Stochastic Process

Paper Code: 
25STT421
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

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 sciences.

 

Course Outcomes: 

Course

Course Outcomes

Learning and teaching strategies

Assessment Strategies

Course Code

Course Title

25STT421

Stochastic Process

(Theory)

CO 112: Identify and formulate fundamental probability distribution and density functions and describe their properties.

CO 113: Apply the concept of finite markov chain in real life problems.

CO 114: Analyze continuous and discrete-time random processes. 

CO 115: Explain the concepts of stationary and wide-sense stationarity, and appreciate their significance.

CO 116: Formulate simple stochastic process models in the time domain and provide qualitative and quantitative analyses of such models.

CO 117: Contribute effectively in course-specific interaction.

Approach in teaching: 

Interactive Lectures, 

Group Discussion, 

Classroom Assignment,

Problem Solving Sessions.

 

Learning activities for the students:

Assignments,

Seminar,

Presentation,

Subject based  Activities.

Classroom Quiz,

Assignments,

Class Test,

Individual Presentation.

 

15.00
Unit I: 
Introduction to Stochastic Processes and Markov Chains

Introduction to stochastic process - 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. 

 

 

15.00
Unit II: 
Transition Probability Matrix and Ergodic Chains

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.

 

15.00
Unit III: 
Continuous Time Markov Processes

Discrete state continuous time Markov process: Kolmogorov difference – differential equations. Birth and death process, pure birth process (Yule- Fury process). Immigration-Emigration process. Linear growth process, pure death process.

 

15.00
Unit IV: 
Renewal process

Renewal process when time is discrete and continuous. Renewal function and renewal density. Statements of Elementary renewal theorem and Key renewal theorem.

 

15.00
Unit V: 
Branching Processes and Wiener Process

Branching process: Galton-Watson branching process. Mean and variance of size of nth generation, probability of ultimate extinction of a branching process. Fundamental theorem of branching process and applications. Introduction of Wiener process.

 

Essential Readings: 
  • Adke, S.R. & Manjunath S.M. (1984) : An Introduction of Finite Markov Processes, Wiley Eastern. 
  • Bhatt ,B.R. (2000): Stochastic Models: Analysis and applications, New Age International, India 
  • Cox,,P.R.(1970): Demography, Cambridge University Press 

 

SUGGESTED READINGS:

  • Harris, T.E. (1963): The Theory of Branching processes, Springer-Verlag. 
  • Medhi, J (1982): Stochastic Processes, Wiley Eastern. 
  • Ballingsley, P (1962) : Statistical Inference for Markov Chains, Chicago University Press, Chicago. 
  • Ross, S.M (1983); Stochastic Processes, Wiley. 

 

e-RESOURCES:

 

JOURNALS:

  • Sankhya The Indian Journal of Statistics, Indian Statistical Institute
  • Aligarh Journal of Statistics, Department of Statistics and Operations Research, Aligarh Muslim University
  • Afrika Statistika, Saint-Louis Senega University
  • International Journal of Statistics and Reliability Engineering, Indian Association for Reliability and Statistic
  • Journal of the Indian Society for Probability and Statistics, Indian Society for Probability and Statistics
  • Journal of the Indian Statistical Association, Indian Statistical Association
  • Statistica, Department of Statistical Sciences Paolo Fortunato, University of Bologna
  • Statistics and Applications, Society of Statistics, Computer and Applications
  • Stochastic Modeling and Applications, MUK Publications and Distributions

 

Academic Year: 
2025-2026
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