M. Sc. Sven Haadem ved Matematisk institutt vil forsvare sin avhandling for graden ph.d.: Stochastic Control and Optimal Stopping for Non-Markov Processes with Infinite Horizon and Related Topics

## Bedømmelseskomité

• Professor Yaozhong Hu , University of Kansas

• Professor José Manuel Corcuera, Universitat de Barcelona

• Professor Inge Helland, Universitetet i Oslo

## Leder av disputas

Professor Tom Louis Lindstrøm, Matematisk institutt, Universitet i Oslo

## Sammendrag

To get the most out of a given resource one must quantify and optimize the system in a satisfying way. This is not easy and the fact that most system has a random component makes this very complex. For time dependent systems one need to decide what happens when the horizon is reached.  Deciding the length of the period and what happens at the end can be difficult. So, it is often preferable to look at infinite horizons to avoid the horizon problem. In this thesis a way to solve infinite horizon problems for non-Markovian systems is developed. It is coherent with the deterministic theory.

Differential equations play an important part in optimization. We need to know when we have a solution to our equations and the properties of the solution. Therefore in the last part of this thesis we study the basic building block for the above: stochastic differential equations.

Infinite-horizon optimal control problems arise in many fields of economics, in particular in models of economic growth. Important examples are the perpetual American option and sustainability issues in optimal resource extraction. When it comes to stochastic control, the possibly infinite horizon does not cause any problems when using dynamic programming.  However, dynamic programming can only be used for Markovian models.  In non-Markovian models the maximum principle need to be used instead.

Since dynamic systems often have some kind of delay an extension to delay equations is of great importance. We therefore include a study of infinite horizon system with delay. We further extend the findings to fractional Brownian motion. Fractional Brownian motion is a generalization of Brownian motion without independent increments. It represents a natural one-parameter extension and has been widely used to model a number of phenomena in diverse fields from biology to finance.

## For mer informasjon

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Publisert 7. mars 2014 16:46 - Sist endret 7. mars 2014 16:49