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

### 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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