1. Introduction to Stochastic Processes
2. Introduction to Stochastic Processes (Contd.)
3. Problems in Random Variables and Distributions
4. Problems in Sequences of Random Variables
5. Definition, Classification and Examples
6. Simple Stochastic Processes
7. Stationary Processes
8. Autoregressive Processes
9. Introduction, Definition and Transition Probability Matrix
10. Chapman-Kolmogrov Equations
11. Classification of States and Limiting Distributions
12. Limiting and Stationary Distributions
13. Limiting Distributions, Ergodicity and Stationary Distributions
14. Time Reversible Markov Chain
15. Reducible Markov Chains
16. Definition, Kolmogrov Differential Equations and Infinitesimal Generator Matrix
17. Limiting and Stationary Distributions, Birth Death Processes
18. Poisson Processes
19. M/M/1 Queueing Model
20. Simple Markovian Queueing Models
21. Queueing Networks
22. Communication Systems
23. Stochastic Petri Nets
24. Conditional Expectation and Filtration
25. Definition and Simple Examples
26. Definition and Properties
27. Processes Derived from Brownian Motion
28. Stochastic Differential Equations
29. Ito Integrals
30. Ito Formula and its Variants
31. Some Important SDE`s and Their Solutions
32. Renewal Function and Renewal Equation
33. Generalized Renewal Processes and Renewal Limit Theorems
34. Markov Renewal and Markov Regenerative Processes
35. Non Markovian Queues
36. Non Markovian Queues Cont,,
37. Application of Markov Regenerative Processes
38. Galton-Watson Process
39. Markovian Branching Process

Probability Review and Introduction to Stochastic Processes (SPs): Probability spaces, random variables and probability distributions, expectations, transforms and generating functions, convergence, LLNs, CLT.
Definition, examples and classification of random processes according to state space and parameter space.
Stationary Processes: Weakly stationary and strongly stationary processes, moving average and auto regressive processes
Discrete-time Markov Chains (DTMCs): Transition probability matrix, Chapman-Kolmogorov equations; n-step transition and limiting probabilities, ergodicity, stationary distribution, random walk and gambler’s ruin problem, applications of DTMCs.
Continuous-time Markov Chains (CTMCs): Kolmogorov differential equations for CTMCs, infinitesimal generator, Poisson and birth-death processes, stochastic Petri net, applications to queueing theory and communication networks.
Martingales: Conditional expectations, definition and examples of martingales, applications in finance.
Brownian Motion: Wiener process as a limit of random walk; process derived from Brownian motion, stochastic differential equation, stochastic integral equation, Ito formula, Some important SDEs and their solutions, applications to finance.
Renewal Processes: Renewal function and its properties, renewal theorems, cost/rewards associated with renewals, Markov renewal and regenerative processes, non Markovian queues, applications of Markov regenerative processes.
Branching Processes: Definition and examples branching processes, probability generating function, mean and variance, Galton-Watson branching process, probability of extinction.