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.
Litteratur:
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.
