Adaptive Barrier Monitor

A five-notebook quantitative-finance project connecting random walks, Brownian motion, geometric Brownian motion, first-passage times, Brownian bridges, and state-dependent monitoring.

The motivating question is:

A stock is monitored for a large move over a short window. Continuous polling is expensive. What probability model describes a hidden barrier crossing, and how can that model inform a sampling schedule?

The project is written for a mathematically mature reader who wants to see how Gaussian processes, conditioning, stochastic calculus, and Monte Carlo methods appear in a practical monitoring problem.

Core results and scope

Under geometric Brownian motion,

\frac{dS_t}{S_t}=\mu\,dt+\sigma\,dW_t,

the relative log-price X_t=\log(S_t/S_0) is arithmetic Brownian motion. A 10% drop corresponds to the lower log barrier B=\log(0.9).

For zero drift, the probability of touching the barrier by time T is

P(\tau_B\leq T)=2\Phi\!\left(\frac{B}{\sigma\sqrt{T}}\right).

At 30% annualised volatility, a 10% move in five trading minutes is roughly a 49-standard-deviation diffusion event. Pure GBM therefore assigns it probability below ordinary floating-point resolution; jumps and market microstructure are essential for realistic extreme-move modelling.

Conditional on two observations x_0,x_T>B, the Brownian-bridge probability that the hidden path crossed the barrier is

P_{\mathrm{cross}}
=\exp\!\left(
-\frac{2(x_0-B)(x_T-B)}{\sigma^2\Delta t}
\right).

If both endpoint distances are set equal to D, inversion gives

\Delta t_{\mathrm{sym}}
=\frac{2D^2}{\sigma^2\log(1/\varepsilon)}.

This inversion is exact conditional on both endpoints being known. In a live scheduler, the future endpoint is unknown; the implementation substitutes the current distance for both endpoints. Thus \varepsilon is a local diffusion-design parameter, not an unconditional miss guarantee, and it does not control jumps. A hard maximum polling interval remains necessary.

Notebooks

#	Notebook	Main topics
01	Random walks to Brownian motion	Log returns, the \min(s,t) covariance kernel, Cholesky sampling, Brownian scaling
02	GBM and Itô's lemma	Multiplicative prices, exact GBM simulation, Itô correction
03	First passage and reflection	Reflection principle, Bachelier–Lévy formula, hitting-time diagnostics
04	Brownian bridges and hidden crossings	Gaussian conditioning, Schur complements, bridge crossing probabilities
05	Adaptive barrier monitoring	Unit-consistent scheduler, practical polling cap, controlled jump stress test, model-risk discussion

The analytical formulae are checked against Monte Carlo simulation in the notebooks.

Interactive web application

The FastAPI/Plotly demo compares two sampling schedules on the same simulated paths. The adaptive schedule uses the local symmetric-endpoint bridge proxy, while the fixed baseline can run in either of two modes:

• Equal budget: the fixed schedule receives exactly the adaptive schedule's sample count on each path, isolating where observations are placed.
• Fixed cadence: the fixed schedule samples at a user-selected interval, so detection quality, lag, and total observation cost can be compared directly.

A barrier event counts as detected only if a sampled point remains beyond the barrier within a configurable number of simulation steps. The comparison is therefore explicit and reproducible rather than based on an unrestricted "eventually detected" definition.

The demo supports GBM and an optional Merton jump-diffusion stress mode. When jumps are enabled, the interface explicitly warns that the Brownian diffusion parameter \varepsilon does not bound jump-event misses.

Run locally

python -m venv .venv
source .venv/bin/activate
python -m pip install -e ".[webapp]"
python -m uvicorn webapp.app:app --host 127.0.0.1 --port 8055

Open http://127.0.0.1:8055.

Docker

docker compose -f docker-compose.webapp.yml up --build

For an existing Caddy Docker network:

docker compose -f docker-compose.webapp.proxy.yml up --build -d

The container runs as a non-root user and includes an HTTP health check.

Run the notebooks

python -m venv .venv
source .venv/bin/activate
pip install -e ".[notebooks]"
jupyter lab notebooks/

Notebooks that request market data cache successful downloads under data/cache/. Their analytical and simulation sections remain usable when the network fetch is unavailable.

Tests

pip install -e ".[dev,webapp]"
pytest

The test suite covers:

• inversion of the Brownian-bridge formula;
• vectorised interval calculations and input validation;
• consistent time/volatility units in the adaptive schedule;
• enforcement of the detection deadline;
• exact per-path sample-budget equality;
• equivalence of zero-intensity jump diffusion and GBM;
• aggregate simulation invariants.

Project structure

adaptive-barrier-monitor/
├── notebooks/                    # five executed research notebooks
├── src/adaptive_barrier/
│   ├── __init__.py
│   └── engine.py                 # samplers, closed forms, scheduler, evaluation
├── tests/
│   └── test_engine.py
├── webapp/
│   ├── app.py                    # FastAPI API
│   ├── Dockerfile
│   └── static/                   # vanilla JS, Plotly, CSS
├── .github/workflows/tests.yml
├── pyproject.toml
├── requirements.txt
├── requirements-webapp.txt
├── requirements-dev.txt
├── bibliography.md
├── LICENSE
└── webapp.md

Model limitations

• Online endpoint uncertainty: the bridge crossing formula is conditional on both endpoints; the scheduler uses a local approximation before the next endpoint exists.
• Jump risk: diffusion-derived polling cannot guarantee detection of sudden jump-and-recovery events.
• No market microstructure model: bid–ask bounce, asynchronous feeds, exchange halts, queueing, and packet latency are not represented.
• Simulation-grid dependence: the web demo's detection deadline is measured in simulated grid steps; changing n_steps changes its physical duration.
• Educational calibration: jump parameters in the sandbox are user-controlled stress parameters, not production estimates.

Tech stack

Python, NumPy, SciPy, pandas, SymPy, Matplotlib, FastAPI, Pydantic, Uvicorn, Plotly.js, Docker, pytest, and GitHub Actions.

License

MIT — see LICENSE.