Stochastic processes §
A stochastic process is a family of random variables $ indexed by time: instead of asking “what is the probability of an outcome?”, it asks “how does a random quantity evolve?”. It is the natural language for queues, population dynamics, financial prices, signal noise, and physical diffusion.
Brief history §
- Bachelier (1900) and Einstein (1905) modeled Brownian motion, five years before Perrin’s experiments confirmed it.
- Kolmogorov (1931) introduced Markov processes and the Chapman–Kolmogorov equations.
- Doob (1940s–1953) developed martingale theory and the optional stopping theorem.
- Itô (1944) built stochastic calculus, making differential equations with noise solvable.
Classes §
The classes below are organized along two first-principles axes: the time index (discrete vs continuous) and the path behavior (counting jumps, continuous diffusion, or both). Two unifying families cut across the process families: Lévy processes generalize the increment structure, and Gaussian processes characterize everything through the covariance kernel. The martingale property is not a family of processes but a structural property a process from any family may have: Brownian motion is a martingale, and the compensated Poisson process $ is a martingale.
| Page | Class | Nature | What it studies | Key objects |
|---|---|---|---|---|
| Markov chains | Markov chains | Process family (discrete time, memoryless) | Step-by-step evolution where the future depends on the present state only | Transition matrix, stationary distribution, hitting times |
| Poisson processes | Poisson processes | Process family (continuous time, counting) | Counts of events occurring independently at a constant rate | Arrival times, inter-arrivals, thinning, superposition |
| Brownian motion | Brownian motion and diffusions | Process family (continuous time, continuous paths) | The scaling limit of random walks; continuous random paths | Wiener process, quadratic variation, Itô calculus, SDEs |
| Lévy processes | Lévy processes | Unifying family (drift + diffusion + jumps) | Processes with stationary independent increments | Lévy–Itô decomposition, Lévy–Khintchine triplet, subordinators |
| Martingales | Martingales | Structural property (cross-cutting) | Fair games: the conditional expectation of the future equals the present | Optional stopping, Doob's inequalities, convergence theorems |
| Gaussian processes | Gaussian processes | Unifying family (all finite-dimensional laws Gaussian) | Processes fully specified by a mean and a covariance kernel; includes spatial random fields | Covariance kernels, kriging, GP regression |
Details
The classes answer different questions and deliberately overlap:
- Markov chains ask “what does the process remember?” Answer: only the present state. The Poisson process is a Markov process in continuous time (its future depends only on the current count), and Brownian motion is a Markov process with continuous paths.
- Poisson processes ask “how do events accumulate?” They are the pure-jump members of the Lévy family: the counting process $ jumps by one at each arrival, and the inter-arrival times are independent exponentials.
- Brownian motion asks “what do continuous random paths look like?” It is the scaling limit of random walks, has continuous but nowhere-differentiable paths, and is both a Markov process and a martingale.
- Lévy processes unify the above: they are exactly the processes with stationary independent increments, and every Lévy process decomposes into a drift, a Brownian part, and a jump part.
- Martingales are a structural property, not a competing family: a fair game has no predictable trend. Brownian motion and the compensated Poisson process $ are both martingales, which is why martingale tools apply across all the families above.
- Gaussian processes are characterized entirely by their mean and covariance functions; Brownian motion is the Gaussian process with covariance $, and the same framework covers spatial random fields indexed by location rather than time.
All of these build on Random variables and expectation, Conditional probability, and Limit theorems.