ABSTRACT
The article proposes a new approach to practical multicriteria (vector) optimization and control of risk processes in discrete time. These processes describe a stochastic evolution of some asset, for example, the evolution of reserves of an insurance company or the state of stock in a warehouse under conditions of random demand. In the case of the insurance model, the so-called level of payments acts as the main random factor, i.e., the ratio of insurance claim payments to premiums under relevant contracts. In inventory theory, the primary random variable is the demand for a product. Control affects the parameters of the risk process, such as the rate of replenishment or consumption of an asset. The quality of management is evaluated according to utility and risk criteria. The expected asset accumulation level, expected cumulative dividends, expected satisfied demand, and other additive indicators can be used as utility criteria. The risk criteria are, for example, the probability of ruin, i.e., the probability that the process falls into some risk area, as well as the expected cost of ruin prevention. The optimal control is sought in the class of state of the process and a finite number of unknown deterministic parameters. The heuristic basis for choosing the class of parametric control strategies are theoretical results on analytical forms of optimal controls, for example, results on the barrier or band structure of optimal controls. When such a parametric control is substituted into the control criteria, the problem of stochastic optimal control is transformed into a finite-dimensional stochastic optimization problem with complex probabilistic criteria in the form of mathematical expectations and ruin event probabilities. This paper shows that there are two options for calculating optimization criteria, these are the solution of some integral equations and the Monte Carlo method. To find the Pareto-optimal frontier, it is proposed to use a deterministic or random discrete approximation of a finite-dimensional region of optimization parameters. As a result, the original vector optimal control problem is reduced to a discrete finite-dimensional vector optimization problem, which is solved by enumeration algorithms. Numerical experiments show the efficiency of this approach.
