We consider a discrete-time version of the popular optimal dividend pay-out problem in risk theory. The novel aspect of our approach is that we allow for a risk averse insurer, i.e., instead of maximising the expected discounted dividends until ruin we maximise the expected utility of discounted dividends until ruin. This task has been proposed as an open problem in H. Gerber and E. Shiu (2004). The model in a continuous-time Brownian motion setting with the exponential utility function has been analysed in P. Grandits, F. Hubalek, W. Schachermayer and M. Zigo (2007). Nevertheless, a complete solution has not been provided. In this work, instead we solve the problem in discrete-time setup for the exponential and the power utility functions and give the structure of optimal history-dependent dividend policies. We make use of certain ideas studied earlier in N. Bäuerle and U. Rieder (2013), where Markov decision processes with general utility functions were treated. Our analysis, however, include new aspects, since the reward functions in this case are not bounded.
In this paper, we consider the mixed ratcheting-periodic dividend strategies for spectrally negative Lévy risk model, in which dividend payments can both be made continuously without falling and discretely at the jump times of an independent Poisson process. The expected net present value(NPV) of dividends paid up to ruin and the Laplace transform of the ruin time are obtained by using Lévy fluctuation theory. All the results are expressed in terms of scale functions. Finally, numerical results for Brownian motion with drift are given.
Optimal Dividend Payout Model with Risk Sensitive Preferences
考虑了具有风险敏感股东的离散时间股利支付问题。假设他们都具有风险厌恶系数，并利用指数保费原理构造他们的折现收益。这导致了股息的非预期递归效用。在这样一个框架内，不仅考虑到股息的期望值，而且还考虑到其可变性。我们的方法是由Gerber and Shiu（2004）中的一句话激发的。我们处理的有限和无限的时间范围的问题，并证明，即使在一般设置，最优股利政策是一个波段政策。我们还表明，策略改进算法可以用来获得最优策略和相应的价值函数。接下来，提供了一个明确的例子，其中的障碍类型的最优策略是存在的。最后，我们给出了一些数值研究，并讨论了风险敏感参数对最优股利政策的影响。
We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ($i=1,2,\dots$) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature, that is, if the surplus process just before the $i$th arrival is at level $u$, then for $a>0$ the capital jumps up to the level $(1+a)u+C_i$. The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arriva…
We consider the optimal dividend problem in the so-called degenerate bivariate risk model under the assumption that the surplus of one branch may become negative. More specific, we solve the stochastic control problem of maximizing discounted dividends until simultaneous ruin of both branches of an insurance company by showing that the optimal value function satisfies a certain Hamilton-Jacobi-Bellman (HJB) equation. Further, we prove that the optimal value function is the smallest viscosity solution of said HJB equation, satisfying certain growth conditions. Under some additional assumptions,…
In this manuscript we consider the dual risk model with financial application, where the random gains occur under a renewal process. We particularly work the Erlang(n) case for common distribution of the inter-arrival times, from there it is easy to understand that our method or procedures can be generalised to other cases under the matrix-exponential family case. We work several and different problems involving future dividends and ruin. We also show that our results are valid even if the usual income condition is not satisfied. In most known works under the dual model, the main target under …
We consider the general class of spectrally positive Lévy risk processes, which are appropriate for businesses with continuous expenses and lump sum gains whose timing and sizes are stochastic. Motivated by the fact that dividends cannot be paid at any time in real life, we study $\textit{periodic}$ dividend strategies whereby dividend decisions are made according to a separate arrival process. In this paper, we investigate the impact of fixed transaction costs on the optimal periodic dividend strategy, and show that a periodic $(b_u,b_l)$ strategy is optimal when decision times arrive accordi…
On optimality of the barrier strategy for a general Levy risk process
在一般Levy过程的背景下，研究了保险风险过程的最优分红问题.我们的目标是找到一个策略，最大限度地提高期望的总折扣红利，直到破产的时间。给出了最优策略是障碍型的充分条件。特别地，我们证明了如果Levy密度是完全单调函数，那么最优分红策略是障碍策略。该方法受到Avram等人（2007）[Annals of Applied Probability 17，156-180]、Loeffen（2008）[Annals of Applied Probability 18，1669-1680]和Kyprianou等人（2010）[Journal of Theoretical Probability 23，547-564]，其中在光谱负Levy过程设置下考虑了相同的问题。
We study the problem of optimal dividend payout from a surplus process governed by Brownian motion with drift under the additional constraint of ratcheting, i.e. the dividend rate can never decrease. We solve the resulting two-dimensional optimal control problem, identifying the value function to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. For finitely many admissible dividend rates we prove that threshold strategies are optimal, and for any finite continuum of admissible dividend rates we establish the $\varepsilon$-optimality of curve strategies. T…
We address a long-standing open problem in risk theory, namely the optimal strategy to pay out dividends from an insurance surplus process, if the dividend rate can never be decreased. The optimality criterion here is to maximize the expected value of the aggregate discounted dividend payments up to the time of ruin. In the framework of the classical Cramér-Lundberg risk model, we solve the corresponding two-dimensional optimal control problem and show that the value function is the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We also show that the value fun…
This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a single spectrally negative \lev process with a final payoff at an exponential terminal time…
In this paper, we investigate the problem of optimal strategies of dividend and reinsurance under the Cramér-Lundberg risk model embedded with the thinning-dependence structure which was firstly introduced by Wang and Yuen (2005), subject to the optimality criteria of maximizing the expected accumulated discounted dividends paid until ruin. To enhance the practical relevance of the optimal dividend and reinsurance problem, non-cheap reinsurance is considered and transaction costs and taxes are imposed on dividends, which converts our optimization problem into a mixed classical-impulse control …
Maximising dividends is one classical stability criterion in actuarial risk theory. Motivated by the fact that dividends are paid periodically in real life, $\textit{periodic}$ dividend strategies were recently introduced (Albrecher, Gerber and Shiu, 2011). In this paper, we incorporate fixed transaction costs into the model and study the optimal periodic dividend strategy with fixed transaction costs for spectrally negative Lévy processes. The value function of a periodic $(b_u,b_l)$ strategy is calculated by means of exiting identities and Itô's excusion when the surplus process is of unboun…
In this paper we develop the Gerber-Shiu theory for the classic and dual discrete risk processes in a Markovian (regime switching) environment. In particular, by expressing the Gerber-Shiu function in terms of potential measures of an upward (downward) skip-free discrete-time and discrete-space Markov Additive Process (MAP), we derive closed form expressions for the Gerber-Shiu function in terms of the so-called (discrete) $\boldsymbol{W}_v$ and $\boldsymbol{Z}_v$ scale matrices, which were introduced in arXiv:2008.06697. We show that the discrete scale matrices allow for a unified approach fo…
This paper enhances the pricing of derivatives as well as optimal control problems to a level comprising risk. We employ nested risk measures to quantify risk, investigate the limiting behavior of nested risk measures within the classical models in finance and characterize existence of the risk-averse limit. As a result we demonstrate that the nested limit is unique, irrespective of the initially chosen risk measure. Within the classical models risk aversion gives rise to a stream of risk premiums, comparable to dividend payments. In this context we connect coherent risk measures with the Shar…
Risk-sensitive control has received considerable interest since the seminal work of Howard and Matheson [120] because of its ability to account for fluctuations about the mean, its connection with $H_\infty$ control, and its application to financial mathematics. In this article, we attempt to put together a comprehensive survey on the research done on ergodic risk-sensitive control over the last four decades.
The paper solves constrained Dynkin games with risk-sensitive criteria, where two players are allowed to stop at two independent Poisson random intervention times, via the theory of backward stochastic differential equations. This generalizes the previous work of [Liang and Sun, Dynkin games with Poisson random intervention times, SIAM Journal on Control and Optimization, 2019] from the risk-neutral criteria and common signal times for both players to the risk-sensitive criteria and two heterogenous signal times. Furthermore, the paper establishes a connection of such constrained risk-sensitiv…
We formulate a probabilistic Markov property in discrete time under a dynamic risk framework with minimal assumptions. This is useful for recursive solutions to risk-sensitive versions of dynamic optimisation problems such as optimal prediction, where at each stage the recursion depends on the whole future. The property holds for standard measures of risk used in practice, and is formulated in several equivalent versions including a representation via acceptance sets, a strong version, and a dual representation.
In this paper, we study two optimisation settings for an insurance company, under the constraint that the terminal surplus at a deterministic and finite time $T$ follows a normal distribution with a given mean and a given variance. In both cases, the surplus of the insurance company is assumed to follow a Brownian motion with drift. First, we allow the insurance company to pay dividends and seek to maximise the expected discounted dividend payments or to minimise the ruin probability under the terminal distribution constraint. Here, we find explicit expressions for the optimal strategies in bo…
