Citation
MacDowell, Thomas William (1968) Boundary Value Problems for Stochastic Differential Equations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/JR7J9W71. https://resolver.caltech.edu/CaltechTHESIS:04022013104632972
Abstract
A theory of twopoint 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 FokkerPlanck equation of the process; and the proof that the transistion probability density of the process is a unique solution of the FokkerPlanck equation.
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 FokkerPlanck 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 FokkerPlanck equations.
This theory is then applied to the problem of a vibrating string with stochastic density.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  (Applied Mathematics) 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Applied Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  9 May 1968 
Record Number:  CaltechTHESIS:04022013104632972 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:04022013104632972 
DOI:  10.7907/JR7J9W71 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  7572 
Collection:  CaltechTHESIS 
Deposited By:  INVALID USER 
Deposited On:  02 Apr 2013 18:34 
Last Modified:  03 Apr 2024 21:18 
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