Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process

In this paper, we study the optimal investment and proportional reinsurance strategy when an insurance company wishes to maximize the expected exponential utility of the terminal wealth. It is assumed that the instantaneous rate of investment return follows an Ornstein-Uhlenbeck process. Using stochastic control theory and Hamilton-Jacobi-Bellman equations, explicit expressions for the optimal strategy and value function are derived not only for the compound Poisson risk model but also for the Brownian motion risk model. Further, we investigate the partially observable optimization problem, and also obtain explicit expressions for the optimal results.     

Optimal proportional reinsurance and investment under partial information

In this paper, we study the optimal proportional reinsurance and investment strategy for an insurer that only has partial information at its disposal, under the criterion of maximizing the expected utility of the terminal wealth. We assume that the surplus of the insurer is governed by a jump diffusion process, and that reinsurance is used by the insurer to reduce risk. In addition, the insurer can invest in financial markets. We give a characterization for the optimal strategy within a non-Markovian setting. Malliavin calculus for Lévy processes is used for the analysis.     

Optimal control of the insurance company with proportional reinsurance policy under solvency constraints

This paper considers the optimal control problem of the insurance company with proportional reinsurance policy under solvency constraints. The management of the company controls the reinsurance rate and dividends payout processes to maximize the expected present value of the dividend until the time of bankruptcy. This is a mixed singular-regular control problem. However, the optimal dividend payout barrier may be too low to be acceptable. The company may be prohibited to pay dividend according to external reasons because this low dividend payout barrier will result in bankruptcy soon. Therefore, some constraints on the insurance company’s dividend policy will be imposed. One reasonable and normal constraint is that if b is the minimum dividend barrier, then the bankrupt probability should not be larger than some predetermined ε within the time horizon T. This paper is to work out the optimal control policy of the insurance company under the solvency constraints.     

Pricing and hedging in the presence of extraneous risks

Given an underlying complete financial market, we study contingent claims whose payoffs may depend on the occurrence of nonmarket events. We first investigate the almost-sure hedging of such claims. In particular, we obtain new representations of the hedging prices and provide necessary and sufficient conditions for a claim to be marketed. The analysis of various examples then leads us to investigate alternative pricing rules. We choose to embed the pricing problem into the agent’s portfolio decision and study reservation prices. We establish the existence and consistency of this pricing rule in a semimartingale model. We characterize the nonlinear dependence of the reservation price with respect to both the agent’s initial capital and the size of her position. The fair price arises as a limiting case.     

Optimal dividend payments in the stochastic Ramsey model

We consider the dividend payments of a self-financing firm in the stochastic Ramsey model. The firm invests in capital stock and its production technology is given by the Cobb–Douglas function. Our objective is to maximize the expected present value of future real dividends subject to a positive constraint on the capital stock. We use the penalization method to obtain a solution for the variational inequality associated with the optimal growth problem and give a synthesis of the optimal dividend policy.     

Constructing sublinear expectations on path space

We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G$G$-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G$G$-expectation, and an optional sampling theorem that holds without exceptional set. Our results also shed light on the inherent limitations to constructing sublinear expectations through aggregation.     

A stochastic partially reversible investment problem on a finite time-horizon: Free-boundary analysis

We study a continuous-time, finite horizon, stochastic partially reversible investment problem for a firm producing a single good in a market with frictions. The production capacity is modeled as a one-dimensional, time-homogeneous, linear diffusion controlled by a bounded variation process which represents the cumulative investment–disinvestment strategy. We associate to the investment–disinvestment problem a zero-sum optimal stopping game and characterize its value function through a free-boundary problem with two moving boundaries. These are continuous, bounded and monotone curves that solve a system of non-linear integral equations of Volterra type. The optimal investment–disinvestment strategy is then shown to be a diffusion reflected at the two boundaries.     

Optimal proportional reinsurance and investment for stochastic factor models

In this work we investigate the optimal proportional reinsurance-investment strategy of an insurance company which wishes to maximize the expected exponential utility of its terminal wealth in a finite time horizon. Our goal is to extend the classical Cramér–Lundberg model introducing a stochastic factor which affects the intensity of the claims arrival process, described by a Cox process, as well as the insurance and reinsurance premia. The financial market is supposed not influenced by the stochastic factor, hence it is independent on the insurance market. Using the classical stochastic control approach based on the Hamilton–Jacobi–Bellman equation we characterize the optimal strategy and provide a verification result for the value function via classical solutions to two backward partial differential equations. Existence and uniqueness of these solutions are discussed. Results under various premium calculation principles are illustrated and a new premium calculation rule is proposed in order to get more realistic strategies and to better fit our stochastic factor model. Finally, numerical simulations are performed to obtain sensitivity analyses.     

Optimal time-consistent investment and reinsurance policies for mean-variance insurers

This paper investigates the optimal time-consistent policies of an investment-reinsurance problem and an investment-only problem under the mean-variance criterion for an insurer whose surplus process is approximated by a Brownian motion with drift. The financial market considered by the insurer consists of one risk-free asset and multiple risky assets whose price processes follow geometric Brownian motions. A general verification theorem is developed, and explicit closed-form expressions of the optimal polices and the optimal value functions are derived for the two problems. Economic implications and numerical sensitivity analysis are presented for our results. Our main findings are: (i) the optimal time-consistent policies of both problems are independent of their corresponding wealth processes; (ii) the two problems have the same optimal investment policies; (iii) the parameters of the risky assets (the insurance market) have no impact on the optimal reinsurance (investment) policy; (iv) the premium return rate of the insurer does not affect the optimal policies but affects the optimal value functions; (v) reinsurance can increase the mean-variance utility.     

Optimal investment,consumption and proportional reinsurance under model uncertainty

This paper considers the optimal investment, consumption and proportional reinsurance strategies for an insurer under model uncertainty. The surplus process of the insurer before investment and consumption is assumed to be a general jump–diffusion process. The financial market consists of one risk-free asset and one risky asset whose price process is also a general jump–diffusion process. We transform the problem equivalently into a two-person zero-sum forward–backward stochastic differential game driven by two-dimensional Lévy noises. The maximum principles for a general form of this game are established to solve our problem. Some special interesting cases are studied by using Malliavin calculus so as to give explicit expressions of the optimal strategies.     

Optimal proportional reinsurance and investment with minimum probability of ruin

The paper concerns a problem of optimal reinsurance and investment in order to minimizing the probability of ruin. In the whole paper, the cedent’s surplus is allowed to invest in a risk-free asset and a risky asset and the company’s risk is reduced through proportional reinsurance, while in addition the claim process is assumed to follow a Brownian motion with drift. By solving the corresponding Hamilton-Jacobi-Bellman equations, the optimal reinsurance-investment strategy is derived. The presented results generalize those by Taksar [1].     

Optimal dividend and investing control of an insurance company with higher solvency constraints

This paper considers the optimal control problem of a large insurance company under a fixed insolvency probability. The company controls proportional reinsurance rate, dividend pay-outs and investing process to maximize the expected present value of the dividend pay-outs until the time of bankruptcy. This paper aims at describing the optimal return function as well as the optimal policy. As a by-product, the paper theoretically sets a risk-based capital standard to ensure the capital requirement that can cover the total risk.     

Transformation of an axialsymmetric disk problem for the helmholtz equation into an ordinary differential equation

The problem of solving the three-dimensional Helmholtz equation in the exterior of a circular disk is considered where radially symmetric Dirichlet data on the disk are assumed to be prescribed. This problem for example arises in the scattering of plane (sound) waves at an infinite plane screen with a circular aperture if the direction of the incident wave is normal to the screen, as well as in the process of diffusion through a circular hole. By applying the factorization technique developed in [N. GORENFLO, M. WERNER,Solution of a finite convolution equation with a Hankel kernel by matrix factorization, SIAM J. Math. Anal., 28 (1997), pp. 434–451] to the disk problem an equivalent ordinary differential equation is derived, whose solution leads directly to the solution of the disk problem. This differential equation belongs to a class of ordinary differential equations which are of higher complexity than the standard ordinary differential equations of mathematical physics. The examination of this new class of differential equations therefore is motivated.     

Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint

In this paper, the basic claim process is assumed to follow a Brownian motion with drift. In addition, the insurer is allowed to invest in a risk-free asset and n risky assets and to purchase proportional reinsurance. Under the constraint of no-shorting, we consider two optimization problems: the problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the probability of ruin. By solving the corresponding Hamilton–Jacobi–Bellman equations, explicit expressions for their optimal value functions and the corresponding optimal strategies are obtained. In particular, when there is no risk-free interest rate, the results indicate that the optimal strategies, under maximizing the expected exponential utility and minimizing the probability of ruin, are equivalent for some special parameter. This validates Ferguson’s longstanding conjecture about the relation between the two problems.     

Switching problem and related system of reflected backward SDEs

This paper studies a system of backward stochastic differential equations with oblique reflections (RBSDEs for short), motivated by the switching problem under Knightian uncertainty and recursive utilities. The main feature of our system is that its components are interconnected through both the generators and the obstacles. We prove existence, uniqueness, and stability of the solution of the RBSDE, and give the expression of the price and the optimal strategy for the original switching problem via a verification theorem.     

Stochastic realization of a Gaussian stochastic control system

The stochastic realization problem is considered of representing a stationary Gaussian process as the observation process of a Gaussian stochastic control system. The problem formulation includes that the lastm components of the observation process form the Gaussian white noise input process to the system. Identifiability of this class of systems motivates the problem. The results include a necessary and sufficient condition for the existence of a stochastic realization. A subclass of Gaussian stochastic control systems is defined that is almost a canonical form for this stochastic realization problem. For a structured Gaussian stochastic control system an equivalent condition for identifiability of the parametrization is stated.The research of this paper is supported in part by the Commission of the European Communities through the SCIENCE Program by the projectSystem Identification with contract number SC1-CT92-0779.     

Optimal risk control and dividend policies under excess of loss reinsurance

We study the optimal reinsurance policy and dividend distribution of an insurance company under excess of loss reinsurance. The objective of the insurer is to maximize the expected discounted dividends. We suppose that in the absence of dividend distribution, the reserve process of the insurance company follows a compound Poisson process. We first prove existence and uniqueness results for this optimization problem by using singular stochastic control methods and the theory of viscosity solutions. We then compute the optimal strategy of reinsurance, the optimal dividend strategy and the value function by solving the associated integro-differential Hamilton–Jacobi–Bellman Variational Inequality numerically.