University of Illinois Urbana-Champaign

The Prediction Process of Step-Processes and Applications

Goswami, Alok Kumar

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Description

Title
The Prediction Process of Step-Processes and Applications
Author(s)
Goswami, Alok Kumar
Issue Date
1985
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
  • Frank B. Knight, in 1975, presented a new viewpoint in the theory of stochastic processes, by introducing what is called the prediction process. This dissertation applies that method to a special kind of
  • process, namely step-processes. The prediction process is constructed for processes of the types (UNFORMATTED TABLE FOLLOWS)
  • (i) X(,t) = X (.) 1(,(t(GREATERTHEQ)T(,*))), 0 (LESSTHEQ) t (LESSTHEQ) (INFIN)(TABLE ENDS)
  • where X (NOT=) 0 and 0 (LESSTHEQ) T(,*) (LESSTHEQ) (INFIN) are random variables on a probability space, and, more generally,
  • (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
  • where X(,0) (ELEM) (//R), J(,k) (NOT=) 0, 0 < S(,k) < (INFIN) are random variables on a probability space with
  • with probability 1 and, T(,0) = 0, T(,n) = S(,1) +...+ S(,n), for n (GREATERTHEQ) 1. The prediction process Z(,t) in either case becomes a Borel right process with unique left limits Z(,t-), and X(,t) is shown to be probabilistically equivalent to a Borel function of Z(,t). Discontinuities of Z(,t) are studied especially with regard to how they reflect the discontinuities of X(,t). A Levy system (N,H) for the Markov process Z(,t) is constructed and used to establish a known result on representation of martingales. Finally, the special case where the conditional distributions of waiting times for jumps given the past are exponential, is studied; and a charac- terisation of such processes in terms of the prediction process is derived. Also, a necessary and sufficient condition for the predic- tion process Z(,t) to be a step-process is proved; and, some simple examples (including non-markovian ones) are discussed.
Graduation Semester
1985
Type of Resource
text
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