GRS MA884- Seminar in probability and statistics with focus on Multiscale Methods for Stochastic Processes and stochastic dynamical systems

Instructor: Konstantinos Spiliopoulos

Office: 111 Cummington Mall, Room 222
Office Hours: Tuesday-Thursday 3:15-4:15
Email: kspiliop_at_math.bu.edu
to send me an email replace _at_ by @.
Meets: Fall 2019, Tuesday-Thursday 12:30-1:45 at PRB 148

Texts:

G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Springer, 2007

Other recommended textbooks:

For multiscale methods and perturbation theory:

G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Springer, 2007
J.-P. Fouque , G. Papanicolaou, R. Sircar, K. Solna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge University Press, 2011
A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures (Studies in mathematics and its applications), Elsevier, 1978
V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functions, Springer-Verlang, 1991
H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering, Luban Press, 2005.
M. H. Holmes, Introduction to Perturbation Methods, Springer, 1998.

For stochastic calculus and the interplay between PDE's and stochastic processes:
- M. Freidlin, Functional Integration and Partial Differential Equations, Princeton University Press, 1985
- B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, 2007 (6th edition)
- I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 2nd edition

Syllabus (pdf)

Course Description: Data obtained from a physical system sometimes possess many characteristic length and time scales. In such cases, it is desirable to construct models that are effective for large-scale structures, while capturing small scales at the same time. Modeling this type of data and physical phenomena via mulitple scale diffusion processes and PDE's with multiple scales may be well-suited in many cases. Thus, such models have been used to describe the behavior of phenomena in scientific areas such as chemistry and biology, ocean-atmosphere sciences, finance and econometrics.

In this course, we will study concepts, analytic and probabilistic tools that are used in various scientific disciplines. Emphasis will be placed on

Review of probability theory, introduction to stochastic calculus (Brownian motion, stochastic differential equations, It\^{o} formula, Fokker-Planck eqs, Feynman-Kac formula, relation to PDE's)
Multiscale analysis (averaging and homogenization) of stochastic processes and differential equations using various deterministic and probabilistic tools.
Backward SDE's and applications to homogenization of non-linear PDEs.
Numerical methods and Monte Carlo methods for multiscale processes.
Applications to various disciplines such that mathematical finance, physics, chemistry and engineering will be discussed.

The course material will be based on theory, methods and examples from various scientific disciplines.

Course Prerequisites: The course will be largly self-contained, accesible to a broad audience and a revision to the basic tools from probability, stochastic processes and differential equations that are needed, will be given. However, students are expected to have the knowlesge equivalent to undergrdauate level probability, stochastic processes and differential equations. PDE's and graduate level probability will be helpful but NOT necessary.