Stokastisk analyse

En spesialisering innen Masterstudiet i Matematikk ved UiO

Studiets oppbygning

Hva er "Stokastisk Analyse"?

Feltet er en blanding av matematisk analyse og sannsynlighetsteori, med mange anvendelser, bl.a. innen matematisk finans. Det fins en aktiv forskningsgruppe innen dette feltet ved Matematisk Institutt, som ogs? er en del av CMA (Centre of Mathematics for Applications). Her f?lger en introduksjon til dette feltet (p? engelsk).

What is Stochastic Analysis?

Many phenomena in nature and society evolve randomly over time. For instance, it is impossible to forecast accurately the weather tomorrow, or to know the future price of financial stocks. By the laws of physics we can give reasonable predictions in many situations, but there will always be an element of randomness. The randomness may be a result of models which are not sufficiently precise, inaccurate measurements, or simply randomness which does not seem to be possible to model by physical laws.

Looking at how financial markets operate, it seems to be a great deal of uncertainty in how stock prices dynamically moves over time. Prices are results of traders¡¯ beliefs in the future, which to a large extent can not be modeled deterministically and therefore has a high degree of uncertainty. The dynamics of different weather parameters can, on the other hand, be well understood by the laws of physics, typically in the language of sophisticated partial differential equations. However, uncertainty in measurements and instability in the solutions may lead to unpredicted deviations in the weather conditions, and thus uncertainty in the future.

Stochastic processes are used to model the dynamics phenomena which has a degree of randomness. A stochastic process is a family of random variables indexed over time, and describes in a mathematical way the dynamics of uncertainty. This may be the stock price for a company or the temperature evolvement. The classical and probably most prominent example of a stochastic process is the Brownian motion, first observed by the botanist Robert Brown in 1827 in the movements of pollen grains suspended in a liquid. Later, in 1900, Louis Bachelier used concepts from probability theory to formalize the definition of a Brownian motion, and applied it to price various financial instruments on the French Stock Exchange. Perhaps more known is the work of Albert Einstein in 1905 on thermodynamics where the use of Brownian motion played a crucial role. This work was among the famous contributions of Einstein to physics in his Annus mirabilis, where the photoelectric effect was discovered and the theory of relativity introduced. In 1921 he was awarded the Nobel Prize in Physics for his achievements.

To model the dynamics of uncertainty by stochastic processes is a challenging issue, depending on sophisticated methods in statistics and probability theory. On the other hand, given such a model, we want to use it for further analysis. For instance, we may ask the question how the Norwegian Petroleum Fund should invest our oil money in the financial markets in order to get the highest return, but at the same time control the risk of loosing money. Another example may be the development of a new oil field. When is the right time to start development of an oil field, given that it is very costly to start production, and that the oil price is evolving in an uncertain way. In biology one often studies examples of harvesting from a population, like for instance fish. The amount of fish is not known exactly, and the influence of harvesting on the population is also uncertain. One may ask what is the optimal way to harvest given all the uncertainty. To answer these questions, we need stochastic analysis.

Stochastic analysis is, put very simply, the theory of differentiation and integration for stochastic processes. Unfortunately, the paths of Brownian motion are so irregular that even though it is continuous, it is nowhere differentiable. This makes integration with respect to Brownian paths a challenge. However, the Ito integral introduced by Kiyoshi Ito in the late 1940s gives the mathematical foundation for integrating stochastic processes with respect to Brownian motion. Even more, he proved the so-called Ito Formula, which is a chain rule for Ito integrals. Ito integration and Ito¡¯s Formula are indispensable tools in the analysis of stochastic processes.

In 1972 Fisher Black and Myron Scholes proved the famous formula for the price of options using the tools of stochastic analysis. An option is a financial contract giving the owner the right, but not the obligation, to buy a stock for an agreed price. By modeling the stock price as an exponential of a Brownian motion, they used the Ito formula to derive the correct price of an option. Scholes was later awarded the Nobel Prize in economy in 1997 (Black died two years earlier). He shared the prize with Robert Merton, who contributed significantly to the theory of option pricing, also using the tools of stochastic analysis.

Stochastic analysis is a vital research field by its own, and our group is active in contributing to the theoretical development. However, we are largely involved in various applications of stochastic analysis, for instance in fields like finance, biology, energy and insurance. By using advanced stochastic processes and techniques from stochastic analysis we investigate various problems in these fields, like for instance optimal management of portfolios and harvesting in biological systems. This involves stochastic control theory, where for instance partial differential equations play a crucial role. Our group also enjoys a close collaboration with insurance mathematics, where stochastic modeling and analysis are used for understanding the risk in insurance.

Om masterstudiet i matematikk med "Stokastisk analyse" som spesialisering.

For ? kunne skrive en masteroppgave innenfor feltet "Stokastisk analyse", m? man ha fullf?rt bachelorgraden i Matematikk, informatikk og teknologi, med studieretningen Matematikk (eller ha en tilvarende godkjent utdanning).

Deretter b?r f?lgende emner taes i l?pet av masterstudiet:

MAT4400 ¨C Line?r analyse med anvendelser obligatorisk dersom ikke MAT3400 ¨C Line?r analyse med anvendelser inng?r i bachelorgraden
MAT4410 ¨C Videreg?ende line?r analyse sterkt anbefalt
MAT4500 ¨C Topologi
MAT4701 ¨C Stokastisk analyse med anvendelser (videref?rt)

Det anbefales at man tar emnet MAT4410 ¨C Videreg?ende line?r analyse i 1. semester.

Man kan velge ? skrive en kort master oppgave (30 studiepoeng) eller en lang oppgave (60 studiepoeng). Dette m? avgj?res i forbindelse med at Masteravtalen skrives sammen med veileder i l?pet av 2.semester i masterstudiet.

Et eksempel p? en studieplan med kort masteroppgave er:

Et eksempel p? en studieplan med lang masteroppgave:

Et annet eksempel p? en studieplan med lang masteroppgave:

Denne studieveiene er bare ment som et eksempel. Den enkelte students studievei p? masterniv? vil bli lagt opp i 澳门葡京手机版app下载 med veilederen med utgangspunkt i studentens bakgrunn og interesser og med tanke p? temaet for masteroppgaven.

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