ThotDjehuty
dd51156174
Each of the eight v2.0 companion notebooks (10_bsde through 17_generative_calibration) now follows the mandatory pedagogical sandwich structure: PRE markdown : theorem / model / pivot equation / what the cell verifies CODE cell : labelled prints + at least one matplotlib figure POST markdown: expected result, graph reading, conclusion Each notebook carries at least one concrete real-world example (heat plate, inverted pendulum, opinion polarization, collective decision, OU drift under Cauchy noise, mixture vs gaussian MMD, etc.) Generator script: scripts/enrich_v2_notebooks.py Doc plots refreshed via scripts/inject_doc_plots.py.
124 lines
3.5 KiB
ReStructuredText
124 lines
3.5 KiB
ReStructuredText
BSDE — θ-scheme and deep-BSDE bridge
|
||
====================================
|
||
|
||
This notebook exercises `optimizr.linear_bsde_constant_coeffs`, the Crank–Nicolson θ-scheme for the BSDE
|
||
`-dY = (a Y + b Z + c) dt - Z dW` with constant coefficients, and verifies the discrete trajectory against the analytic solution `Y_t = exp(-ρ (T - t))`.
|
||
|
||
.. note:: Companion executed notebook: `10_bsde.ipynb <../../examples/notebooks/10_bsde.ipynb>`_
|
||
|
||
10 — BSDE θ-scheme
|
||
==================
|
||
|
||
Generic CPU-only Crank–Nicolson scheme for linear backward stochastic differential equations. Reference doc page: [bsde.rst](../../docs/source/algorithms/bsde.rst).
|
||
|
||
.. code-block:: python
|
||
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from optimizr import _core as opt
|
||
plt.rcParams['figure.figsize'] = (7, 4)
|
||
plt.rcParams['figure.dpi'] = 110
|
||
|
||
Exponential ground-truth check
|
||
------------------------------
|
||
|
||
With $a(t) \equiv -\rho$, $b = c = 0$ and $Y_T = 1$ the analytic deterministic solution is $Y_t = e^{-\rho (T-t)}$.
|
||
|
||
.. code-block:: python
|
||
|
||
rho = 0.3
|
||
T = 1.0
|
||
res = opt.linear_bsde_constant_coeffs(
|
||
a_const=-rho, b_const=0.0, c_const=0.0,
|
||
terminal=1.0, n_steps=200, t_horizon=T, theta=0.5,
|
||
)
|
||
tg = np.array(res['time_grid'])
|
||
yg = np.array(res['y'])
|
||
analytic = np.exp(-rho * (T - tg))
|
||
print('Y0 =', yg[0], ' exp(-rho T) =', analytic[0])
|
||
print('max abs error =', float(np.max(np.abs(yg - analytic))))
|
||
|
||
.. code-block:: python
|
||
|
||
fig, ax = plt.subplots()
|
||
ax.plot(tg, yg, label='θ-scheme', lw=2)
|
||
ax.plot(tg, analytic, '--', label='analytic exp(-ρ(T-t))')
|
||
ax.set_xlabel('t'); ax.set_ylabel('Y_t')
|
||
ax.set_title('Linear BSDE — Crank–Nicolson vs analytic')
|
||
ax.legend(); ax.grid(alpha=0.3)
|
||
fig.tight_layout(); plt.show()
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
.. AUTO-PLOT-BEGIN
|
||
.. image:: ../_static/auto/algorithms__bsde/block_03_fig_01.png
|
||
:align: center
|
||
:width: 80%
|
||
|
||
.. AUTO-PLOT-END
|
||
.. image:: ../_static/v2/bsde/plot_01.png
|
||
:align: center
|
||
:width: 80%
|
||
|
||
Convergence rate study
|
||
----------------------
|
||
|
||
Crank–Nicolson is second-order in `Δt`.
|
||
|
||
.. code-block:: python
|
||
|
||
errs = []
|
||
ns = [25, 50, 100, 200, 400, 800]
|
||
for n in ns:
|
||
r = opt.linear_bsde_constant_coeffs(-rho, 0.0, 0.0, 1.0, n, T, 0.5)
|
||
errs.append(abs(r['y'][0] - np.exp(-rho * T)))
|
||
print(list(zip(ns, errs)))
|
||
|
||
.. code-block:: python
|
||
|
||
fig, ax = plt.subplots()
|
||
ax.loglog(ns, errs, 'o-')
|
||
ax.loglog(ns, [errs[0] * (ns[0] / n) ** 2 for n in ns],
|
||
':', label='O(Δt²) reference')
|
||
ax.set_xlabel('n_steps'); ax.set_ylabel('|Y0 − analytic|')
|
||
ax.set_title('Crank–Nicolson convergence'); ax.grid(which='both', alpha=0.3); ax.legend()
|
||
fig.tight_layout(); plt.show()
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
.. AUTO-PLOT-BEGIN
|
||
.. image:: ../_static/auto/algorithms__bsde/block_05_fig_01.png
|
||
:align: center
|
||
:width: 80%
|
||
|
||
.. AUTO-PLOT-END
|
||
.. image:: ../_static/v2/bsde/plot_02.png
|
||
:align: center
|
||
:width: 80%
|
||
|
||
**Verified against analytic ground truth:** `Y_t = exp(-ρ (T - t))` — relative error at `t = 0` below `1e-3` for `n_steps = 200`.
|
||
|
||
API
|
||
---
|
||
|
||
.. code-block:: rust
|
||
|
||
pub fn solve_linear_bsde<A, B, C>(
|
||
a: A, b: B, c: C, terminal: f64, cfg: &ThetaSchemeConfig
|
||
) -> Result<ThetaSchemeResult>
|
||
where A: Fn(f64) -> f64, B: Fn(f64) -> f64, C: Fn(f64) -> f64;
|
||
|
||
pub struct ThetaSchemeConfig { pub n_steps: usize, pub t_horizon: f64, pub theta: f64 }
|
||
pub struct ThetaSchemeResult { pub y: Array1<f64>, pub z: Array1<f64>, pub time_grid: Array1<f64> }
|
||
|
||
pub trait ConditionalExpectation { /* deep-BSDE bridge */ }
|
||
pub struct DeepBsdeBridge { /* ... */ }
|