Disputas: Kristina Rognlien Dahl
M. Sc. Kristina Rognlien Dahl ved Matematisk institutt vil forsvare sin avhandling for graden ph.d.:
Information and Memory in Stochastic Optimal Control
Kristina Rognlien Dahl
Tid og sted for prøveforelesning
Associate Professor Albina Danilova, London School of Economics
Professor Peter Imkeller, Humboldt-Universität zu Berlin
Professor Tom Louis Lindstrøm, Universitetet i Oslo
Leder av disputas
Instituttleder Arne Huseby, Matematisk institutt, Universitetet i Oslo
- Professor Bernt Øksendal, Matematisk institutt, Universitetet i Oslo
- Professor Fred Espen Benth, Matematisk institutt, Universitetet i Oslo
The world is an uncertain place. Therefore, mathematical models incorporating uncertainty, or stochasticity, have become increasingly important over the last decades. In this thesis, we study such models with the goal of making optimal decisions under uncertainty while incorporating that agents may have different levels of information and memory. Possible applications include optimal trading strategies in finance, minimizing the spreading of viruses and optimal harvesting.
Finance is a field where applications of stochastic analysis are essential. In this thesis, we derive the price of contingent claims (a kind of financial asset) under some general level of partial information. We do this using so-called duality theory, which essentially involves transforming an initial problem to some other related problem which is easier to solve or provides a more informative interpretation of the problem. We also apply a form of duality theory to solve the problem of an agent trying to determine his optimal consumption when his wage is a Lévy process, meaning his wage is a stochastic process which may jump. Such jumps in wage can for instance occur when losing a job.
Determining optimal consumption over time in an uncertain world is an example of a stochastic optimal control problem. A possible way to solve such problems is using stochastic maximum principles. We derive such maximum principles connected to singular recursive utility and situations where one has noisy memory. Singular recursive utility is an alternative way of measuring total utility, and we consider the situation where the agent can have a singular consumption process. Noisy memory means that the agent has a memory which is slightly disturbed by a Brownian motion.
In order to use the maximum principles to solve the stochastic optimal control problems, we must solve backward stochastic differential equations (BSDEs). Therefore, we have derived existence and uniqueness properties of the BSDEs corresponding to our optimal control problems, as well as solutions of these equations in some special cases.
In general, stochastic differential equations (SDEs) describe the evolution over time of some system with uncertainty. Both SDEs and BSDEs can only be solved analytically in a few special cases. Therefore, one needs numerical methods to approximate solutions. We derive such a numerical method for SDEs involving noisy memory.
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