Processes as first-class algebraic objects with exact Lévy–Khintchine triplet arithmetic.

spxa — Stochastic Process Algebra

spxa is a Python library where stochastic processes are first-class algebraic objects.

from spxa.zoo.levy import VarianceGamma, BrownianMotion, GammaProcess
from spxa.zoo.beyond import FractionalBrownianMotion

X = VarianceGamma(sigma=0.2, nu=0.1, theta=-0.1)
Y = GammaProcess(a=1.0, b=2.0)

Z = X + 0.5 * Y          # exact: triplets add
W = X @ Y                 # exact: subordination via Bernstein functions

Z.triplet                 # LevyTriplet(b=..., sigma=..., nu=...)
Z.cumulants(order=4)      # exact symbolic cumulants
Z.char_func(u=1.0)        # characteristic function E[e^{iuZ_t}]
Z.__story__()             # human-readable derivation of Z's properties

Write Z = X + c*Y, get back a new process with its Lévy–Khintchine triplet computed exactly, cumulants derived symbolically, and a property lattice tracking what remains true (stationarity of increments, martingale property, tail class, self-similarity index) and what was invalidated by the operation. The library is loudly honest when you leave the Lévy regime, degrading gracefully to moment propagation rather than silently lying.

Why spxa

Every stochastic modeling library treats processes as simulation engines. spxa treats them as algebraic objects. The Lévy–Khintchine bijection — between Lévy processes and infinitely divisible distributions — means triplet arithmetic is exact for independent Lévy processes. This is a mathematical fact that no existing software exploits compositionally.

Researchers in quantitative finance, statistical physics, and computational biology currently re-derive combinations by hand each time, or simulate everything at the cost of closed-form insight. spxa closes that gap.

Core guarantees

• Every operation returns a new immutable process object — no mutation
• Every process carries an ExactnessLevel: EXACT, MOMENT_PROPAGATION, or SIMULATION_ONLY
• Operations between exactness levels degrade to the lower level with an explicit warning and mathematical reason
• The property lattice propagates automatically and never makes silent incorrect claims
• tests/exactness/ verifies closed-form results against published formulas and blocks any PR that breaks them

Installation

pip install spxa

Requires Python ≥ 3.10.

Documentation

• Theory: Lévy–Khintchine and triplet arithmetic
• Theory: Triplet arithmetic under composition
• Theory: Beyond the Lévy regime
• Tutorial: First composition
• Tutorial: Finance models
• Tutorial: Physics applications
• Process registry

Process zoo

See PROCESS_REGISTRY.md for the full table of implemented processes with exactness levels, available cumulants, simulation algorithms, and primary references.

Contributing

See CONTRIBUTING.md. Contribution tiers:

• New process to the zoo: requires triplet, cumulants, one primary citation
• New operation: requires exactness proof or explicit approximation justification with cited error bounds
• Changes to core/: requires RFC document and two maintainer approvals

License

MIT