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