Portfolio optimization with special restrictions

U ovom radu bavimo se, kao što i naslov kaže, optimizacijom portfelja uz neke posebne uvjete. Ovaj problem promatramo na financijskom tržištu i ideja je naći portfelj koji dugoročno donosi pozitivne prinose te promatramo njemu pridružen proces bogatstva koji opisuje naše prinose s obzirom na portfelj koji smo odabrali te postavljamo uvjete na sam proces. Prvo poglavlje je uvod gdje izlažemo uvjete na proces bogatstva, odnosno, htjeli bismo da ne padne ispod postotka maksimalne vrijednosti koju je postigao od početka ulaganja. U drugom poglavlju navodimo defnicije, koncepte i rezultate koji su potrebni za razumijevanje i praćenje samog rada. U centru su pojmovi martingala, Itovog integrala te njihovih svojstava. U trećem poglavlju je strogo izložen problem koji proučavamo, koji je zapravo problem koji su proučavali J. Cvitanić i I. Karatzas a koji je svojevrsno poopćenje onoga što su proučavali Grossman i Zhou, a to je zapravo proširenje tog problema na više financijskih imovina. Nadalje je dan model kojim opisujemo kretanje financijskih imovina na tržištu te slučajni proces bogatstva u obliku stohastičke diferencijalne jednadžbe. Zatim, kako bi se došlo do rješenja, uvodi se pomoćni slučajni proces te na kraju kroz primjer, uz konkretnu funkciju korisnosti, dano je rješenje problema, odnosno portfelj koji optimizira ulaganja te pritom pripadajući proces bogatstva s obzirom na dani portfelj koji zadovoljava otprije zadani uvjet. U četvrtom poglavlju pokaže se da postoji i kako izgleda portfelj koji maksimizira dugoročnu stopu srednje vrijednosti korisnosti. Potom se i dokaže sama tvrdnja. U petom poglavlju je pokazano da portfelj iz četvrtog poglvalja maksimizira i dugoročnu stopu investicije.In this thesis we observe the problem of optimization of portfolio with special conditions imposed on our process. This problem is observed on financial markets and the idea is to find a portfolio which brings long term positive income and simultaneously observing wealth process with respect to portfolio that we chose thus imposing conditions on our wealth process. First chapter is introduction where we present conditions restricting our wealth process and we don’t want it to fall under the fixed fraction of maximum value achieved at or before the observed time. In second chapter we give the definitions of concepts used in this paper needed for understanding the materia as well as theorems and others results that are natural extension of before given definitions. In the center of interest are the concepts such as martingales, Ito integral and their properties. In the third chapter is strictly determined the problem we study which is a problem studied by J. Cvitanić and I. Karatzas which is a generalization of problem studied by Grossmann and Zhou. It is the expansion of the original problem on the several risky assets. Furthermore, we define the stochastic model which describes dynamic of financial assets on the financial markets and random wealth process in terms of differential stochastic equations. Then, to find a solution of our problem, we define an auxiliary stochastic control problem with finite horizon so we could, through example, with explicit utility function, construct the portfolio which optimizes our investments and satisfy the equation representing our wealth process. In the fourth chapter we discuss and find portfolio that maximize long term average growth rate of utility and then we prove existence and above statement of maximization. We also show that the same portfolio maximize the long term growth rate of investment