Computer Construction of Quasi Optimal Portfolio for Stochastic Models with Jumps of Financial Markets

In the paper we propose a purely computational new method of construction of a quasi–optimal portfolio for stochastic models of a financial market with jumps. Here we present the method in the framework of a Black–Scholes–Merton model of an incomplete market (see, eg. [5], [7]), considering a well known optimal investment and consumption problem with the HARA type optimization functional. Our method is based on the idea to maximize this functional, taking into account only some subsets of possible portfolio and consumption processes. We show how to reduce the main problem to the construction of a portfolio maximizing a deterministic function of a few real valued parameters but under purely stochastic constraints. It is enough to solve several times an indicated system of stochastic differential equations (SDEs) with properly chosen parametrs. This is a generalization of an approach presented in [4] in connection with a well known classical Black–Scholes model. Results of computer experiments presented here were obtained with the use of the SDE–Solver software package. This is our own professional C++ application to Windows system, designed as a scientific computing tool based on Monte Carlo simulations and serving for numerical and statistical construction of solutions to a wide class of systems of SDEs, including a broad class of diffusions with jumps driven by non-Gaussian random measures (consult [1], [4], [6], [9]). The approach to construction of approximate optimal portfolio presented here should be useful in a stock market analysis, eg. for evolution based computer methods.