Diffusions and stochastic differential equations

• 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.

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, including boundary conditions/boundary behavior. We consider stochastic filtering in sde’s with discrete time measurements, stochastic stability, and stochastic control. The theory is illustrated with applications in engineering, physics, biology, and oceanography.

02407/02417/02443/01035, The course assumes some exposure to stochastic processes, for example obtained through 02407, 02417 or 02443. Programming using Matlab, R or similar is assumed.

Lectures and exercises

The course may be offered also at Ph.D. level. Contact the course responsible person for details.