Marc Mangel (msmangel@ucsc.edu)
MWF, 11-12:10, Baskin Engineering 169
Stochastic differential equations (SDE) arise in many branches of science and engineering. This is an introduction to SDE, requiring only upper division probability and differential equations, since we will approach the analysis of questions about SDE through the associated differential equations. Approximate topical outline:
Brownian Motion
and White Noise
The Gambler’s Ruin
Ornstein Uhlenbeck Process
Ito and Stratonovich Calculi
Kolmogorov backward and
forward (aka Fokker Planck) equations
Feynman-Kac formula and path
integrals
Poisson Increment SDE and the
Generalized Ornstein Uhlenbeck Process
Applications (mainly based on approximate and
asymptotic solutions of SDE)
The Einstein-Smoluchowski
theory of Brownian motion
The diffusion (Kramers)
theory of reaction rates and escape from a domain of attraction
Gillespie’s tau-method for
chemical kinetics
Fluctuations in threshold
systems
Black-Scholes theory of
option pricing (possible, depending upon the audience and time left in the
quarter)
There is no text, but I will suggest some books that might
interest you. Grades will be determined by homework (assigned one class period,
due the next, with no late homework accepted), a take home final and
participation (I expect students to attend class).
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But then, in any science, the
'noise' might prove to be not merely something to get rid of, but the essential
phenomenon of interest. It seems curious (at least, to a physicist) that this
was first seen clearly not in physics, but in biology. In the late 19th century
many biologists saw it as the major task confronting them to confirm Darwin's
theory by exhibiting the detailed mechanism by which evolution takes
place...Biologists have a mechanistic picture of the world because, being
trained to believe in causes, they continue to use the full power of their
brains to search for them--and so they find them.
Jaynes, E.T.. 2003. Probability Theory. The logic of science.
Cambridge University Press (pg 230...328)
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