95 lines
2.7 KiB
Plaintext
95 lines
2.7 KiB
Plaintext
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# Optimal Control
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Hamilton–Jacobi–Bellman (HJB) solvers, regime-switching thresholds, OU parameter estimation, and Kalman filtering utilities backed by Rust.
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## HJB switching boundaries (OU process)
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```python
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from optimizr import solve_hjb_py, solve_hjb_full_py
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lower, upper, residual, iters = solve_hjb_py(
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kappa=3.0,
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theta=0.0,
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sigma=0.2,
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rho=0.04,
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transaction_cost=0.001,
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n_points=400,
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max_iter=4000,
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tolerance=1e-7,
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n_std=5.0,
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)
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print(f"bounds=({lower:.3f}, {upper:.3f}), residual={residual:.2e}, iters={iters}")
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```
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- Model: $dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t$ with quadratic transaction costs.
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- Output: optimal buy/sell thresholds; `solve_hjb_full_py` also returns $V, V_x, V_{xx}$ for diagnostics.
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- Diagnostics: plot $V_x$ for smoothness near thresholds; monitor `residual` and increase `max_iter` if not converged.
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## Backtesting optimal switching
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```python
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from optimizr import backtest_optimal_switching_py
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metrics = backtest_optimal_switching_py(
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spread=spread,
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lower_bound=lower,
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upper_bound=upper,
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transaction_cost=0.001,
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)
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(
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total_return,
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sharpe,
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max_dd,
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n_trades,
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win_rate,
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pnl_path,
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) = metrics
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```
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Inspect `win_rate` vs `max_dd` to tune aggressiveness; combine with HMM regimes for state-aware controls.
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## OU parameter estimation
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```python
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import numpy as np
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from optimizr import estimate_ou_params_py
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spread = np.random.randn(10_000)
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kappa, theta, sigma, half_life = estimate_ou_params_py(spread, dt=1/252)
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```
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Method-of-moments / MLE fit returns $(\kappa, \theta, \sigma, \text{half-life})$. Use a few thousand samples for stability; winsorize heavy tails if needed.
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## Kalman filtering (linear, EKF, UKF)
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```python
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import numpy as np
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from optimizr import LinearKalmanFilter
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F = [[1.0, 1.0], [0.0, 1.0]]
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H = [[1.0, 0.0]]
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Q = [[1e-4, 0.0], [0.0, 1e-4]]
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R = [[1e-2]]
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kf = LinearKalmanFilter(
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f_matrix=F,
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h_matrix=H,
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q_matrix=Q,
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r_matrix=R,
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initial_state=[0.0, 0.0],
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initial_covariance=[[1.0, 0.0], [0.0, 1.0]],
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)
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kf.predict(control=[0.0, 0.0])
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kf.update(observation=[1.2])
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state = kf.get_state()
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```
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- Interfaces: `LinearKalmanFilter`, `UnscentedKalmanFilter`, and `KalmanState` for batch `filter` and smoothing.
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- Concept: prediction (dynamics prior) + correction (measurement residual); RTS smoother refines past states.
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## Practical notes
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- Rust backend (`optimizr._core`) must be present for control utilities; install from source if wheels are unavailable.
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- Grids: for HJB, `n_points≈400` is stable; widen `n_std` for volatile spreads.
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- Combine with Mean Field Games: see `mean_field_games.md` for population dynamics; use Kalman estimates as control inputs if needed.
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