A stochastic process \(X(t), t \in T\) is a collection of random variables. For each \(t \in T\), \(X(t)\) is a random variable. The index \(t\) is often interpreted as time, and as a result, we refer to \(X(t)\) as the state of the process at time \(t\).

The set \(T\) is called the index set of the process. When \(T\) is a countable set, the stochastic is a discrete-time process. If \(T\) is an interval of the real line, the process is said to be a continuous-time process.

Focus: Discrete time, discrete state space Markov Chain

Each \(X_t\) is a random variable.

Example of stochastic process: Gambler’s ruin

  1. A gambler starts with an initial fortune of \(k\) dollars.
  2. The gambler plays against \(B\) with an initial fortune of \(N-k\) dollars.
  3. Each game he bets $1, wins with probability \(p\)
  4. Let \({X_t = t = 0,1,2 \dots}\) represent his fortune as the betting goes on.
  5. Game only stops when either gambler or \(B\) is ruined.
  6. Here \(\mathcal{S} = {0,1,\dots,N}\)
  7. For a realization of the results of the first 10 games, (here \(p=12\)):
sample(c(-1, 1), 10, replace=T)

Reference Textbooks

(Ross, 2014),(Pinsky, 2010)

Bibliography

Ross, S. M., Introduction to probability models (2014), : Academic press.

Pinsky, M., & Karlin, S., An introduction to stochastic modeling (2010), : Academic press.