Supervisors info:
Ιωάννης Στρατής Καθηγ. ΕΚΠΑ (Επιβλέπων), Αθανάσιος Γιαννακόπουλος Καθηγ. Ο.Π.Α, Απόστολος Μπουρνέτας Καθηγ. Ε.Κ.Π.Α
Summary:
The theory of large deviations deals with the probabilities of rare events (or
fluctuations) that are exponentially small as a function of some parameter,
e.g., the number of random components of a system, the time over which a
stochastic system is observed, the amplitude of the noise perturbing a
dynamical system or the temperature of a chemical reaction. The theory has
applications in many different scientific fields, ranging from queuing theory
to statistics and from finance to engineering. It is also increasingly used in
statistical physics for studying both equilibrium and nonequilibrium systems.
In this context, deep analogies can be made between familiar concepts of
statistical physics, such as the entropy and the free energy, and concepts of
large deviation theory having more technical names, such as the rate function
and the scaled cumulant generating function. The first part of this Master
thesis introduces the basic elements of large deviation theory in mathematics.
The focus there is on the simple but powerful ideas behind large deviation
theory, stated in non-technical terms, and on the application of these ideas in
simple stochastic processes, such as sums of independent and identically
distributed random variables and Markov processes. In the second part, the
problem of numerically evaluating large deviation probabilities is treated at a
basic level. The fundamental idea of importance sampling is introduced there
together with its sister idea, the exponential change of measure. Also, there
is a basic reference about the applications of large deviation theory in
physics, finance and insurance. Finally, in the third part, we revisit large
deviation theory for stochastic differential equations in the small-noise
limit.
Keywords:
Large Deviations, Probabilities Distribution, Stochastic process, Simulation, Stochastic Differential Equation
