Courses

Course

Stochastic Processes

Indian Institute of Technology Delhi

This course provides a comprehensive introduction to Stochastic Processes (SPs), emphasizing key concepts and practical applications. The curriculum includes:

  • Probability Foundations: Review of probability spaces, random variables, probability distributions, expectations, and convergence.
  • Random Processes: Definition, examples, and classification based on state space and parameter space.
  • Stationary Processes: Exploration of weakly and strongly stationary processes, moving averages, and autoregressive processes.
  • Discrete-time Markov Chains (DTMCs): Transition probability matrix, Chapman-Kolmogorov equations, ergodicity, random walk, and real-world applications.
  • Continuous-time Markov Chains (CTMCs): Kolmogorov differential equations, infinitesimal generator, and applications to queueing theory and communication networks.
  • Martingales: Conditional expectations, martingale definitions, and applications in finance.
  • Brownian Motion: Wiener process, stochastic differential equations, and financial applications.
  • Renewal Processes: Renewal functions and theorems, cost/reward analysis, and applications of regenerative processes.
  • Branching Processes: Definitions, examples, and properties, including the Galton-Watson process.
Course Lectures
  • Mod-01 Lec-01 Introduction to Stochastic Processes
    Mod-01 Lec-01 Introduction to Stochastic Processes
    Dr. S. Dharmaraja
    Play

    This module provides an overview of stochastic processes, laying the groundwork for understanding the subject. It covers fundamental concepts in probability, including:

    • Probability spaces
    • Random variables
    • Probability distributions
    • Expectations and their calculations
    • Convergence concepts
    • Law of Large Numbers (LLN)
    • Central Limit Theorem (CLT)

    Emphasis is placed on the relation of these concepts to stochastic processes and their significance in various applications.

  • Mod-01 Lec-02 Introduction to Stochastic Processes (Contd.)
    Mod-01 Lec-02 Introduction to Stochastic Processes (Contd.)
    Dr. S. Dharmaraja
    Play

    This module continues the introduction to stochastic processes, focusing on the classification of random processes. It includes:

    • Definition of stochastic processes
    • Examples of various stochastic processes
    • Classification based on state space and parameter space

    Students will gain insights into how different processes can be categorized and the implications of these classifications.

  • Mod-01 Lec-03 Problems in Random Variables and Distributions
    Mod-01 Lec-03 Problems in Random Variables and Distributions
    Dr. S. Dharmaraja
    Play

    This module delves into problems associated with random variables and distributions. Key topics include:

    • Solving practical problems involving random variables
    • Understanding different probability distributions
    • Examples illustrating the application of various distributions

    Students will engage in hands-on problem-solving to solidify their understanding of concepts introduced in the previous modules.

  • Mod-01 Lec-04 Problems in Sequences of Random Variables
    Mod-01 Lec-04 Problems in Sequences of Random Variables
    Dr. S. Dharmaraja
    Play

    This module covers problems in sequences of random variables, emphasizing:

    • Convergence properties of sequences
    • Distributional aspects of sequences
    • Applications of sequences in stochastic processes

    Students will learn to analyze and solve problems related to sequences, preparing them for more complex concepts in later modules.

  • Mod-02 Lec-01 Definition, Classification and Examples
    Mod-02 Lec-01 Definition, Classification and Examples
    Dr. S. Dharmaraja
    Play

    This module provides a comprehensive overview of definitions, classifications, and examples of stochastic processes. Key aspects include:

    • Different types of stochastic processes
    • How to classify processes based on their characteristics
    • Real-world examples illustrating various processes

    Students will engage with the content through discussions and case studies to enhance their practical understanding.

  • Mod-02 Lec-02 Simple Stochastic Processes
    Mod-02 Lec-02 Simple Stochastic Processes
    Dr. S. Dharmaraja
    Play

    This module introduces simple stochastic processes, providing an insight into:

    • Basic definitions and concepts
    • Examples of simple processes
    • How simple processes serve as building blocks for complex models

    Students will learn to recognize simple processes in real-world applications and their importance in stochastic modeling.

  • Mod-03 Lec-01 Stationary Processes
    Mod-03 Lec-01 Stationary Processes
    Dr. S. Dharmaraja
    Play

    This module focuses on stationary processes, detailing:

    • Definitions of weakly and strongly stationary processes
    • Characteristics of stationary processes
    • Applications of stationary processes in various fields

    Students will learn to identify stationary processes and understand their relevance in stochastic modeling and analysis.

  • Mod-03 Lec-02 Autoregressive Processes
    Mod-03 Lec-02 Autoregressive Processes
    Dr. S. Dharmaraja
    Play

    This module covers autoregressive processes, a core concept in time series analysis. It includes:

    • Definition and formulation of autoregressive models
    • Understanding the dynamics of time series
    • Applications in finance and economics

    Students will engage in practical modeling exercises to apply autoregressive concepts to real-world data sets.

  • Mod-04 Lec-01 Introduction, Definition and Transition Probability Matrix
    Mod-04 Lec-01 Introduction, Definition and Transition Probability Matrix
    Dr. S. Dharmaraja
    Play

    This module introduces discrete-time Markov chains (DTMCs), emphasizing:

    • Understanding the transition probability matrix
    • Chapman-Kolmogorov equations
    • Applications of DTMCs in various fields

    Students will explore various models and their real-world implications through case studies and examples.

  • Mod-04 Lec-02 Chapman-Kolmogrov Equations
    Mod-04 Lec-02 Chapman-Kolmogrov Equations
    Dr. S. Dharmaraja
    Play

    This module continues the exploration of DTMCs, focusing on Chapman-Kolmogorov equations. Key topics include:

    • The significance of these equations in Markov processes
    • Applications in predicting future states
    • Real-world examples illustrating their use

    Students will engage in problem-solving exercises to strengthen their understanding of these equations.

  • Mod-04 Lec-03 Classification of States and Limiting Distributions
    Mod-04 Lec-03 Classification of States and Limiting Distributions
    Dr. S. Dharmaraja
    Play

    This module delves deeper into the classification of states and limiting distributions for DTMCs. It includes:

    • Various classifications of states in Markov chains
    • Understanding limiting distributions and their importance
    • Practical examples and applications

    Students will analyze case studies to understand the implications of state classification in real-world scenarios.

  • Mod-04 Lec-04 Limiting and Stationary Distributions
    Mod-04 Lec-04 Limiting and Stationary Distributions
    Dr. S. Dharmaraja
    Play

    This module focuses on limiting and stationary distributions, providing insights into:

    • Understanding the concepts of limiting and stationary distributions
    • Mathematical derivation and practical implications
    • Applications in various stochastic processes

    Students will engage with exercises that illustrate the calculation and application of these distributions in real-world scenarios.

  • Mod-04 Lec-05 Limiting Distributions, Ergodicity and Stationary Distributions
    Mod-04 Lec-05 Limiting Distributions, Ergodicity and Stationary Distributions
    Dr. S. Dharmaraja
    Play

    This module discusses limiting distributions, ergodicity, and stationary distributions, covering:

    • Definitions and key properties of ergodicity
    • The relationship between ergodicity and stationary distributions
    • Applications in various fields, including finance and communication

    Students will work on examples that showcase the importance of these concepts in real-world applications.

  • Mod-04 Lec-06 Time Reversible Markov Chain
    Mod-04 Lec-06 Time Reversible Markov Chain
    Dr. S. Dharmaraja
    Play

    This module covers time-reversible Markov chains, including:

    • Definitions and key characteristics of time-reversible processes
    • Mathematical formulations and implications
    • Applications in statistical mechanics and other fields

    Students will engage with exercises to deepen their understanding of the significance of time-reversibility in Markov processes.

  • Mod-04 Lec-07 Reducible Markov Chains
    Mod-04 Lec-07 Reducible Markov Chains
    Dr. S. Dharmaraja
    Play

    This module discusses reducible Markov chains, focusing on:

    • Identification of reducible vs. irreducible chains
    • The implications of reducibility on state transitions
    • Applications and examples from various domains

    Students will analyze real-world examples to understand the concept of reducibility and its relevance in Markov chain applications.

  • Mod-05 Lec-01 Definition, Kolmogrov Differential Equations and Infinitesimal Generator Matrix
    Mod-05 Lec-01 Definition, Kolmogrov Differential Equations and Infinitesimal Generator Matrix
    Dr. S. Dharmaraja
    Play

    This module introduces continuous-time Markov chains (CTMCs), emphasizing:

    • Key definitions and properties of CTMCs
    • Kolmogorov differential equations
    • Applications in modeling real-world systems

    Students will explore various examples to understand the applications of CTMCs in different fields.

  • Mod-05 Lec-02 Limiting and Stationary Distributions, Birth Death Processes
    Mod-05 Lec-02 Limiting and Stationary Distributions, Birth Death Processes
    Dr. S. Dharmaraja
    Play

    This module focuses on the Kolmogorov differential equations for CTMCs, detailing:

    • The form and significance of these equations
    • Applications in predicting system behavior
    • Real-world examples illustrating their use

    Students will engage in practical exercises to strengthen their understanding of these vital equations.

  • Mod-05 Lec-03 Poisson Processes
    Mod-05 Lec-03 Poisson Processes
    Dr. S. Dharmaraja
    Play

    This module introduces Poisson processes, a key concept in stochastic processes. Key topics include:

    • Definition and characteristics of Poisson processes
    • Applications in queuing theory and telecommunications
    • Examples illustrating their practical significance

    Students will participate in exercises to understand the application of Poisson processes in real-world scenarios.

  • Mod-05 Lec-04 M/M/1 Queueing Model
    Mod-05 Lec-04 M/M/1 Queueing Model
    Dr. S. Dharmaraja
    Play

    This module covers the M/M/1 queueing model, focusing on:

    • Definition and structure of the M/M/1 queue
    • Performance metrics and analysis
    • Applications in real-world queueing scenarios

    Students will engage with practical examples to reinforce their understanding of queueing theory concepts.

  • Mod-05 Lec-05 Simple Markovian Queueing Models
    Mod-05 Lec-05 Simple Markovian Queueing Models
    Dr. S. Dharmaraja
    Play

    This module discusses simple Markovian queueing models, emphasizing:

    • Different types of Markovian queues
    • Analysis of their performance metrics
    • Applications in various fields, including operations research

    Students will work on case studies to illustrate the significance of these models in real-world applications.

  • Mod-05 Lec-06 Queueing Networks
    Mod-05 Lec-06 Queueing Networks
    Dr. S. Dharmaraja
    Play

    This module explores queueing networks, detailing:

    • Understanding of networked queueing systems
    • Performance analysis techniques
    • Applications in logistics and telecommunications

    Students will engage in exercises to analyze queueing networks in real-world scenarios, enhancing their practical skills.

  • Mod-05 Lec-07 Communication Systems
    Mod-05 Lec-07 Communication Systems
    Dr. S. Dharmaraja
    Play

    This module covers communication systems in the context of stochastic processes, highlighting:

    • Modeling communication networks using stochastic processes
    • Analyzing performance metrics in communication systems
    • Real-world applications and case studies

    Students will explore various communication systems to understand the relevance of stochastic modeling in this field.

  • Mod-05 Lec-08 Stochastic Petri Nets
    Mod-05 Lec-08 Stochastic Petri Nets
    Dr. S. Dharmaraja
    Play

    This module delves into Stochastic Petri Nets, which are a powerful modeling tool for concurrent systems. Students will learn the fundamental concepts and properties of Petri nets and how they can be applied to analyze complex systems involving stochastic behavior.

    Key topics include:

    • Basic definitions and structure of Stochastic Petri Nets.
    • Markovian properties of the nets and their implications.
    • Applications of Stochastic Petri Nets in various fields, including computer science and network communications.
  • Mod-06 Lec-01 Conditional Expectation and Filtration
    Mod-06 Lec-01 Conditional Expectation and Filtration
    Dr. S. Dharmaraja
    Play

    This module introduces the concept of Conditional Expectation and Filtration in stochastic processes. Students will explore how conditional expectations provide insights into the behavior of random variables given certain conditions.

    Topics include:

    • Definition and properties of conditional expectations.
    • The role of filtration in probability theory.
    • Applications of conditional expectation in various fields such as finance and statistics.
  • Mod-06 Lec-02 Definition and Simple Examples
    Mod-06 Lec-02 Definition and Simple Examples
    Dr. S. Dharmaraja
    Play

    This module presents various definitions and simple examples of stochastic processes. Students will understand the foundational concepts essential for analyzing more complex stochastic processes.

    Key components include:

    • Basic definitions of stochastic processes.
    • Examples illustrating different types of stochastic processes.
    • Introduction to the properties and classifications of these processes.
  • Mod-07 Lec-01 Definition and Properties
    Mod-07 Lec-01 Definition and Properties
    Dr. S. Dharmaraja
    Play

    This module focuses on the definition and properties of stochastic processes. It dives deep into the mathematical foundation necessary for understanding the behavior of various stochastic models.

    Students will explore:

    • Theoretical foundations and definitions.
    • Properties of different types of stochastic processes.
    • Applications in real-world scenarios.
  • Mod-07 Lec-02 Processes Derived from Brownian Motion
    Mod-07 Lec-02 Processes Derived from Brownian Motion
    Dr. S. Dharmaraja
    Play

    This module covers processes derived from Brownian motion, emphasizing their significance in stochastic calculus and real-world applications. Students will learn how Brownian motion serves as a fundamental building block for various stochastic models.

    Key learning points include:

    • Understanding the Wiener process and its properties.
    • Applications of Brownian motion in finance and physics.
    • Connections to other stochastic processes.
  • Mod-07 Lec-03 Stochastic Differential Equations
    Mod-07 Lec-03 Stochastic Differential Equations
    Dr. S. Dharmaraja
    Play

    This module introduces Stochastic Differential Equations (SDEs), a critical area in stochastic processes that extends ordinary differential equations to account for randomness. Students will learn about the foundational concepts of SDEs and their applications.

    Topics covered include:

    • Basic definitions and examples of SDEs.
    • The role of noise in modeling dynamic systems.
    • Applications in finance, biology, and engineering.
  • Mod-07 Lec-04 Ito Integrals
    Mod-07 Lec-04 Ito Integrals
    Dr. S. Dharmaraja
    Play

    This module focuses on Ito integrals, which are essential in the study of stochastic calculus. Students will understand how Ito's lemma and integrals are used in various applications, especially in financial mathematics.

    Key topics include:

    • The definition and properties of Ito integrals.
    • Applications of Ito integrals in modeling financial instruments.
    • Connections to stochastic differential equations.
  • Mod-07 Lec-05 Ito Formula and its Variants
    Mod-07 Lec-05 Ito Formula and its Variants
    Dr. S. Dharmaraja
    Play

    This module introduces Ito's formula and its variants, which are critical for transforming stochastic processes into more tractable forms. Students will explore the implications of these transformations in various applications.

    Key learning points include:

    • The derivation of Ito's formula and its significance.
    • Variants of Ito's formula and their applications.
    • Real-world examples demonstrating the use of Ito's formula in finance and other fields.
  • Mod-07 Lec-06 Some Important SDE`s and Their Solutions
    Mod-07 Lec-06 Some Important SDE`s and Their Solutions
    Dr. S. Dharmaraja
    Play

    This module presents some important SDEs and their solutions, illustrating how these equations are utilized in different contexts. Students will discover various methods for solving SDEs and analyzing their behavior.

    Key areas of focus include:

    • Commonly encountered SDEs in practice.
    • Techniques for finding solutions to SDEs.
    • Applications of SDEs in finance and other fields.
  • Mod-08 Lec-01 Renewal Function and Renewal Equation
    Mod-08 Lec-01 Renewal Function and Renewal Equation
    Dr. S. Dharmaraja
    Play

    This module explores the renewal function and renewal equation, key components in the study of renewal processes. Students will learn about the properties of renewal functions and how they relate to practical applications.

    Key topics include:

    • Definition and significance of the renewal function.
    • Renewal equations and their applications.
    • Connections to queueing theory and operational research.
  • Mod-08 Lec-02 Generalized Renewal Processes and Renewal Limit Theorems
    Mod-08 Lec-02 Generalized Renewal Processes and Renewal Limit Theorems
    Dr. S. Dharmaraja
    Play

    This module covers generalized renewal processes and renewal limit theorems, enhancing students' understanding of renewal theory. Students will explore how these concepts apply to various stochastic models.

    Key components include:

    • Generalized renewal processes and their properties.
    • Limit theorems related to renewal processes.
    • Applications in reliability theory and queuing systems.
  • Mod-08 Lec-03 Markov Renewal and Markov Regenerative Processes
    Mod-08 Lec-03 Markov Renewal and Markov Regenerative Processes
    Dr. S. Dharmaraja
    Play

    This module introduces Markov renewal and Markov regenerative processes, focusing on their definitions and applications. Students will learn about these processes' unique properties and how they differ from traditional renewal processes.

    Key topics include:

    • Definitions of Markov renewal processes.
    • Properties of Markov regenerative processes.
    • Applications in operations research and system reliability.
  • Mod-08 Lec-04 Non Markovian Queues
    Mod-08 Lec-04 Non Markovian Queues
    Dr. S. Dharmaraja
    Play

    This module discusses non-Markovian queues, providing insights into systems where the Markovian assumption does not hold. Students will explore the complexities of analyzing such queues and their implications in various fields.

    Key areas of focus include:

    • Understanding the characteristics of non-Markovian queues.
    • Analysis techniques for non-Markovian systems.
    • Applications in telecommunications and service systems.
  • Mod-08 Lec-05 Non Markovian Queues Cont,,
    Mod-08 Lec-05 Non Markovian Queues Cont,,
    Dr. S. Dharmaraja
    Play

    This module continues the discussion on non-Markovian queues, diving deeper into analysis and real-world applications. Students will learn advanced techniques for modeling and analyzing these complex systems.

    Key topics include:

    • Advanced analysis techniques for non-Markovian queues.
    • Case studies illustrating practical applications.
    • Connections to operational efficiency and performance metrics.
  • Mod-08 Lec-06 Application of Markov Regenerative Processes
    Mod-08 Lec-06 Application of Markov Regenerative Processes
    Dr. S. Dharmaraja
    Play

    This module examines the application of Markov regenerative processes, emphasizing their usefulness in analyzing systems with regenerative properties. Students will gain insights into their practical applications across different fields.

    Topics covered include:

    • Properties and definitions of Markov regenerative processes.
    • Real-world applications in operational research and queuing theory.
    • Case studies demonstrating their effectiveness in system analysis.
  • Mod-09 Lec-01 Galton-Watson Process
    Mod-09 Lec-01 Galton-Watson Process
    Dr. S. Dharmaraja
    Play

    This module introduces the Galton-Watson process, which models the evolution of populations in a stochastic manner. Students will learn about the characteristics and applications of this branching process.

    Key components include:

    • Definition and properties of the Galton-Watson process.
    • Applications in biology, demography, and epidemiology.
    • Probability of extinction and its implications.
  • Mod-09 Lec-02 Markovian Branching Process
    Mod-09 Lec-02 Markovian Branching Process
    Dr. S. Dharmaraja
    Play

    This module focuses on the Markovian branching process, examining its properties and applications in various fields. Students will understand how this process extends the Galton-Watson model and its significance in stochastic analysis.

    Key areas of focus include:

    • Definition and characteristics of Markovian branching processes.
    • Applications in genetics, ecology, and network theory.
    • Modeling techniques and their implications in real-world scenarios.
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