Overordnede kursusmål
The course connects the theory of molecular diffusion with the
theory of differential equations driven by noise. It enables the
student to build and examine models of how noise and uncertainty
propagates in dynamic systems. This can be used for time series
analysis, of for dynamic decision making under uncertainty, i.e.
stochastic control. The course contains theory for stochastic
differential equations, stochastic calculus, and applications to
engineering problems.
Læringsmål
En studerende, der fuldt ud har opfyldt kursets mål, vil kunne:
- Define and describe diffusion and Brownian motion
- Compute stochastic integrals, most importantly the Ito
integral, and apply the rules of stochastic calculus
- Build stochastic dynamic models by combining ordinary
differential equations with models of how noise affects the
system
- Explain how a given stochastic differential equation
corresponds to a certain advective-diffusive transport
equation
- Investigate the properties of a stochastic differential
equation in terms of sample paths and transition probabilities,
both analytically and numerically.
- Implement a state estimation filter for analyzing time series
data based on a stochastic differential equation
- Identify the optimal strategy to control a system in presence
of noise
- 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
Lecture notes: U.H. Thygesen: Lecture notes on diffusion and
stochastic differential equations.
Supplementary literature is B. Øksendal: Stochastic differential
equations: An introduction with applications. Springer, 2010.
Bemærkninger
The course may be offered also at Ph.D. level. Contact the course
responsible person for details.
Sidst opdateret
26. juni, 2018
