Probability Theory Refresher: Axiomatic construction of probability spaces, random variables and vectors, probability distributions, functions of random variables; mathematical expectations, transforms and generating functions, modes of convergence of sequences of random variables, laws of large numbers, central limit theorem;Introduction to Stochastic Processes (SPs): Definition and examples of SPs, classification of random processes according to state space and parameter space, types of SPs, elementary problems;Stationary Processes: Weakly stationary and strongly stationary processes, moving average and auto regressive processes;Discrete-time Markov Chains (DTMCs): Definition and examples of MCs, transition probability matrix, Chapman-Kolmogorov equations; calculation of n-step transition probabilities, limiting probabilities, classification of states, ergodicity, stationary distribution, transient MC; random walk and gambler’s ruin problem, applications;Continuous-time Markov Chains (CTMCs): Kolmogorov- Feller differential equations, infinitesimal generator, Poisson process, birth-death process, stochastic Petri net, applications to queueing theory and communication networks;Martingales: Conditional expectations, definition and examples of martingales.

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

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Course Curriculum

Introduction to Stochastic Processes Details 55:11
Introduction to Stochastic Processes (Contd.) Details 59:10
Problems in Random Variables and Distributions Details 48:40
Problems in Sequences of Random Variables Details 41:18
Definition, Classification and Examples Details 50:35
Simple Stochastic Processes Details 57:2
Stationary Processes Details 54:37
Autoregressive Processes Details 1:2:14
Introduction, Definition and Transition Probability Matrix Details 56:1
Chapman-Kolmogrov Equations Details 56:45
Classification of States and Limiting Distributions Details 51:14
Limiting and Stationary Distributions Details 59:39
Limiting Distributions, Ergodicity and Stationary Distributions Details 48:25
Time Reversible Markov Chain Details 56:31
Reducible Markov Chains Details 55:41
Definition, Kolmogrov Differential Equations and Infinitesimal Generator Matrix Details 55:24
Limiting and Stationary Distributions, Birth Death Processes Details 58:36
Poisson Processes Details 56:9
M/M/1 Queueing Model Details 56:23
Simple Markovian Queueing Models Details 58:3
Queueing Networks Details 58:43
Communication Systems Details 51:18
Stochastic Petri Nets Details 58:1
Conditional Expectation and Filtration Details 48:45
Definition and Simple Examples Details 55:51
Definition and Properties Details 46:41
Processes Derived from Brownian Motion Details 39:29
Stochastic Differential Equations Details 47:38
Ito Integrals Details 50:15
Ito Formula and its Variants Details 39:53
Some Important SDE`s and Their Solutions Details 39:31
Renewal Function and Renewal Equation Details 46:48
Generalized Renewal Processes and Renewal Limit Theorems Details 37:58
Markov Renewal and Markov Regenerative Processes Details 1:1:8
Non Markovian Queues Details 39:39
Non Markovian Queues Cont,, Details 44:25
Application of Markov Regenerative Processes Details 47:43
Galton-Watson Process Details 43:48
Markovian Branching Process Details 46:6

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