Stochastic Processes & Martingale Theory

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Abstractions about the formal machinery of stochastic processes, covering martingales and filtrations (Doob martingale, local martingale, natural filtration, stopping times), point-process and state-estimation models (Cox processes, filtering problems, projection filters), and structural properties like stationarity and ergodicity.

19 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Compound Poisson process — A jump process formed by summing independent random jump sizes at event times of a Poisson counting process.
• Continuous-time stochastic process — A collection of random variables indexed by a continuous parameter set, usually a real time interval, without implying that its sample paths are continuous.
• Cox process — A point process that is conditionally Poisson given a random intensity measure, thereby representing clustered or environment-driven event rates.
• Doob martingale — Track the evolving conditional expectation E[Y|F_t] of an integrable target as a filtration reveals information, producing a martingale of progressively refined best predictions.
• Ergodic process — A stochastic process for which specified long-run time averages along almost every realization equal the corresponding ensemble expectations, allowing one sufficiently long trajectory to represent the regime.
• Event (probability theory) — A measurable subset of a sample space representing the collection of outcomes for which a probabilistic proposition holds.
• Fano factor — A count-dispersion ratio equal to variance divided by mean, with one marking a Poisson baseline.
• Filtering problem (stochastic processes) — The sequential inference problem of estimating a hidden stochastic state from noisy partial observations available up to the present time.
• Fourier number — A dimensionless elapsed time for diffusion, equal to diffusivity times time divided by the square of a characteristic length.
• Local martingale — A stochastic process that becomes a martingale when stopped along an increasing sequence of stopping times tending to the time horizon.
• Martingale (probability theory) — Model an adapted integrable stochastic process whose conditional expected future value, given present information, equals its current value.
• Natural filtration — The smallest filtration that makes a given stochastic process adapted by recording exactly the events observable from its history up to each time.
• Palm calculus — The probability calculus relating a stationary point process as seen from a typical event to its ordinary time- or space-average law.
• Progressively measurable process — A stochastic process whose restriction through every time t is jointly measurable with respect to Borel time and the information available by t.
• Projection filters — Nonlinear state-estimation algorithms that approximate an evolving conditional probability density by projecting infinite-dimensional filtering dynamics onto a finite-dimensional statistical manifold.
• Stationary process — A stochastic process whose probabilistic law is invariant under shifts of its time index, with weaker forms preserving selected moments instead.
• Stochastic drift — The systematic time-directed component of a stochastic process, commonly represented by the conditional mean rate of change apart from random fluctuation.
• Stopping time — A random time whose occurrence can be determined from information available up to that time, without access to future states of the stochastic process.
• Σ-Algebra of τ-past — The stopped sigma-algebra Fτ containing exactly those events whose truth is knowable by a stopping time τ, defined by compatibility of each event with {τ≤t} and the filtration Ft.