Definition

1. Noun:
   • A mathematical model for a system that evolves over time in a way that involves randomness or probability: A "stochastic process" is a collection of random variables, typically indexed by time, that describes the potential states of a system and the probabilities of transitioning between these states. It is used to model systems where outcomes are uncertain.

Examples of Usage

• Noun:
  • The movement of a stock price is often modeled as a stochastic process.
  • Scientists use a stochastic process to simulate the spread of a disease through a population.
  • Analyzing the stochastic process of customer arrivals helps optimize queue management.

Advanced Usage

• "Markov process": A specific type of stochastic process where the future state depends only on the present state, not on the sequence of events that preceded it.

  • The model assumes the system's behavior follows a Markov process.

• "Stationary stochastic process": A process whose statistical properties (like mean and variance) do not change over time.

  • For the analysis to be valid, we must assume the data represents a stationary stochastic process.

Variants and Related Words

• Stochastic (adj): Having a random probability distribution or pattern that may be analyzed statistically but not predicted precisely.

  • The model accounts for stochastic fluctuations in demand.

• Random process (n): A common synonym for "stochastic process."

Synonyms

• Random process: A process involving random variables.
• Probabilistic process: A process governed by probability laws.

Related Phrases

• Sample path or realization: A single observed outcome or trajectory of a stochastic process.

  • Each simulation run produces a different sample path of the stochastic process.

• State space: The set of all possible values (states) that the random variables in the process can take.

  • Defining the state space is the first step in modeling the stochastic process.

Noun

1. a statistical process involving a number of random variables depending on a variable parameter (which is usually time)