optimiz-rs/examples/notebooks/10_bsde.ipynb

{
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  {
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   "id": "848441ca",
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   "source": [
    "# 10 — BSDE θ-scheme (linear, constant coefficients)\n",
    "\n",
    "CPU-only Crank–Nicolson scheme for linear backward stochastic\n",
    "differential equations. Doc page:\n",
    "[bsde.rst](../../docs/source/algorithms/bsde.rst).\n",
    "\n",
    "Each computational cell is sandwiched between a *PRE* markdown\n",
    "(theoretical reminder) and a *POST* markdown (expected result and\n",
    "graph reading).\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d4d08da6",
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from optimizr import _core as opt\n",
    "plt.rcParams['figure.figsize'] = (8.5, 4.5)\n",
    "plt.rcParams['figure.dpi'] = 110\n",
    "plt.rcParams['axes.grid'] = True\n",
    "plt.rcParams['grid.alpha'] = 0.3\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ad34d21a",
   "metadata": {},
   "source": [
    "## Cellule 1 — Vérification du schéma θ contre la solution analytique\n",
    "\n",
    "**Théorème (Pardoux-Peng 1990) — BSDE linéaire à coefficients constants.**\n",
    "Pour $a$, $b$, $c$ déterministes constants et terminal $\\xi$, la\n",
    "solution déterministe (lorsque $b = c = 0$) de\n",
    "$$-dY_t = (a Y_t)\\,dt - Z_t\\,dW_t,\\qquad Y_T = \\xi$$\n",
    "est $Y_t = \\xi e^{a(T-t)}$.\n",
    "\n",
    "**Équation pivot.**\n",
    "$$Y_t = \\xi\\,e^{a(T-t)}.$$\n",
    "\n",
    "**Démonstration (esquisse).** En l'absence de bruit ($b = 0$) et de\n",
    "forçage ($c = 0$), l'EDS rétrograde devient l'EDO $\\dot Y = -aY$\n",
    "intégrée en $Y_t = \\xi e^{a(T-t)}$.  $\\square$\n",
    "\n",
    "**Ce que la cellule vérifie.** Le schéma $\\theta = 0.5$ (Crank–Nicolson)\n",
    "reproduit la décroissance exponentielle pour $a = -0.3$, $T = 1$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "62662239",
   "metadata": {
    "execution": {
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     "shell.execute_reply": "2026-05-12T14:05:13.155633Z"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Y0 numérique  = 0.740818\n",
      "Y0 analytique = 0.740818\n",
      "Erreur max sur la grille : 4.17e-08\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 935x495 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rho, T, n = 0.3, 1.0, 200\n",
    "res = opt.linear_bsde_constant_coeffs(-rho, 0.0, 0.0, 1.0, n, T, 0.5)\n",
    "tg = np.array(res['time_grid'])\n",
    "yg = np.array(res['y'])\n",
    "analytic = np.exp(-rho * (T - tg))\n",
    "err = float(np.max(np.abs(yg - analytic)))\n",
    "print(f\"Y0 numérique  = {yg[0]:.6f}\")\n",
    "print(f\"Y0 analytique = {analytic[0]:.6f}\")\n",
    "print(f\"Erreur max sur la grille : {err:.2e}\")\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(tg, yg, lw=2, label='θ-scheme')\n",
    "ax.plot(tg, analytic, '--', lw=1.5, label=r'$\\xi e^{-\\rho(T-t)}$')\n",
    "ax.set_xlabel('t'); ax.set_ylabel(r'$Y_t$')\n",
    "ax.set_title(\"BSDE linéaire — Crank–Nicolson vs analytique\")\n",
    "ax.legend()\n",
    "fig.tight_layout(); plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a06e1e55",
   "metadata": {},
   "source": [
    "**Résultat attendu.** $Y_0 \\approx 0.7408$, erreur max $< 10^{-3}$.\n",
    "\n",
    "**Lecture du graphique.** Les deux courbes se superposent visuellement.\n",
    "\n",
    "**Conclusion.** Le primitive `linear_bsde_constant_coeffs` est calibré\n",
    "sur le test analytique exponentiel ; il est utilisé comme brique pour\n",
    "les BSDE non-linéaires (cellule 3).\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "54ed7d4f",
   "metadata": {},
   "source": [
    "## Cellule 2 — Étude de convergence en $\\Delta t$\n",
    "\n",
    "**Théorème (Crank–Nicolson, ordre 2).** Pour une EDO linéaire\n",
    "$\\dot y = -a y$, le schéma $\\theta = 1/2$ vérifie\n",
    "$|y_n - y(t_n)| = O(\\Delta t^2)$.\n",
    "\n",
    "**Équation pivot.**\n",
    "$$\\log\\,\\text{erreur}(n) \\;\\approx\\; -2\\log n + C.$$\n",
    "\n",
    "**Ce que la cellule vérifie.** On trace $|Y_0^{(n)} - e^{-\\rho T}|$\n",
    "en log-log et on compare à la pente $-2$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "025c2ffb",
   "metadata": {
    "execution": {
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     "shell.execute_reply": "2026-05-12T14:05:13.729675Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n =   25  ->  erreur = 2.667e-06\n",
      "n =   50  ->  erreur = 6.667e-07\n",
      "n =  100  ->  erreur = 1.667e-07\n",
      "n =  200  ->  erreur = 4.167e-08\n",
      "n =  400  ->  erreur = 1.042e-08\n",
      "n =  800  ->  erreur = 2.604e-09\n"
     ]
    },
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 935x495 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ns = [25, 50, 100, 200, 400, 800]\n",
    "errs = []\n",
    "for n in ns:\n",
    "    r = opt.linear_bsde_constant_coeffs(-rho, 0.0, 0.0, 1.0, n, T, 0.5)\n",
    "    errs.append(abs(r['y'][0] - np.exp(-rho * T)))\n",
    "for n, e in zip(ns, errs):\n",
    "    print(f\"n = {n:4d}  ->  erreur = {e:.3e}\")\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.loglog(ns, errs, 'o-', lw=2, label='erreur empirique')\n",
    "ax.loglog(ns, [errs[0] * (ns[0] / n) ** 2 for n in ns], ':',\n",
    "          label=r'pente $-2$ (référence)')\n",
    "ax.set_xlabel('n_steps'); ax.set_ylabel(r'$|Y_0 - e^{-\\rho T}|$')\n",
    "ax.set_title(\"Convergence de Crank–Nicolson\")\n",
    "ax.legend()\n",
    "fig.tight_layout(); plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2586561",
   "metadata": {},
   "source": [
    "**Résultat attendu.** Les points s'alignent sur la pente $-2$.\n",
    "\n",
    "**Lecture du graphique.** Parallélisme avec la droite pointillée.\n",
    "\n",
    "**Conclusion.** Précision $10^{-6}$ atteinte avec $n \\approx 800$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f1586c1a",
   "metadata": {},
   "source": [
    "## Cellule 3 — Exemple concret : actualisation d'une espérance terminale\n",
    "\n",
    "**Modèle utilisé.** Pour un brownien $W_t$, la valeur actualisée\n",
    "$$Y_t = \\mathbb{E}\\!\\left[e^{-\\rho(T-t)}\\,W_T^2\\,\\big|\\,\\mathcal{F}_t\\right]$$\n",
    "satisfait l'EDP de Feynman–Kac\n",
    "$\\partial_t u + \\tfrac{1}{2}\\partial_{xx} u - \\rho u = 0$,\n",
    "$u(T, x) = x^2$.\n",
    "\n",
    "**Équation pivot (cas déterministe).** $\\mathbb{E}[W_T^2] = T$, donc\n",
    "$$Y_0 = e^{-\\rho T}\\,T.$$\n",
    "\n",
    "**Ce que la cellule vérifie.** On compare le primitive `linear_bsde`\n",
    "(avec terminal $\\xi = T$ — espérance déterministe de $W_T^2$) à une\n",
    "simulation Monte Carlo de $10^4$ trajectoires.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "dcfe4109",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2026-05-12T14:05:13.735165Z",
     "iopub.status.busy": "2026-05-12T14:05:13.734851Z",
     "iopub.status.idle": "2026-05-12T14:05:14.271954Z",
     "shell.execute_reply": "2026-05-12T14:05:14.270549Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Monte Carlo (M=10000) : Y0 = 0.738001\n",
      "BSDE primitive      : Y0 = 0.740818\n",
      "Écart relatif       : 0.38%\n"
     ]
    },
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 1210x440 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rng = np.random.default_rng(0)\n",
    "M = 10_000\n",
    "W_T = rng.standard_normal(M) * np.sqrt(T)\n",
    "mc_value = np.exp(-rho * T) * float(np.mean(W_T ** 2))\n",
    "\n",
    "# valeur déterministe via le primitive (xi = E[W_T^2] = T)\n",
    "res = opt.linear_bsde_constant_coeffs(-rho, 0.0, 0.0, T, n, T, 0.5)\n",
    "y0_pde = float(res['y'][0])\n",
    "\n",
    "print(f\"Monte Carlo (M={M}) : Y0 = {mc_value:.6f}\")\n",
    "print(f\"BSDE primitive      : Y0 = {y0_pde:.6f}\")\n",
    "print(f\"Écart relatif       : {abs(y0_pde - mc_value)/mc_value:.2%}\")\n",
    "\n",
    "ts = np.linspace(0, T, 50)\n",
    "paths = np.cumsum(rng.standard_normal((20, len(ts))) *\n",
    "                  np.sqrt(T / len(ts)), axis=1)\n",
    "fig, axes = plt.subplots(1, 2, figsize=(11, 4))\n",
    "for p in paths:\n",
    "    axes[0].plot(ts, p, alpha=0.5)\n",
    "axes[0].set_title(\"20 trajectoires browniennes\")\n",
    "axes[0].set_xlabel('t'); axes[0].set_ylabel(r'$W_t$')\n",
    "\n",
    "tg = np.array(res['time_grid'])\n",
    "yg = np.array(res['y'])\n",
    "axes[1].plot(tg, yg, lw=2, color='C3', label='BSDE')\n",
    "axes[1].axhline(mc_value, ls='--', color='C0',\n",
    "                label=f'Monte Carlo Y0 = {mc_value:.3f}')\n",
    "axes[1].set_title(\"Valeur actualisée déterministe\")\n",
    "axes[1].set_xlabel('t'); axes[1].set_ylabel(r'$Y_t$')\n",
    "axes[1].legend()\n",
    "fig.tight_layout(); plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "10290d98",
   "metadata": {},
   "source": [
    "**Résultat attendu.** $Y_0 = T \\cdot e^{-\\rho T} \\approx 0.7408$,\n",
    "estimateur Monte Carlo à moins de 3 %.\n",
    "\n",
    "**Lecture du graphique.** Faisceau brownien à gauche ; valeur actualisée\n",
    "et niveau Monte Carlo à droite — ils coïncident.\n",
    "\n",
    "**Conclusion.** La primitive BSDE évite la simulation Monte Carlo\n",
    "quand le terminal est une fonctionnelle simple du brownien.\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "rhftlab",
   "language": "python",
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