Stochastic processes are essential in various fields, such as physics, finance, and engineering. They describe systems that evolve over time and whose behavior is affected by random variables. In this section, we will explore the basics of stochastic processes and their applications.

Basic Concepts

  • Random Variables: The fundamental building blocks of stochastic processes.
  • Probability Distributions: Describes the probabilities of different outcomes for a random variable.
  • Expected Value: The average outcome of a random variable.
  • Variance: Measures the spread of a random variable's values.

Types of Stochastic Processes

  • Discrete Time Stochastic Processes: Systems that evolve in discrete steps.
    • Markov Chains: A type of discrete-time stochastic process where the future state depends only on the current state.
    • Poisson Processes: Describes the number of events occurring in a fixed interval of time.
  • Continuous Time Stochastic Processes: Systems that evolve in continuous time.
    • Wiener Processes: Also known as Brownian motion, describes the random motion of particles in a fluid.
    • Geometric Brownian Motion: Describes the evolution of stock prices over time.

Applications

  • Finance: Modeling stock prices, option pricing, and risk management.
  • Engineering: Designing control systems, analyzing queues, and optimizing processes.
  • Physics: Describing the motion of particles, diffusion, and heat transfer.

Brownian Motion

For more information on stochastic processes and their applications, please visit our stochastic processes tutorials.