Stochastic Processes

This course explanations and expositions of stochastic processes concepts which they need for their experiments and research. It also covers theoretical concepts pertaining to handling various stochastic modeling. This course provides classification and properties of stochastic processes, discrete and continuous time Markov chains, simple Markovian queueing models, applications of CTMC, martingales, Brownian motion, renewal processes, branching processes, stationary and autoregressive processes.

INTENDED AUDIENCE: Under-graduate, Post-graduate and PhD students of mathematics, electrical engineering, computer engineeringPRE REQUISITES : A basic course on ProbabilityINDUSTRY SUPPORT: Goldman Sachs, FinMachenics, Deutsche Bank and other finance companies.

COURSE LAYOUT

Week 1:Probability theory refresher

1. Introduction to stochastic process
   
2. Introduction to stochastic process (contd.)
   

Week 2:Probability theory refresher (contd.)

1. Problems in random variables and distributions
   
2. Problems in Sequence of random variables

Week 3:Definition and simple stochastic process

1. Definition, classification and Examples
   
2. Simple stochastic processes
   

Week 4:Discrete-time Markov chains

1. Introduction, Definition and Transition Probability Matrix
   
2. Chapman-Kolmogorov Equations
   
3. Classification of States and Limiting Distributions
   

Week 5:Discrete-time Markov chains (contd.)

1. Limiting and Stationary Distributions
   
2. Limiting Distributions, Ergodicity and stationary distributions
   
3. Time Reversible Markov Chain, Application of Irreducible Markov chains in Queueing Models
   
4. Reducible Markov Chains
   

Week 6:Continuous-time Markov chains

1. Definition, Kolmogrov Differential Equation and Infinitesimal Generator Matrix
   
2. Limiting and Stationary Distributions, Birth Death Processes
   
3. Poisson processes
   

Week 7:Continuous-time Markov Chains (contd.)

1. M/M/1 Queueing model
   
2. Simple Markovian Queueing Models
   

Week 8:Applications of CTMC

1. Queueing networks
   
2. Communication systems
   
3. Stochastic Petri Nets
   

Week 9:Martingales

1. Conditional Expectation and filteration
   
2. Definition and simple examples
   

Week 10:Brownian Motion

1. Definition and Properties
   
2. Processes Derived from Brownian Motion
   
3. Stochastic Differential Equation
   

Week 11:Renewal Processes

1. Renewal Function and Equation
   
2. Generalized Renewal Processes and Renewal Limit Theorems
   
3. Markov Renewal and Markov Regenerative Processes
   
4. Non Markovian Queues
   
5. Application of Markov Regenerative Processes
   

Week 12:Branching Processes, Stationary and Autoregressive Processes