Field · 6 models
Stochastic Process Simulations
A stochastic process is a sequence of random values indexed by time: a walker’s position, a chain’s state, the count of arrivals so far. These models show single sample paths and the statistics of many paths side by side, because both matter.
Models in this field
- OS-012Random walkCoin-flip steps on a line or grid: distance grows like √n, and in 1-D and 2-D every walker comes back.
- OS-013Brownian motionParticles jiggle with Gaussian steps; their mean squared displacement grows as 4Dt, as Einstein predicted.
- OS-014Markov chainA process that remembers only its current state: stationary distribution, mixing and absorption, live.
- OS-015Poisson processEvents at random times with a constant rate: Poisson counts, exponential gaps and the waiting-time paradox.
- OS-016Gambler's ruinBet 1 unit until broke or at N: exact odds, expected game length, and why a small edge matters.
- OS-017Geometric Brownian motionThe standard stock-price model: log-normal paths whose mean and typical outcome drift apart.
One path versus the ensemble
A single random walk looks like it is going somewhere; a thousand of them reveal a symmetric spread growing like √t. Most questions about a stochastic process are about the ensemble — expected values, variances, probabilities of hitting a level — while intuition is built from single paths. Each page draws both.
How the models connect
- The random walk is the discrete starting point. Shrink its steps and speed up time and it becomes Brownian motion; exponentiate a Brownian motion with drift and you get geometric Brownian motion, the textbook stock-price model.
- Stop a random walk at two boundaries and you have gambler’s ruin, one of the few processes with exact formulas for both the outcome and its duration.
- A random walk is also the simplest Markov chain: the next step depends only on the current state. Markov chains add arbitrary transition probabilities and a stationary distribution to converge to.
- The Poisson process models events arriving at random times; it drives the arrivals in every queueing model.
Discrete time, continuous time
Computers step in discrete time, so continuous-time processes are simulated either exactly at chosen times (Brownian increments are normal, so a path can be sampled on any grid without error) or event by event (Poisson arrivals are generated as exponential gaps). The pages state which approach they use and what it costs in accuracy.