Stochastic Calculus with Applications to Stochastic Portfolio Optimisation

Magisterarbeit von Daniel Michelbrink

Introduction:

The present paper is about continuous time stochastic calculus and its application to stochastic portfolio selection problems. The paper is divided into two parts:

The first part provides the mathematical framework and consists of Chapters 1 and 2, where it gives an insight into the theory of stochastic process and the theory of stochastic calculus. The second part, consisting of Chapters 3 and 4, applies the first part to problems in stochastic portfolio theory and stochastic portfolio optimisation.

Chapter 1, "Stochastic Processes", starts with the construction of stochastic process. The significance of Markovian kernels is discussed and some examples of process and emigroups will be given. The simple normal-distribution will be extended to the multi-variate normal distribution, which is needed for introducing the Brownian motion process. Finally, another class of stochastic process is introduced which plays a central role in mathematical finance: the martingale.

Chapter 2, "Stochastic Calculus", begins with the introduction of the stochastic integral. This integral is different to the Lebesgue-Stieltjes integral because of the randomness of the integrand and integrator. This is followed by the probably most important theorem in stochastic calculus: Itˆo’s formula. Itˆo’s formula is of central importance and most of the proofs of Chapters 3 and 4 are not possible without it. We continue with the notion of a stochastic differential equations. We introduce strong and weak solutions and a way to solve stochastic differential equations by removing the drift. The last section of Chapter 2 applies stochastic calculus to stochastic control. We will need stochastic control to solve some portfolio problems in Chapter 4.

Chapter 3, "Stochastic Portfolio Theory", deals mainly with the problem of introducing an appropriate model for stock prices and portfolios. These models will be needed in Chapter 4. The first section of Chapter 3 introduces a stock market model, portfolios, the risk-less asset, consumption and labour income processes. The second section, Section 3.2, introduces the notion of relative return as well as portfolio generating functions. Relative return finds application in Chapter 4 where we deal with benchmark optimisation. Benchmark optimisation is optimising a portfolio with respect to a given benchmark portfolio. The final section of Chapter 3 contains some considerations about the long-term behaviour of portfolios and markets. It finishes with the surprising result that under some apparently irrelevant assumption the market is not diverse.

Chapter 4, "Stochastic Portfolio Optimisation", is dedicated to apply all prior groundwork to the problem of finding optimum stock portfolios. Section 4.1 intro-duces two ways to make decisions under risk. The first is the mean-variance approach used by Markowitz and the second is by using an utility function. In Section 4.2 we solve various optimisation problems mainly in the mean-variance sense. Section 4.3 uses utility functions to maximise an investors terminal wealth, consumption, and a combination of both. It generalises the typical investment-consumption problem by introducing labour earnings to the portfolio value process. Section 4.4 solves the problem from Section 4.3 by using stochastic control theory. We finish the paper by solving the problem of maximising expected utility from relative return, which is the problem of maximising the expected utility from a portfolio with respect to a given benchmark portfolio.

Table of Contents:

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