Optimal investment for insurers

In this paper, we consider a Sparre Andersen risk model perturbed by a spectrally negative Lévy process (SNLP). Assuming that the interclaim times follow a Coxian distribution, we show that the Laplace transforms and defective renewal equations for the Gerber–Shiu functions can be obtained by employing the roots of a generalized Lundberg equation. When the SNLP is a combination of a Brownian motion and a compound Poisson process with exponential jumps, explicit expressions and asymptotic formulas for the Gerber–Shiu functions are obtained for exponential claim size distribution and heavy-tailed claim size distribution, respectively.

This paper considers the optimal investment strategy for an insurer under the criterion of mean-variance. The risk process is a compound Poisson process and the insurer can invest in a risk-free asset and multiple risky assets. We obtain the optimal investment policy using the stochastic liner-quadrant (LQ) control theory. Then the efficient strategy (optimal investment strategy) and efficient frontier are derived explicitly by a verification theorem with the classical solution of Hamilton-Jacobi-Bellman (HJB) equation.

This work considers optimal investment and reinsurance strategies for an insurer with stochastic economic factor. In our mathematical model, a risk-free asset and a risky asset are assumed to rely on a stochastic economic factor which is described by a diffusion process. We generalize the claim process to a compound Poisson process with the stochastic economic factor. Using expected utility maximization, we characterize the optimal strategy of investment-reinsurance under the power utility function. We use dynamic programming principle to derive the Hamilton–Jacobi–Bellman (HJB) equation. Then, by analysing the solution of the HJB equation, the optimal investment-reinsurance strategy is obtained and given in the verification theorem. Finally, sensitivity analysis is given to show the economic behavior of the optimal investment and reinsurance strategies.

This work deals with an optimal dividend payment problem for a piecewise-deterministic compound Poisson insurance risk model. The objective is to maximize the expected discounted dividend payout up to the time of ruin. When the dividend payment rate is restricted, the value function is shown to be a solution of the corresponding Hamilton-Jacobi-Bellman equation, which in turn leads to a tractable methodology to find an optimal threshold dividend payment policy. For the case of unrestricted payment rate, the value function and an optimal barrier strategy are determined explicitly with exponential claim size distributions. A comparison of two examples is provided to illustrate the main results.

A fundamental challenge for insurance companies (insurers) is to strike the best balance between optimal investment and risk management of paying insurance liabilities, especially in a low interest rate environment. The stochastic interest rate becomes a critical factor in this asset-liability management (ALM) problem. This paper derives the closed-form solution to the optimal investment problem for an insurer subject to the insurance liability of compound Poisson process and the stochastic interest rate following the extended CIR model. Therefore, the insurer’s wealth follows a jump-diffusion model with stochastic interest rate when she invests in stocks and bonds. Our problem involves maximizing the expected constant relative risk averse (CRRA) utility function subject to stochastic interest rate and Poisson shocks. After solving the stochastic optimal control problem with the HJB framework, we offer a verification theorem by proving the uniform integrability of a tight upper bound for the objective function.

This paper investigates an optimal investment problem for an open-end fund with dynamic flows. The accumulated fund inflow and outflow processes are modeled by two independent compound Poisson processes, respectively. The fund manager allocates the fund capital between a risk-free asset and a risky asset. The correlation between the fund inflow process and the risky asset's price process is under consideration. Inspired by Berk and Van Binsbergen [(2015). Measuring skill in the mutual fund industry. Journal of Financial Economics, 118(1), 1–20], we first introduce the accumulated value added process to measure the fund performance. The manager aims to maximize the expected utility of the fund's accumulated value added. We derive the optimal investment strategy explicitly via dynamic programming approach. Furthermore, a verification theorem is provided and some properties of the optimal strategy are analyzed. Finally, sensitivity analysis is given to illustrate the effects of model parameters on the optimal strategy.

In this paper, we investigate the optimal dividend problem under Parisian ruin with affine penalty payments at Parisian ruin time. The underlying risk process is assumed to be a spectrally negative Levy risk process. With the help of the dynamic programming principle, we prove that the value function associated to our optimal control problem is the smallest solution with certain characteristics to the corresponding Hamilton–Jacobi–Bellman (HJB) equation. In addition, the form of the performance function under barrier dividend strategy is expressed in terms of various extended scale functions. Then we identify a condition under which the performance function under certain barrier strategy is also a solution to the HJB equation, which in turn illustrates the optimalilty of such barrier dividend strategy among all admissible strategies. Various numerical examples are also given when the underlying risk process is compound Poisson process, Brownian motion with drift and jump-diffusion process.

This paper studies a class of renewable resource allocation problems for the processing of dynamically arriving tasks with deterministic deadlines. This class of problems has many applications, however, it conforms to neither the standard resource allocation model nor the standard optimal control model. A new problem formulation has to be developed and analyzed to provide a satisfactory solution. In this paper, a new formulation is presented. By augmenting the state variabls, the problem is converted into a Markovian decision problem. It can then be treated, at least in principle, by using the stochastic dynamic programming (SDP) method. However, since the system dynamics involves the evolution of sets (of tasks and resources), the implementation of the dynamic programming equation is by no means straightforward. For a problem with infinite planning horizon, the optimal strategy is shown to be stationary under mild conditions. An SDP algorithm based on a successive approximation technique is developed to obtain the optimal stationary strategy. The implementation of the algorithm employs a special coding scheme to handle set variables, and utilizes a dominance property for computational efficiency. Effects of key system parameters on optimal decisions are investigated and analyzed through numerical examples. As the computational complexity of the algorithm is of exponential increase, practical applications of the algorithm is limited to problems of moderate size. Two heuristic rules are therefore investigated and compared to the optimal policy.

Asymptotic and numerical analysis of the optimal investment strategy for an insurer

(a) Definitions Consideration is now extended from the claim number processes to processes which operate the claim amounts, concerning both the individual claims and their sums, the aggregate claims. A primary building block is the randomly varying size Z of an individual claim, i.e. the sum to be paid by the insurer at occurrence of fire, accident or any other event insured against. It is assumed that the claim sizes Z arising from different claim causing events are mutually independent and equally distributed, having a d.f. $$ S(Z) = prob\{ Z \le Z\} . $$ (3.1.1) Following the collective approach outlined in item 2.1(b) no regard is paid to the risk unit (policy) from which the claim has arisen. The d.f. S describes the variability of sizes of the continual flow of Claims. The aspects discussed in Section 2.3 suggest that payments due to one and the same event should be united as one claim irrespective of whether or not they formally concern different policies (e.g. different owners of a property in a damaged building complex or in a vehicle). Furthermore, if two or more Claims are Coming from the same risk units they are considered as different Claims if they are caused by different events.

This paper is concerned with the optimal portfolio problem for a company that can invest in two risky assets, where a novel Lévy-process-driven model is constructed to describe the dynamics of the wealth process by using a constant elasticity of variance model and a jump-diffusion process. A delicately designed value function is proposed under the mean–variance criterion to reflect the optimal portfolio for the stochastic volatility model. By using the verification theorem, the desired optimal portfolio strategy is proposed by the solution to certain Hamilton–Jacobi–Bellman equations. Furthermore, the corresponding expressions are achieved by using the stochastic analysis theory. Finally, a numerical simulation example is provided to verify the effectiveness of the proposed optimal portfolio strategy.

This work focuses on optimal harvesting problems for a stochastic competitive ecosystem subject to Levy noise. A verification theorem for corresponding harvesting strategy is established, which offers sufficient conditions for deriving an optimal harvesting strategy and an upper bound of the value function. For a given instantaneous marginal yields function, a concrete upper bound of value function is constructed by applying the verification theorem obtained in this paper. Meanwhile, the monotonicity of value function is investigated. Also, an e-optimal harvesting strategy is designed to find an approximate optimal harvesting strategy for those harvesting problems with no exact optimal harvesting strategy. Finally, by choosing appropriately Markov decision process defined on a discrete state space, a computational method for an optimal harvesting strategy is designed and a concrete example is also given to show the implementation of the algorithm.

Inconsistent Investment and Consumption Problems

We analyse an optimal portfolio and consumption problem with stochastic factor and delay over a finite time horizon. The financial market includes a risk-free asset, a risky asset and a stochastic factor. The price process of the risky asset is modelled as a stochastic differential delay equation whose coefficients vary according to the stochastic factor; the drift also depends on its historical performance. Employing the stochastic dynamic programming approach, we establish the associated Hamilton–Jacobi–Bellman equation. Then we solve the optimal investment and consumption strategies for the power utility function. We also consider a special case in which the price process of the stochastic factor degenerates into a Cox–Ingersoll–Ross model. Finally, the effects of the delay variable on the optimal strategies are discussed and some numerical examples are presented to illustrate the results. doi:10.1017/S1446181119000014

A stochastic model describing the impact of daily rainfall depth distribution on the soil water balance