. We show that the value function fulfils a Bellman equation. As a special case, we consider the case of premia of size one. In this case we show that the optimal strategy is a two barrier strategy. That is, there is a barrier if a next dividend of size one can be paid without tax and a barrier if the next dividend of size one will be taxed. In both models, we illustrate the findings by de Finetti’s example.

scholarly journals ON THE OPTIMAL DIVIDEND PROBLEM FOR A SPECTRALLY POSITIVE LÉVY PROCESS

2014 ◽  
Vol 44 (3) ◽  
pp. 635-651 ◽  
Author(s):  
Chuancun Yin ◽  
Yuzhen Wen ◽  
Yongxia Zhao
Keyword(s):  
Lévy Process ◽  
Risk Model ◽  
Levy Process ◽  
Dual Model ◽  
Threshold Strategy ◽  
Classical Risk Model ◽  
Surplus Process ◽  
Optimal Dividend ◽  
Special Cases ◽  
The Classical Risk Model

AbstractIn this paper we study the optimal dividend problem for a company whose surplus process evolves as a spectrally positive Lévy process before dividends are deducted. This model includes the dual model of the classical risk model and the dual model with diffusion as special cases. We assume that dividends are paid to the shareholders according to an admissible strategy whose dividend rate is bounded by a constant. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. We show that the optimal dividend strategy is formed by a threshold strategy.

scholarly journals Some Optimal Dividends Problems

2004 ◽  
Vol 34 (1) ◽  
pp. 49-74 ◽  
Author(s):  
David C.M. Dickson ◽  
Howard R. Waters
Keyword(s):  
Discrete Time ◽  
Risk Model ◽  
Present Value ◽  
Classical Risk Model ◽  
Optimal Dividends ◽  
Surplus Process ◽  
Dividend Payments ◽  
Constant Dividend Barrier ◽  
De Finetti ◽  
The Classical Risk Model

We consider a situation originally discussed by De Finetti (1957) in which a surplus process is modified by the introduction of a constant dividend barrier. We extend some known results relating to the distribution of the present value of dividend payments until ruin in the classical risk model and show how a discrete time risk model can be used to provide approximations when analytic results are unavailable. We extend the analysis by allowing the process to continue after ruin.

AN OPTIMAL DIVIDEND POLICY WITH DELAYED CAPITAL INJECTIONS

2013 ◽  
Vol 55 (2) ◽  
pp. 129-150 ◽  
Author(s):  
ZHUO JIN ◽  
GEORGE YIN
Keyword(s):  
Value Function ◽  
Closed Form Solution ◽  
Form Solution ◽  
Delay Period ◽  
Programming Approach ◽  
Dividend Payment ◽  
Capital Injection ◽  
Optimal Dividend ◽  
Capital Injections ◽  
The Value Function

AbstractThis work focuses on finding optimal dividend payment and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections with delays. Starting from the classical Cramér–Lundberg process, using the dynamic programming approach, the value function obeys a quasi-variational inequality. With delays in capital injections, the company will be exposed to the risk of financial ruin during the delay period. In addition, the optimal dividend payment and capital injection strategy should balance the expected cost of the possible capital injections and the time value of the delay period. In this paper, the closed-form solution of the value function and the corresponding optimal policies are obtained. Some limiting cases are also discussed. A numerical example is presented to illustrate properties of the solution. Some economic insights are also given.

scholarly journals Optimal Surplus-Dependent Reinsurance Under Regime-Switching in a Brownian Risk Model

Risks ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 73
Author(s):  
Julia Eisenberg ◽  
Lukas Fabrykowski ◽  
Maren Diane Schmeck
Keyword(s):  
Regime Switching ◽  
Risk Model ◽  
Numerical Approach ◽  
Hjb Equation ◽  
Recursive Procedure ◽  
Optimal Reinsurance ◽  
Surplus Process ◽  
Hamilton Jacobi Bellman ◽  
Capital Injections ◽  
A Company

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.

scholarly journals Optimising dividends and consumption under an exponential CIR as a discount factor

2020 ◽  
Vol 92 (2) ◽  
pp. 285-309
Author(s):  
Julia Eisenberg ◽  
Yuliya Mishura
Keyword(s):  
Brownian Motion ◽  
Optimal Strategy ◽  
Value Function ◽  
Insurance Company ◽  
Parameter Choice ◽  
Surplus Process ◽  
Dividend Payments ◽  
Cir Process ◽  
The Value Function ◽  
Special Parameter

AbstractWe consider an economic agent (a household or an insurance company) modelling its surplus process by a deterministic process or by a Brownian motion with drift. The goal is to maximise the expected discounted spending/dividend payments under a discounting factor given by an exponential CIR process. In the deterministic case, we are able to find explicit expressions for the optimal strategy and the value function. For the Brownian motion case, we are able to show that for a special parameter choice the optimal strategy is a constant-barrier strategy.

scholarly journals A note on the optimal dividends paid in a foreign currency

2016 ◽  
Vol 11 (1) ◽  
pp. 67-73 ◽  
Author(s):  
Julia Eisenberg ◽  
Paul Krühner
Keyword(s):  
Brownian Motion ◽  
Optimal Strategy ◽  
Value Function ◽  
Foreign Currency ◽  
Initial Capital ◽  
Brownian Motion With Drift ◽  
Optimal Dividends ◽  
Surplus Process ◽  
Dividend Payments ◽  
The Value Function

AbstractWe consider an insurance entity endowed with an initial capital and a surplus process modelled as a Brownian motion with drift. It is assumed that the company seeks to maximise the cumulated value of expected discounted dividends, which are declared or paid in a foreign currency. The currency fluctuation is modelled as a Lévy process. We consider both cases: restricted and unrestricted dividend payments. It turns out that the value function and the optimal strategy can be calculated explicitly.

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