IIT Delhi Course , Prof. S. Dharmaraja

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IIT Delhi Course , Prof. S. Dharmaraja

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 gamblers 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

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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Stochastic Processes by Dr. S. Dharmaraja, Department of Mathematics, IIT Delhi. For more details on NPTEL visit httpnptel.iitm.ac.in

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- 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.MM1 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

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