The course aims to enable the student to examine – analytically and
numerically - diffusion processes and models based on stochastic
differential equations. Central in the course is the connection
between the physical notion of diffusive transport, diffusion as a
Markov process, and stochastic differential equations as dynamic
systems driven by noise.
Læringsmål:
En studerende, der fuldt ud har opfyldt kursets mål, vil kunne:
Describe Brownian motion and its role in stochastic integrals
and stochastic differential equations
Compare different stochastic integrals, most importantly the
Ito integral and the Stratonovich integral.
Apply Ito's lemma and similar formulas from stochastic
analysis.
Convert a stochastic differential equation to an
advective-diffusive transport equation, and conversely.
Analyse a stochastic differential equation in terms of sample
paths and transition probabilities, both analytically and
numerically.
Synthesise a stochastic model of a system by combining
deterministic differential equations with assumed noise
properties.
Implement a filter for a scalar stochastic differential
equation with discrete-time measurements
Evaluate the importance of including noise in a study of a
given system.
Kursusindhold:
The course starts with advective and diffusive transport, and Monte
Carlo simulation of a molecule in flow. We then turn to Brownian
motion and stochastic integrals, and establish the Ito integral. We
define stochastic differential equations (sde's), and cover
analytical and numerical techniques to solve them. We describe the
transition probabilities of solutions to sde's, and establish
the forward and backward Kolmogorov equations. We consider
stochastic filtering in sde's with discrete time measurements.
Additional topics may include stochastic stability, optimal
stopping, stochastic control, boundary conditions, or diffusion on
manifolds. The theory is illustrated with applications in
engineering, physics, biology, and oceanography.
Litteraturhenvisninger:
B. Øksendal: Stochastic differential equations: An introduction
with applications. Springer, 2005. Also, lecture notes.
Bemærkninger:
The course is offered also at Ph.D. level, which requires and extra
assignment. Contact the course responsible person for
details.
