stochastic-processes

A standalone stochastic processes laboratory: from-scratch Python implementations of random walks, Markov chains, Poisson processes, Brownian motion, martingales, and first-passage problems, each with its own tests, a verification CLI checked against numpy/scipy oracles, and an application to real EUR/USD exchange-rate data.

This repository is independent of math-implementations but follows the same standards.

Project structure

stochastic-processes/
    README.md
    data/DEXUSEU_returns.csv
    project1_random_walks/
        random_walks.py
        test_random_walks.py
        cli.py
        README.md
    project2_markov_chains/
        markov_chains.py
        test_markov_chains.py
        cli.py
        README.md
    project3_poisson_processes/
        poisson_processes.py
        test_poisson_processes.py
        cli.py
        README.md
    project4_brownian_motion/
        brownian_motion.py
        test_brownian_motion.py
        cli.py
        README.md
    project5_martingales/
        martingales.py
        test_martingales.py
        cli.py
        README.md
    project6_first_passage/
        first_passage.py
        test_first_passage.py
        cli.py
        README.md

Status

#	Project	Status
1	Random walks	Complete (58 tests, 14/14 verify checks)
2	Markov chains	Complete (55 tests, 15/15 verify checks)
3	Poisson processes	Complete (47 tests, 11/11 verify checks)
4	Brownian motion	Complete (42 tests, 16/16 verify checks)
5	Martingales	Complete (56 tests, 16/16 verify checks)
6	First-passage problems	Complete (27 tests, 12/12 verify checks)

All 6 projects complete: 285 tests, 84 verify checks, all passing. A repository-wide synthesis of what the EUR/USD experiments across all six projects add up to together is in project6_first_passage/README.md, at the bottom.

Setup

Python 3.11+. Each project's implementation needs only the standard library. The verify command in every project's CLI needs the scientific stack as an independent oracle:

pip install --break-system-packages numpy scipy

Running tests

python3 -m pytest project1_random_walks/test_random_walks.py -v

Running the oracle

cd project1_random_walks && python3 cli.py verify

The data

data/DEXUSEU_returns.csv holds daily EUR/USD exchange rates sourced from the Federal Reserve (FRED series DEXUSEU), 1999-01-04 to 2026-08-21 (6,931 price levels, 6,930 daily returns). Columns:

• date: observation date
• price: U.S. dollars per one euro
• return: simple daily return, $(P_t - P_{t-1}) / P_{t-1}$
• log_return: daily log return, $\ln(P_t / P_{t-1})$

The file was retrieved once and is committed as-is; no external fetch is needed to run any test or CLI command.

Design rules

• Standard library only in implementations. numpy and scipy are used exclusively inside each cli.py verify command as an independent oracle, never in the core mathematics.
• No hardcoded values. Every number in CLI output is computed from the data or from a formula at runtime.
• Every formula is verified against an exact analytical value, a brute-force enumeration, or a numpy/scipy oracle, with the verify command printing PASS/FAIL per check.
• One module fully complete before the next. Each project ships its implementation, tests, CLI, verify command, and README together before work starts on the following project.

What each project README covers

Every project folder documents the same things, in the same order: the underlying math and where it comes from, how the algorithm follows from that math, notes on the from-scratch implementation, what the tests check, how the verify command establishes correctness independently, what the EUR/USD experiment showed, and where the method breaks down or falls short. Derivation comes before code, and code is checked against an independent source before it gets applied to real data.