Putyatina O. Models of stochastic processes applied to portfolio optimization

The object of research is the complicated dynamic systems with random noise, particularly, investment management systems. The aim of research is the improvement of the financial and investment activity by developing the methods of portfolio optimitzation for asset price models with jumps. The methods of research are the methods of filtering theory, the methods of approximation and linearization of stochastic processes and the methods of optimal control theory for stochastic processe, Monte-Carlo method. Equipment is personal computer. Theoretical and practical results of the research include the new scientifically proven results, which solve the important problem of mathematical modelling of stochastic processes in portfolio optimization problems, practical importance of the developed mathematical models and numerical methods is that they allow to evaluate the observable and non-observable parameters of stochastic financial processes with required accuracy by using filtering, including the abrupt movements such as shot-noise and non-linear asset price dynamics. The scientific novelty consist of the theoretical solution (for the first time) to the portfolio optimization problem for the asset price models driven by Brownian motion and shot-noise process under full information. Verification theorem was formulated and proved (for shot-noise driven asset price model); the new model was proposed, that is driven by Brownian motion and the compound Poisson process instead of shot-noise. This model is convenient for modelling the market processes, because the jumps of the compound Poisson process, unlike the shot-noise process, do not decay as time passes. The solution to the portfolio optimiztion problem for the asset price model driven by the compound Poisson process was obtained under full and partial information; the approximate method for solving the infinite-dimensional filtering problem for the processes driven by Brownian motion and combined noise was proposed. As the result, instead of the diffusion process with jumps the diffusion process without jumps was obtained, that allowed to apply Kalman filter for studing the process of such type; the linearized model was for the first time proposed as a modification of the Heston model and the method for solving the infinite-dimensional non-linear filtering problem for the Heston model with stochastic volatility and drift was proposed; the method for solving the portfolio optimization problem for the Heston model under partial information (asset prices are observable, but not the volatility) was further developed. The numerical results showed that the mean deviation between the optimal strategy under full information and the optimal strategy under partial information is not significant. The results of the dissertation were implemented into the test project of the bank "Mercury" and into the study program of Kharkiv National University of Radioelectronics. Scientific theoretical and practical results of the dissertation can be used in the following institutions: in the universities for the students in mathematical modelling; in the universities with the departments of financial and actuarial mathematics; in the institutions that do research in portfolio optimization; in the investment institutions.