Citation
Darling, Donald Allan (1947) Continuous Stochastic Processes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5M0SFQ29. https://resolver.caltech.edu/CaltechTHESIS:06042015160752368
Abstract
A general review of stochastic processes is given in the introduction; definitions, properties and a rough classification are presented together with the position and scope of the author's work as it fits into the general scheme.
The first section presents a brief summary of the pertinent analytical properties of continuous stochastic processes and their probabilitytheoretic foundations which are used in the sequel.
The remaining two sections (II and III), comprising the body of the work, are the author's contribution to the theory. It turns out that a very inclusive class of continuous stochastic processes are characterized by a fundamental partial differential equation and its adjoint (the FokkerPlanck equations). The coefficients appearing in those equations assimilate, in a most concise way, all the salient properties of the process, freed from boundary value considerations. The writer’s work consists in characterizing the processes through these coefficients without recourse to solving the partial differential equations.
First, a class of coefficients leading to a unique, continuous process is presented, and several facts are proven to show why this class is restricted. Then, in terms of the coefficients, the unconditional statistics are deduced, these being the mean, variance and covariance. The most general class of coefficients leading to the Gaussian distribution is deduced, and a complete characterization of these processes is presented. By specializing the coefficients, all the known stochastic processes may be readily studied, and some examples of these are presented; viz. the Einstein process, Bachelier process, OrnsteinUhlenbeck process, etc. The calculations are effectively reduced down to ordinary first order differential equations, and in addition to giving a comprehensive characterization, the derivations are materially simplified over the solution to the original partial differential equations.
In the last section the properties of the integral process are presented. After an expository section on the definition, meaning, and importance of the integral process, a particular example is carried through starting from basic definition. This illustrates the fundamental properties, and an inherent paradox. Next the basic coefficients of the integral process are studied in terms of the original coefficients, and the integral process is uniquely characterized. It is shown that the integral process, with a slight modification, is a continuous Markoff process.
The elementary statistics of the integral process are deduced: means, variances, and covariances, in terms of the original coefficients. It is shown that an integral process is never temporally homogeneous in a nondegenerate process.
Finally, in terms of the original class of admissible coefficients, the statistics of the integral process are explicitly presented, and the integral process of all known continuous processes are specified.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics and Physics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Minor Option:  Physics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  1947 
Record Number:  CaltechTHESIS:06042015160752368 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:06042015160752368 
DOI:  10.7907/5M0SFQ29 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  8990 
Collection:  CaltechTHESIS 
Deposited By:  INVALID USER 
Deposited On:  05 Jun 2015 14:45 
Last Modified:  17 Mar 2023 18:16 
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