文档介绍：Stochastic Optimization in Finance
Krastyu Gumnerov
Institute of Information Technologies – B A S
******@
Introduction
The financial activity like many other activities has two characteristics:
The decision-making is under uncertainty, ., it depends on the future values of parameters, unknown at the moment of the decision-making, so they are random quantities with respect to the information at the moment.
2) The decisions are optimal with respect to some objective.
Thus it is natural a system for financial planning (portfolio management) to have two modules:
1. a module describing the random quantities of the model and their evolution (scenario generator).
2. an optimization module for given objective function and variables evolution.
This review examines some methods for building the second module. The purpose of this review is to offer a brief description of the main approaches for dynamic stochastic optimization and to show some of their applications in finance. The author hopes in this way to attract the attention of the Bulgarian experts engaged in financial mathematics to these approaches unpopular (up to now) in Bulgaria.
I. General Statement of the Dynamic Stochastic Optimization Problem (Stochastic Control)
To understand the stochastic control it is useful to keep in mind the analogy with some more simple problems:
The elementary problem of finding the conditional extremum of a function;
The problems of the calculus of variations and the variational approach in the classical mechanics and mathematical physics;
The problems of the deterministic optimal control.
In general, the solution of these problems, including the stochastic control problems, is reduced to some optimality conditions, in the form of equations (more often). These are the equations of the considered system: they describe the evolution of the parameters defining the system. Characteristic examples are: the equation for the stationary points of a func