Phd records
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 31 Jan 2024 19:24:17 +0000Boundary value problems for stochastic differential equations
https://resolver.caltech.edu/CaltechTHESIS:04022013-104632972
Authors: {'items': [{'id': 'MacDowell-T-W', 'name': {'family': 'MacDowell', 'given': 'Thomas William'}, 'show_email': 'NO'}]}
Year: 1968
DOI: 10.7907/JR7J-9W71
<p>A theory of two-point boundary value problems analogous
to the theory of initial value problems for stochastic ordinary
differential equations whose solutions form Markov processes is
developed. The theory of initial value problems consists of
three main parts: the proof that the solution process is
markovian and diffusive; the construction of the Kolmogorov
or Fokker-Planck equation of the process; and the proof that
the transistion probability density of the process is a unique
solution of the Fokker-Planck equation. </p>
<p>It is assumed here that the stochastic differential equation
under consideration has, as an initial value problem, a diffusive
markovian solution process. When a given boundary value problem
for this stochastic equation almost surely has unique solutions,
we show that the solution process of the boundary value problem
is also a diffusive Markov process. Since a boundary value
problem, unlike an initial value problem, has no preferred
direction for the parameter set, we find that there are two
Fokker-Planck equations, one for each direction. It is shown
that the density of the solution process of the boundary value
problem is the unique simultaneous solution of this pair of
Fokker-Planck equations. </p>
<p>This theory is then applied to the problem of a vibrating
string with stochastic density. </p>
https://thesis.library.caltech.edu/id/eprint/7572