Stochastic Processes

This is a course on Stochastic Processes that aims at describing some advanced level topics in this area of research with a very strong potential of applications. This course also prepares students for undertaking research in this area. This also helps prepare students for applications of this important subject to agricultural statistics.

 

  • Basics of stochastic processes. Classification according to state space and time domain. Finite and countable state Markov chains, time-homogeneity, Chapman-Kolmogorov equations, marginal distribution and finite dimensional distributions. Classification of Markov chain.  Canonical form of transition probability matrix of a Markov chain.  Fundamental matrix, probabilities of absorption from transient states into recurrent classes in a finite Markov chain, mean time for absorption. Ergodic state and Ergodic chain. Stationary distribution of a Markov chain, existence and evaluation of stationary distribution.  Random walk and gamblers ruin problem.
  • Birth and death processes like pure birth process, linear birth and death process, immigration-birth-death process. Discrete state continuous time Markov process: Kolmogorov difference – differential equations. Pure birth process  (Yule-Fury process).  Immigration-Emigration process.  Linear growth process, pure death process.
  • Renewal process: Renewal process when time is discrete and continuous. Renewal function and renewal density. Statements of Elementary renewal theorem and Key renewal theorem.
  • Elements of queuing processes - queues in series, queuing networks. Applications of queuing theory.

 

Epidemic processes: Simple deterministic and stochastic epidemic model. General epidemic models - Kermack and McKendrick’s threshold theorem. Recurrent epidemics. Chain binomial models. Diffusion processes. Diffusion limit of a random walk and Discrete branching process. Forward and backward Kolmogorov diffusion equations and their applications.