Stochastic Processes (Poisson, Markov, Random Walk)

Stochastic processes model collections of random variables across time or another index. Homogeneous Poisson processes have constant intensity and independent increments; Markov models condition future transitions on the specified current state; random walks accumulate steps under stated rules.

A stochastic process is a collection of random variables indexed by time or another ordered parameter. Its structure specifies more than uncertainty at one moment: it defines how observations relate across steps. A homogeneous Poisson process models counts with constant intensity and independent increments; other Poisson processes can use changing intensity.

The Markov property means the specified current state contains the information from the past needed for the next-state distribution. It does not mean history never matters in reality; the state may need to include relevant history. A random walk accumulates random steps under stated assumptions. Select a process by checking these conditions rather than assigning a familiar label to any fluctuating series.

When to use it

When modeling arrivals, transitions, or cumulative random changes and the dependence across observations affects a decision.

How it can help

Choose a process by checking state, dependence, timing, and boundary assumptions, then compare its predictions with relevant observations.