100 lines
3.3 KiB
ReStructuredText
100 lines
3.3 KiB
ReStructuredText
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BSDE — θ-scheme and deep-BSDE bridge
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====================================
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This notebook exercises `optimizr.linear_bsde_constant_coeffs`, the Crank–Nicolson θ-scheme for the BSDE
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`-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))`.
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.. note:: Companion executed notebook: `10_bsde.ipynb <../../examples/notebooks/10_bsde.ipynb>`_
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10 — BSDE θ-scheme
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==================
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Generic CPU-only Crank–Nicolson scheme for linear backward stochastic differential equations. Reference doc page: [bsde.rst](../../docs/source/algorithms/bsde.rst).
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.. code-block:: python
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import numpy as np
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import matplotlib.pyplot as plt
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from optimizr import _core as opt
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plt.rcParams['figure.figsize'] = (7, 4)
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plt.rcParams['figure.dpi'] = 110
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Exponential ground-truth check
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------------------------------
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With $a(t) \equiv -\rho$, $b = c = 0$ and $Y_T = 1$ the analytic deterministic solution is $Y_t = e^{-\rho (T-t)}$.
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.. code-block:: python
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rho = 0.3
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T = 1.0
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res = opt.linear_bsde_constant_coeffs(
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a_const=-rho, b_const=0.0, c_const=0.0,
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terminal=1.0, n_steps=200, t_horizon=T, theta=0.5,
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)
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tg = np.array(res['time_grid'])
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yg = np.array(res['y'])
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analytic = np.exp(-rho * (T - tg))
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print('Y0 =', yg[0], ' exp(-rho T) =', analytic[0])
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print('max abs error =', float(np.max(np.abs(yg - analytic))))
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.. code-block:: python
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fig, ax = plt.subplots()
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ax.plot(tg, yg, label='θ-scheme', lw=2)
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ax.plot(tg, analytic, '--', label='analytic exp(-ρ(T-t))')
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ax.set_xlabel('t'); ax.set_ylabel('Y_t')
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ax.set_title('Linear BSDE — Crank–Nicolson vs analytic')
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ax.legend(); ax.grid(alpha=0.3)
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fig.tight_layout(); plt.show()
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.. image:: ../_static/v2/bsde/plot_01.png
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:align: center
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:width: 80%
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Convergence rate study
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----------------------
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Crank–Nicolson is second-order in `Δt`.
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.. code-block:: python
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errs = []
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ns = [25, 50, 100, 200, 400, 800]
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for n in ns:
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r = opt.linear_bsde_constant_coeffs(-rho, 0.0, 0.0, 1.0, n, T, 0.5)
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errs.append(abs(r['y'][0] - np.exp(-rho * T)))
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print(list(zip(ns, errs)))
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.. code-block:: python
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fig, ax = plt.subplots()
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ax.loglog(ns, errs, 'o-')
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ax.loglog(ns, [errs[0] * (ns[0] / n) ** 2 for n in ns],
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':', label='O(Δt²) reference')
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ax.set_xlabel('n_steps'); ax.set_ylabel('|Y0 − analytic|')
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ax.set_title('Crank–Nicolson convergence'); ax.grid(which='both', alpha=0.3); ax.legend()
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fig.tight_layout(); plt.show()
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.. image:: ../_static/v2/bsde/plot_02.png
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:align: center
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:width: 80%
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**Verified against analytic ground truth:** `Y_t = exp(-ρ (T - t))` — relative error at `t = 0` below `1e-3` for `n_steps = 200`.
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API
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---
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.. code-block:: rust
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pub fn solve_linear_bsde<A, B, C>(
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a: A, b: B, c: C, terminal: f64, cfg: &ThetaSchemeConfig
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) -> Result<ThetaSchemeResult>
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where A: Fn(f64) -> f64, B: Fn(f64) -> f64, C: Fn(f64) -> f64;
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pub struct ThetaSchemeConfig { pub n_steps: usize, pub t_horizon: f64, pub theta: f64 }
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pub struct ThetaSchemeResult { pub y: Array1<f64>, pub z: Array1<f64>, pub time_grid: Array1<f64> }
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pub trait ConditionalExpectation { /* deep-BSDE bridge */ }
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pub struct DeepBsdeBridge { /* ... */ }
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