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Register a new userOptimal Dividend Payout under Stochastic Discounting
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Adopting a probabilistic approach we determine the optimal dividend payout policy of a firm whose surplus process follows a controlled arithmetic Brownian motion and whose cash-flows are discounted at a stochastic dynamic rate. Dividends can be paid to shareholders at unrestricted rates so that the problem is cast as one of singular stochastic control. The stochastic interest rate is modelled by a Cox-Ingersoll-Ross (CIR) process and the firms objective is to maximize the total expected flow of discounted dividends until a possible insolvency time. We find an optimal dividend payout policy which is such that the surplus process is kept below an endogenously determined stochastic threshold expressed as a decreasing continuous function $r mapsto b(r)$ of the current interest rate value. We also prove that the value function of the singular control problem solves a variational inequality associated to a second-order, non-degenerate elliptic operator, with a gradient constraint.
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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 mixed singular-classical control problem and the corresponding Hamilton-Jacobi-Bellman equation is a variational inequality with fully nonlinear operator and with gradient constraint. The $C^{2,1}$ smoothness of the value function and a comparison principle for its gradient function are established by penalty approximation method. We find that the surplus-time space can be divided into three non-overlapping regions by a risk-magnitude-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 the risks 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.
In this paper, we investigate a sparse optimal control of continuous-time stochastic systems. We adopt the dynamic programming approach and analyze the optimal control via the value function. Due to the non-smoothness of the $L^0$ cost functional, in general, the value function is not differentiable in the domain. Then, we characterize the value function as a viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation. Based on the result, we derive a necessary and sufficient condition for the $L^0$ optimality, which immediately gives the optimal feedback map. Especially for control-affine systems, we consider the relationship with $L^1$ optimal control problem and show an equivalence theorem.
Motivated by recent developments in risk management based on the U.S. bankruptcy code, we revisit De Finetti optimal dividend problems by incorporating the reorganization process and regulators intervention documented in Chapter 11 bankruptcy. The resulting surplus process, bearing financial stress towards the more subtle concept of bankruptcy, corresponds to non-standard spectrally negative Levy processes with endogenous regime switching. In both models without and with fixed transaction costs, some explicit expressions of the expected net present values under a barrier strategy, new to the literature, are established in terms of scale functions. With the help of these expressions, when the tail of the Levy measure is log-convex, the optimal dividend control in each problem is verified to be of the barrier type and the associated optimal barrier can be obtained in analytical form.
We consider an optimal control problem of a property insurance company with proportional reinsurance strategy. The insurance business brings in catastrophe risk, such as earthquake and flood. The catastrophe risk could be partly reduced by reinsurance. The management of the company controls the reinsurance rate and dividend payments process to maximize the expected present value of the dividends before bankruptcy. This is the first time to consider the catastrophe risk in property insurance model, which is more realistic. We establish the solution of the problem by the mixed singular-regular control of jump diffusions. We first derive the optimal retention ratio, the optimal dividend payments level, the optimal return function and the optimal control strategy of the property insurance company, then the impacts of the catastrophe risk and key model parameters on the optimal return function and the optimal control strategy of the company are discussed.
Stochastic games, introduced by Shapley, model adversarial interactions in stochastic environments where two players choose their moves to optimize a discounted-sum of rewards. In the traditional discounted reward semantics, long-term weights are geometrically attenuated based on the delay in their occurrence. We propose a temporally dual notion -- called past-discounting -- where agents have geometrically decaying memory of the rewards encountered during a play of the game. We study past-discounted weight sequences as rewards on stochastic game arenas and examine the corresponding stochastic games with discounted and mean payoff objectives. We dub these games forgetful discounted games and forgetful mean payoff games, respectively. We establish positional determinacy of these games and recover classical complexity results and a Tauberian theorem in the context of past discounted reward sequences.
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