Optimal risk/Dividend distribution control models. Applications to insurance

The current paper presents a short survey of stochastic models of risk control and dividend optimization techniques for a financial corporation. While being close to consumption/investment models of Mathematical Finance, dividend optimization models possess special features which do not allow them to be treated as a particular case of consumption/investment models. In a typical model of this sort, in the absence of control, the reserve (surplus) process, which represents the liquid assets of the company, is governed by a Brownian motion with constant drift and diffusion coefficient. This is a limiting case of the classical Cramer-Lundberg model in which the reserve is a compound Poisson process, amended by a linear term, representing a constant influx of the insurance premiums. Risk control action corresponds to reinsuring part of the claims the cedent is required to pay simultaneously diverting part of the premiums to a reinsurance company. This translates into controlling the drift and the diffusion coefficient of the approximating process. The dividend distribution policy consists of choosing the times and the amounts of dividends to be paid put to shareholders. Mathematically, the cumulative dividend process is described by an increasing functional which may or may not be continuous with respect to time. The objective in the models presented here is maximization of the dividend pay-outs. We will discuss models with different types of conditions imposed upon a company and different types of reinsurances available, such as proportional, noncheap, proportional in a presence of a constant debt liability, excess-of-loss. We will show that in most cases the optimal dividend distribution scheme is of a barrier type, while the risk control policy depends substantially on the nature of reinsurance available.