Optimal investment policy and dividend payment strategy in an insurance company

We consider in this paper the optimal dividend problem for an insurance company whose uncontrolled reserve process evolves as a classical Cram\'{e}r--Lundberg process. The firm has the option of investing part of the surplus in a Black--Scholes financial market. The objective is to find a strategy consisting of both investment and dividend payment policies which maximizes the cumulative expected discounted dividend pay-outs until the time of bankruptcy. We show that the optimal value function is the smallest viscosity solution of the associated second-order integro-differential Hamilton--Jacobi--Bellman equation. We study the regularity of the optimal value function. We show that the optimal dividend payment strategy has a band structure. We find a method to construct a candidate solution and obtain a verification result to check optimality. Finally, we give an example where the optimal dividend strategy is not barrier and the optimal value function is not twice continuously differentiable.Comment: Published in at http://dx.doi.org/10.1214/09-AAP643 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal dividend strategies for a catastrophe insurer

In this paper we study the problem of optimally paying out dividends from an insurance portfolio, when the criterion is to maximize the expected discounted dividends over the lifetime of the company and the portfolio contains claims due to natural catastrophes, modelled by a shot-noise Cox claim number process. The optimal value function of the resulting two-dimensional stochastic control problem is shown to be the smallest viscosity supersolution of a corresponding Hamilton-Jacobi-Bellman equation, and we prove that it can be uniformly approximated through a discretization of the space of the free surplus of the portfolio and the current claim intensity level. We implement the resulting numerical scheme to identify optimal dividend strategies for such a natural catastrophe insurer, and it is shown that the nature of the barrier and band strategies known from the classical models with constant Poisson claim intensity carry over in a certain way to this more general situation, leading to action and non-action regions for the dividend payments as a function of the current surplus and intensity level. We also discuss some interpretations in terms of upward potential for shareholders when including a catastrophe sector in the portfolio

Optimal dividend strategies for a catastrophe insurer

In this paper we study the problem of optimally paying out dividends from an insurance portfolio, when the criterion is to maximize the expected discounted dividends over the lifetime of the company and the portfolio contains claims due to natural catastrophes, modelled by a shot-noise Cox claim number process. The optimal value function of the resulting two-dimensional stochastic control problem is shown to be the smallest viscosity supersolution of a corresponding Hamilton-Jacobi-Bellman equation, and we prove that it can be uniformly approximated through a discretization of the space of the free surplus of the portfolio and the current claim intensity level. We implement the resulting numerical scheme to identify optimal dividend strategies for such a natural catastrophe insurer, and it is shown that the nature of the barrier and band strategies known from the classical models with constant Poisson claim intensity carry over in a certain way to this more general situation, leading to action and non-action regions for the dividend payments as a function of the current surplus and intensity level. We also discuss some interpretations in terms of upward potential for shareholders when including a catastrophe sector in the portfolio

Optimal strategies in a production-inventory control model

We consider a production-inventory control model with finite capacity and two different production rates, assuming that the cumulative process of customer demand is given by a compound Poisson process. It is possible at any time to switch over from the different production rates but it is mandatory to switch-off when the inventory process reaches the storage maximum capacity. We consider holding, production, shortage penalty and switching costs. This model was introduced by Doshi, Van Der Duyn Schouten and Talman in 1978. Our aim is to minimize the expected discounted cumulative costs up to infinity over all admissible switching strategies. We show that the optimal cost functions for the different production rates satisfy the corresponding Hamilton-Jacobi-Bellman system of equations in a viscosity sense and prove a verification theorem. The way in which the optimal cost functions solve the different variational inequalities gives the switching regions of the optimal strategy, hence it is stationary in the sense that depends only on the current production rate and inventory level. We define the notion of finite band strategies and derive, using scale functions, the formulas for the different costs of the band strategies with one or two bands. We also show that there are examples where the switching strategy presented by Doshi et al. is not the optimal strategy.Comment: 31 pages, 15 figure

Optimal Ratcheting of Dividends in a Brownian Risk Model

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 ε-optimality of curve strategies. This work is a counterpart of [2], where the ratcheting problem was studied for a compound Poisson surplus process with drift. In the present Brownian setup, calculus of variation techniques allow to obtain a much more explicit analysis and description of the optimal dividend strategies. We also give some numerical illustrations of the optimality results

Optimal Reinsurance to Minimize the Probability of Drawdown under the Mean-Variance Premium Principle: Asymptotic Analysis

In this paper, we consider an optimal reinsurance problem to minimize the probability of drawdown for the scaled Cram´er-Lundberg risk model when the reinsurance premium is computed according to the mean-variance premium principle. We extend the work of Liang et al. [16] to the case of minimizing the probability of drawdown. By using the comparison method and the tool of adjustment coefficients, we show that the minimum probability of drawdown for the scaled classical risk model converges to the minimum probability for its diffusion approximation, and the rate of convergence is of order O(n−1/2). We further show that using the optimal strategy from the diffusion approximation in the scaled classical risk model is O(n−1/2)-optimalEste documento es una versión del artículo publicado en SIAM Journal on Financial Mathematics, 14(1), 279–313Universidad Torcuato Di TellaHebei University of TechnologyDepartment of Mathematics, University of Michiga

Incursionando en el Mercado Pet Care

En actualidad (2022) el mercado de las mascotas, ha tomado relevancia porque cada vez son más las personas que se interesan por los animales y su cuidado; incluso hay ciertos nichos que los han re-nombrando como perrhijos o gathijos, haciendo alusión de que son su mejor compañía en el hogar y sustituyendo el amor que se le tiene a un hijo por el que se le tiene también a las mascotas. Por tanto, se pretende estudiar al mercado Pet Care para analizar la viabilidad de lanzar un producto innovador que permita a las personas consentir a sus mascotas y que además las mantenga felices y relajadas. Pet High es la propuesta de marca para el producto propuesto y resultado de esta investigación, con la finalidad de que los clientes formen un mejor lazo con su mascota