{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "$\\newcommand\\E{{\\mathbf E}}$\n", "$\\newcommand\\indi[1]{{\\mathbf 1}_{\\displaystyle #1}}$\n", "$\\newcommand\\inde[1]{{\\mathbf 1}_{\\displaystyle\\left\\{ #1 \\right\\}}}$\n", "$\\newcommand{\\ind}{\\inde}$\n", "$\\newcommand{\\N}{{\\mathbb N}}$\n", "$\\newcommand{\\P}{{\\mathbb P}}$\n", "$\\newcommand{\\R}{{\\mathbb R}}$\n", "$\\newcommand{\\Z}{{\\mathbb Z}}$\n", "$\\newcommand{\\Var}{{\\mathbf Var}}$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Exemples d'utilisation de l'algorithme de Robbins et Monro\n", "## Bernard Lapeyre\n", "### Novembre 2019\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# La loi forte comme un algorithme stochastique" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On considère $(G_n,n\\geq 0)$ une suite de gaussiennes centrées réduites\n", "indépendantes. On pose $M_n=(G_1+\\cdots+G_n)/n$\n", "\n", "__Question 1.__ Vérifier que \n", " $$\n", " M_{n+1} = M_n -\\gamma_n \\left(M_n - G_{n+1}\\right),\n", " $$\n", " où $\\gamma_n=1/(n+1)$. \n", " \n", " Interpréter cette équation comme un\n", " algorithme stochastique. \n", " \n", " Vérifier que l'on peut aussi prendre\n", " $\\gamma_n=C /(n+1)^\\beta$, si $1/2<\\beta\\leq 1$. Etudier informatiquement la convergence\n", " pour $\\beta=0.75$ et $C=10$, $C=1$, $C=1/10$, $C=1/100$. Que se passe t'il lorsque\n", " $C$ est (trop) petit ?\n" ] }, { "cell_type": "code", "execution_count": 196, "metadata": {}, "outputs": [], "source": [ "import numpy as np;\n", "import math;\n", "import scipy.stats as stats;\n", "import random;\n", "import matplotlib.pyplot as plt;\n", "\n", "def gamma(n,C,beta):\n", " return C / ((n+1)**beta);\n", "\n", "def algorithme_rm(x_0,n,C,beta,F,G):\n", " X=np.zeros(n)\n", " X[0] = x_0\n", " for k in range(n-1):\n", " A_FAIRE\n", " return X" ] }, { "cell_type": "code", "execution_count": 197, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Dans le cas de la loi forte, F(x,g)= (x-g) et f(x)=x-E(g)=x.\n", "def F(x,g):\n", " return x - g\n", "\n", "n=10000;\n", "\n", "# Beaucoup de variance, la condition initiale est vite oubliee \n", "C=10;\n", "beta=0.75;\n", "x_0=1\n", "G=np.random.normal(size=n);\n", "X=algorithme_rm(x_0,n,C,beta,F,G);\n", "plt.plot(X);\n", "plt.plot(np.zeros(n));plt.ylim(-0.25,+1);plt.show()\n" ] }, { "cell_type": "code", "execution_count": 198, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Trés peu de variance, mais la condition initiale est moins vite oubliee \n", "A_FAIRE\n", "X=...\n", "plt.plot(X);\n", "plt.plot(np.zeros(n));plt.ylim(-0.25,+1);plt.show()" ] }, { "cell_type": "code", "execution_count": 199, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Pas de variance, mais la condition initiale \n", "# a beaucoup de mal a etre oubliee !\n", "A_FAIRE\n", "X=...\n", "plt.plot(X);\n", "plt.plot(np.zeros(n));plt.ylim(-0.25,+1);plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "L'optimum est obtenu pour $\\beta=1$ et $C=1$ (La loi forte habituelle). Mais les performances se dégrade lorsque\n", "l'on prend $C$ trop petit." ] }, { "cell_type": "code", "execution_count": 200, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Le cas optimum.\n", "A_FAIRE\n", "X=...\n", "plt.plot(X);\n", "plt.plot(np.zeros(n));plt.ylim(-0.25,+1);plt.show()" ] }, { "cell_type": "code", "execution_count": 201, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# C est trop petit. Ca marche pas du tout.\n", "A_FAIRE\n", "X=...\n", "plt.plot(X)\n", "plt.plot(np.zeros(n));plt.ylim(-0.25,+1);plt.show()" ] }, { "cell_type": "code", "execution_count": 202, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Il vaut mieux avoir C trop grand que trop petit.\n", "A_FAIRE\n", "X=...\n", "plt.plot(X);\n", "plt.plot(np.zeros(n));plt.ylim(-0.25,+1);plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Question 2.0__ Le but est de vérifier le TCL pour les algorithme stochastiques. Lorsque $C=1$ et $\\beta=1$, il s'agit dans ce cas du TCL \"classique\" i.i.d. On a vu en cours son domaine de validité. Donner la condition sur $C$ qui fait que\n", "le TCL est vrai et vérifier que la variance asymptotique est donnée par $\\frac{C^2}{2C-1}$ dans le cas particulier étudié.\n", "\n", "Tester le TCL avec $C=0.6$, $C=1.0$, $C=1.5$, $C=2.0$." ] }, { "cell_type": "code", "execution_count": 224, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.8282866594385528\n", "x_0 = 0 , C = 0.6 , Beta = 1.0 , n = 1000\n" ] }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "Beta=1.0;\n", "C=0.6\n", "x_0=0;\n", "#Beta=0.75;\n", "#C=1.0;\n", "\n", "n=1000;\n", "N=500;\n", "z=np.zeros(N);\n", "for i in range(N):\n", " G=np.random.normal(size=n);\n", " X=algorithme_rm(x_0,n,C,Beta,F,G);\n", " z[i]=X[n-1];\n", "\n", "sigma=(C/math.sqrt(2*C-1))/math.sqrt(n)\n", "z=z/sigma;# on normalise\n", "print(np.var(z))\n", "print('x_0 =',x_0,',','C =',C,',','Beta =',Beta,',','n =',n);\n", "\n", "# La densité gaussienne de référence\n", "x = np.linspace(-5,5,100)\n", "densiteGaussienne = 1./np.sqrt(2*np.pi)*np.exp(-0.5*x**2)\n", "plt.plot(x, densiteGaussienne, color=\"red\", label=\"densite gaussienne\")\n", "plt.legend(loc=\"best\")\n", "\n", "plt.hist(z, density=\"True\", bins=30, label=\"erreur normalisee\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Calcul de fractile" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pour $p$ un seuil compris entre $0$ et $1$, on cherche à\n", "calculer la valeur d'un fractile $x_p$ d'ordre $p$ d'une\n", "gaussienne centrée réduite $G$. Montrer que $x_p$ est solution\n", "d'un équation du type\n", "$$\n", "f(x_p) =\\E\\left(F(x_p,G)\\right)=0,\n", "$$\n", "où $F$ est une fonction de $x$ et $G$ que l'on précisera." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " Ceci suggère d'utiliser l'algorithme de Robbins et Monro suivant:\n", " $$\n", " X_{n+1}=X_n - \\gamma_n F(X_n,G_{n+1}),\n", " $$ \n", " où $(G_n,n\\geq 1)$ est une suite de variables aléatoires gaussiennes\n", " centrées réduites et $\\gamma_n=C/n^\\beta$, où $0<\\beta\\leq 1$.\n", " \n", " On prendra pour $p=0.95$ (on cherche donc à approximer\n", " $x_p \\approx 1.96$).\n", "\n", "__Question 2.__ Implémenter cet algorithme et étudier la convergence pour\n", " $C=1$, $\\beta=0.6$ et $C=1$, $\\beta=0.8$.\n", "\n", " Dans le cas $\\beta=0.75$, tester les valeurs $C=0.5$, $C=1.0$,\n", " $C=2.0$, $C=4.0$.\n", " \n", " Commenter les difficultés que l'on peut rencontrer." ] }, { "cell_type": "code", "execution_count": 166, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n=10000;\n", "\n", "C=2;\n", "Beta=0.75;\n", "proba=0.95;\n", "x_0=1.5;\n", "\n", "def F(x,g):\n", " A_FAIRE\n", "\n", "# Beaucoup de variance, la condition initiale est vite oubliée \n", "X=algorithme_rm(x_0,n,C,Beta,F,np.random.normal(size=n));\n", "plt.plot(X);\n", "plt.plot(1.96*np.ones(n));plt.ylim(1.5,+2.2);plt.show()" ] }, { "cell_type": "code", "execution_count": 167, "metadata": {}, "outputs": [ { "data": { "image/png": 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SZt65uo3W+sQp7a/IqaBwl3kzNJZl16FheofTvD44zuBYllf7Rnn+9SGGJ7KT0+omsvkTXmBsb0iwrLX4bkEzY2QiS0dTkrOXNmFAruC01sfpaqljWWuK9oYEZy1uZElTiojmSstpTOEus5YvONv2D/LUK/3s6DnC64PjjKaLbyUfTueOm7YXMehqqWPdkka6WuqIRorj2hEzVrXXs35ZM91ntDE8kSOTL9BSFycV1/CHyGwo3KWkQ0cmGBzPsvPgMK/1jfLUq/08u3+IiWyeXDDs0dWSYnFzilXt9Zy/opXmVJzmuhjnLmuhKRVj9aKGsseq2xo0TCIyVwp3mXRoeIIfPNvD5tcG2H1ohN6RNLng3Y9Trelo4Lpzl9DRmGTd4kYuWtXGWYsbq1S1iExH4X4acndeHxzniT39/PrVAXYeHKZnaIK+kTQAHY3J4hDJ6jZyeaerJcWSlhRrOxrpbEpyZmfDaf2BTCK1QOEeQplcgXjUeGJPPw9sOUDvSLr4eSSjmcnPtj4qYrC2s5HL1rZzRns971u/hAtXtiq8RWqcwr1G7ekdYevew4xl8vQE0wVHg4uYW/cdnpwamIhGOGtxIx2NSS5c2UpLXZx4NEJ7Y4ILV7TSvbqdRExvsBEJG4V7jRhN53hm3yA/3N7DL3f1sW9g7C3bE9EInU1JIhHoPqOd85a3sLgpye+9cyWNSR1mkdON/tUvEO7Ok3v6aW9MMJouvkln695BjkwU33n5852HKHhxGOXspc3cduUarj57MWcubiAejbCoIaGhFBGZpHCvInfnS4/s5Bs/33PSdkubU/yLi1dw1ds6uXxNO4ubU6eoQhGpVQr3UyRfcF7tG+WBrQd4+eAwe3pHeG3K55tcta6DzsYkV5y5iIlcgbUdDbQ3JDh7SZPeaSkiM6Zwr5BDRyb4y5/t5oZ3LGV5ax3b9g9y6Eia54OPin3s5d63tG+rj3PpmnauOquDP/qts/S1YiJSUQr3WegZGudHL7zJ47v7WNlez49ffHPyAud3nto77WM6GpO8/9wlXHHmIq5bv1QzVERkXincy7B/YIwN39nCjp4jJ23XfUYbH7/6TH65q48Dh8f5ve4VvL2rmRVtdbrYKSKnVMlwN7N7gA8Ah9z9vBO0uRr4ChAH+tz9vZUsslq++pNd/PefvDzttsvXtnPTpavoXt3OgYExzl3eMjnl8Nq3LzmVZYqIHKecM/d7ga8D355uo5m1AncD17v7PjNbXLnyqufzD73AvU+8Nnn/livO4NPXnT3t154tb607hZWJiJRWMtzd/TEzW32SJr8PPOju+4L2hypT2qnzPx57hR+9eJDfvHb4uG0Pf/JKzlveUoWqRERmrxJj7m8D4mb2c6AJ+Kq7n+gsfwOwAWDVqlUVeOq56R9Jc+1dv2BwLDvt9u9/4t0KdhGpSZUI9xhwCXAtUAc8aWZPuftxg9XuvgnYBNDd3T39V++cIkPjWS75Lz+ZvH/nb6/n/BWtXLCihZi+zFhEalwlwv0AxYuoo8ComT0GXABMfyVyAdjXP8Z7vvzo5P3nPn8dzanjx9JFRGpVJU5Rvw9cZWYxM6sHLgN2VGC/86JQ8MlgT0QjvPbFf6pgF5HQKWcq5P3A1UCHmR0A7qQ45RF33+juO8zsH4HngALwTXffPn8lz17fSJqP/c3myfvPff66KlYjIjJ/ypktc1MZbb4MfLkiFc2Tw6MZuqeMsT95xzX68mURCa3T5srh9555fXL5u//2CrpaNDddRMIr1B8/sHXfYT509xNcsLKVZ/cPArDrCzcQ12wYEQm50Ib7H35nM4+88CbAZLDfdOkqBbuInBZCGe69w+nJYP/T972Ndyxv4chElg9euLzKlYmInBqhC/f9A2Nc9aXiVMdrzlnMH1+7rsoViYiceqEbozga7ADfvKW7ipWIiFRPqML9u5v3Ty4/ecc1+no6ETlthSrc73n8NQAe/8w1muooIqe10IT7fU/vnfymJH2+uoic7kIT7p/7XvETD65a11HlSkREqq/mw31wLMO/+9utk/fv/sjFVaxGRGRhqPlw/19P7eXh53oA2HTzJTTpEx5FRGo/3P/iR8WPjT9naRPXnbu0ytWIiCwMNR/uR/3j7e+pdgkiIgtGTYd7OpcH4LI17VWuRERkYanpcH98dx8Al69dVOVKREQWlpoO9z+4t/itSms7G6pciYjIwlKz4d43kp5cvmhlWxUrERFZeGo23PcNjAHQWh9n1aL6KlcjIrKw1Gy4f+juJwD44ofOr3IlIiILT82G+1HvWNFS7RJERBacmg/3ZS2papcgIrLglAx3M7vHzA6Z2fYTbL/azIbMbFtw+/PKl/lWh4YnSMUj3HblGsz0me0iIscq52v27gW+Dnz7JG1+6e4fqEhFJQyMZrj0Cz8FIBGr+T88RETmRcl0dPfHgIFTUEtZfvDsG5PLP3vpUBUrERFZuCr1BdlXmNmzwBvAn7n7C9M1MrMNwIbg7oiZ7Zzl83UAfXsB+5NZ7qH2dAB91S7iFFOfTw/q88ycUU4jc/fSjcxWAw+7+3nTbGsGCu4+YmY3Al9193Uzq3VmzGyzu59W336tPp8e1OfTw6no85wHrd39iLuPBMv/AMTNTF+HJCJSRXMOdzNbasGUFTO7NNhn/1z3KyIis1dyzN3M7geuBjrM7ABwJxAHcPeNwO8CHzezHDAOfNjLGeuZm03zvP+FSH0+PajPp4d573NZY+4iIlJbNFFcRCSEFO4iIiFUc+FuZteb2U4z221mn6l2PbNlZivN7FEz22FmL5jZp4L17Wb2YzPbFfxsC9abmX0t6PdzZnbxlH19NGi/y8w+Wq0+lcvMomb2jJk9HNxfY2ZPB/X/nZklgvXJ4P7uYPvqKfu4I1i/08zeX52elMfMWs3s783speB4XxH242xmfxL8Xm83s/vNLBW24zzdR7NU8ria2SVm9nzwmK8dnbhSNnevmRsQBfYAa4EE8Cywvtp1zbIvXcDFwXIT8DKwHvgS8Jlg/WeA/xos3wj8EDDgcuDpYH078Erwsy1Ybqt2/0r0/U+Bv6X43gmA/0PxQjzARuDjwfIfARuD5Q8Dfxcsrw+OfRJYE/xORKvdr5P092+A24LlBNAa5uMMLAdeBeqmHN9/E7bjDLwHuBjYPmVdxY4r8GvgiuAxPwRumFF91X6BZvhiXgE8MuX+HcAd1a6rQn37PvA+YCfQFazrAnYGy38N3DSl/c5g+03AX09Z/5Z2C+0GrAB+ClwDPBz84vYBsWOPMfAIcEWwHAva2bHHfWq7hXYDmoOgs2PWh/Y4B+G+PwisWHCc3x/G4wysPibcK3Jcg20vTVn/lnbl3GptWOboL81RB4J1NS34M/Qi4Glgibv3AAQ/FwfNTtT3WntNvgL8B6AQ3F8EDLp7Lrg/tf7JvgXbh4L2tdTntUAv8D+DoahvmlkDIT7O7v468BfAPqCH4nHbQriP81GVOq7Lg+Vj15et1sJ9ujGnmp7LaWaNwAPA7e5+5GRNp1nnJ1m/4JjZB4BD7r5l6uppmnqJbTXTZ4pnohcD33D3i4BRin+un0jN9zkYZ/4gxaGUZUADcMM0TcN0nEuZaR/n3PdaC/cDwMop91dQ/LCymmRmcYrBfp+7PxisftPMuoLtXcDRj748Ud9r6TV5N/DPzOw14H9THJr5CtBqZkffUDe1/sm+BdtbKH5CaS31+QBwwN2fDu7/PcWwD/Nx/ifAq+7e6+5Z4EHgXYT7OB9VqeN6IFg+dn3Zai3cfwOsC666JyhefHmoyjXNSnDl+1vADne/a8qmh4CjV8w/SnEs/uj6W4Kr7pcDQ8GffY8A15lZW3DGdF2wbsFx9zvcfYW7r6Z47H7m7h8BHqX4Tmc4vs9HX4vfDdp7sP7DwSyLNcA6ihefFhx3PwjsN7Ozg1XXAi8S4uNMcTjmcjOrD37Pj/Y5tMd5iooc12DbsJldHryGt0zZV3mqfUFiFhcwbqQ4s2QP8Llq1zOHflxJ8c+s54Btwe1GimONPwV2BT/bg/YG/FXQ7+eB7in7+gNgd3C7tdp9K7P/V/P/Z8uspfiPdjfwXSAZrE8F93cH29dOefzngtdiJzOcRVCFvl4IbA6O9f+lOCsi1McZ+E/AS8B24DsUZ7yE6jgD91O8ppCleKb9sUoeV6A7eP32UPzCJJtJffr4ARGREKq1YRkRESmDwl1EJIQU7iIiIaRwFxEJIYW7iEgIKdxFREJI4S4iEkL/DwB8dOU5P4lSAAAAAElFTkSuQmCC\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Moins de variance, la condition initiale est moins vite oubliee \n", "A_FAIRE\n", "\n", "proba=0.95;\n", "x_0=1.5;\n", "\n", "X=algorithme_rm(x_0,n,C,Beta,F,np.random.normal(size=n));\n", "plt.plot(X);\n", "plt.plot(1.96*np.ones(n));plt.ylim(1.5,+2.2);plt.show()" ] }, { "cell_type": "code", "execution_count": 168, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX4AAAD8CAYAAABw1c+bAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4zLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvIxREBQAAFpdJREFUeJzt3X+QXWV9x/H3Z3eT8EN+JGSlaZIloc04UkcB7wQoTkUrITCU2KmdJmMlWug6Vlp/9MdAnZFpGGf80dGWikLUFHU0qPzQlAFjKihtLTQbRSBAYAnWbEObYBARMMnufvvHeTZ79u7d3bObu9nsPp/XzJ0953mec87z3LP72bNnzz1HEYGZmeWjZao7YGZmR5aD38wsMw5+M7PMOPjNzDLj4Dczy4yD38wsMw5+M7PMOPjNzDLj4Dczy0zbVHegkfnz58eSJUumuhtmZtPGtm3bno2I9iptj8rgX7JkCV1dXVPdDTOzaUPSf1dt61M9ZmaZcfCbmWXGwW9mlhkHv5lZZhz8ZmaZcfCbmWXGwW9mlhkHv5lZZhz8ZmaZcfCbmWXGwW9mlhkHv5lZZhz8ZmaZcfCbmWVmzOCXtFjSvZIek7Rd0vsatJGk6yV1S3pI0tmlurWSnkyvtc0egJmZjU+V+/H3An8ZET+UdAKwTdKWiHi01OZiYFl6nQN8FjhH0jzgWqAGRFp2U0Q819RRmJlZZWMe8UfEMxHxwzT9AvAYsLCu2SrgS1G4HzhZ0gLgImBLROxLYb8FWNnUEZiZ2biM6xy/pCXAWcADdVULgV2l+Z5UNlK5mZlNkcrBL+kVwG3A+yPiF/XVDRaJUcobrb9TUpekrr1791btlpmZjVOl4Jc0iyL0vxIRtzdo0gMsLs0vAnaPUj5MRKyPiFpE1NrbKz0v2MzMJqDKVT0CvgA8FhGfHKHZJuDydHXPucDzEfEMsBlYIWmupLnAilRmZmZTpMpVPecD7wAelvRgKvtboAMgIm4E7gIuAbqBl4B3pbp9kq4Dtqbl1kXEvuZ1v85tfwp9+ydt9WZmk+qYk+Cyf5r0zYwZ/BHx7zQ+V19uE8B7R6jbAGyYUO/G69knoPdXR2RTZmZNd+y8I7KZKkf808e7vz/VPTAzO+r5lg1mZplx8JuZZcbBb2aWGQe/mVlmHPxmZplx8JuZZcbBb2aWGQe/mVlmHPxmZplx8JuZZcbBb2aWGQe/mVlmHPxmZplx8JuZZcbBb2aWmTHvxy9pA3ApsCciXtOg/q+Bt5fW92qgPT196yfAC0Af0BsRtWZ13MzMJqbKEf/NwMqRKiPiExFxZkScCVwDfL/u8YpvSvUOfTOzo8CYwR8R9wFVn5O7Bth4WD0yM7NJ1bRz/JKOo/jL4LZScQDfkbRNUmeztmVmZhPXzGfu/h7wH3Wnec6PiN2SXglskfR4+gtimPSLoROgo6Ojid0yM7OyZl7Vs5q60zwRsTt93QPcASwfaeGIWB8RtYiotbe3N7FbZmZW1pTgl3QS8EbgW6Wy4yWdMDANrAAeacb2zMxs4qpczrkRuACYL6kHuBaYBRARN6Zmvw98JyJeLC16KnCHpIHtfDUivt28rpuZ2USMGfwRsaZCm5spLvssl+0EXjfRjpmZ2eTwJ3fNzDLj4Dczy4yD38wsMw5+M7PMOPjNzDLj4Dczy4yD38wsMw5+M7PMOPjNzDLj4Dczy4yD38wsMw5+M7PMOPjNzDLj4Dczy4yD38wsMw5+M7PMVHkC1wbgUmBPRLymQf0FFI9cfDoV3R4R61LdSuAfgVbg8xHx0Sb126aJiCACYmAaiID+iPQqyge+DtZBMDhf/kpax0B9/bqHTA+0GWEahi/PsLrh6xyo7B+hH9Rtr7/B+uv7ODC+of0srbN+vQ3eh4E3p/w+l+siDrU69D6mHtXNj15f3r+Nlivvo2H1A8s02Eaj7Zff74EvI223ft8O6eMo2xrYL4fWMWq7oX2qf2+H9qVB3UjvecCJx7bx8bdN/vOrxgx+iidrfRr40iht/i0iLi0XSGoFbgAuBHqArZI2RcSjE+xrU/T3Bwf7+znYF/T1BX0R9PUXIdTbH/T29XOwr58DvcH+3j4O9BZtD/b1s7+3nwN9/fT29dPbV6ynr79YfsgPafrm6y/90B/sG2w78OrtL8r7o+jX0CAM+oJiW/1xqL4v1Ze32yhE+2OwHor+DG1Pmh9s318KpXJQ9afBBXUhXDQfMYDNRlI8kRV0aF6H5gfrBhtp2HI6NN9oHdSVl7fXaFugunWNvo0hfalrWz+u0iYa9qG83NzjZnMkVHn04n2Slkxg3cuB7vQIRiTdAqwCJi34P3ffTh7s+Tkv7e/lYF/wwv5e9r24n+dfOsjLB/tSwE/W1scmQVuLaJFoaxGtLWJ2WwstUnoV3xAtLRwqG2jXmpZraRGtGqxvaYFZLcU6lNYxZF2lb8rWlmL5gTYStKb1SIM/IAPLKLUb6HtL+iYut1XpB6Z+nnL7+nYaGBOH+i6Kbbe06NAPx6G+kfqiwR+cwXWV1j/KNmHwfSkvW9Q0XvdA3eCYBueBIf0b2Ibq1jPQhvp+DXmfy9sdPgbK/RlhjAN9G9xOuX+D6TVa6A6tHxpmjFHfMBzr+m1HhypH/FWcJ+nHwG7gryJiO7AQ2FVq0wOc06TtDfPciwf4yF2PAfAb7cdz8nGzOfGYNpaechwnHTuL4+a0FSHXIua0tdDWItpaW2jVYCC2tYhZrS20tbYwu1XMaWtldmo7u62F2W0tadkW2lpT25bhwVn/A3conFv8jW9mU68Zwf9D4LSI+KWkS4BvAssYPAAoG/F4W1In0AnQ0dEx7k7MPX42X3/3eZx64hxOO+X4cS9vZpaLw76qJyJ+ERG/TNN3AbMkzac4wl9carqI4i+CkdazPiJqEVFrb2+fUF+WL53n0DczG8NhB7+kX1M6eSdpeVrnz4CtwDJJSyXNBlYDmw53e2ZmdniqXM65EbgAmC+pB7gWmAUQETcCbwPeI6kXeBlYHcW1S72SrgI2U1zOuSGd+zczsymk+ut6jwa1Wi26urqmuhtmZtOGpG0RUavS1p/cNTPLjIPfzCwzDn4zs8w4+M3MMuPgNzPLjIPfzCwzDn4zs8w4+M3MMuPgNzPLjIPfzCwzDn4zs8w4+M3MMuPgNzPLjIPfzCwzDn4zs8w4+M3MMjNm8EvaIGmPpEdGqH+7pIfS6weSXleq+4mkhyU9KMlPVjEzOwpUOeK/GVg5Sv3TwBsj4rXAdcD6uvo3RcSZVZ8MY2Zmk2vMZ+5GxH2SloxS/4PS7P3AosPvlpmZTZZmn+O/Ari7NB/AdyRtk9Q52oKSOiV1Serau3dvk7tlZmYDxjzir0rSmyiC/w2l4vMjYrekVwJbJD0eEfc1Wj4i1pNOE9VqtaPvCfBmZjNEU474Jb0W+DywKiJ+NlAeEbvT1z3AHcDyZmzPzMwm7rCDX1IHcDvwjoh4olR+vKQTBqaBFUDDK4PMzOzIGfNUj6SNwAXAfEk9wLXALICIuBH4MHAK8BlJAL3pCp5TgTtSWRvw1Yj49iSMwczMxqHKVT1rxqi/EriyQflO4HXDlzAzs6nkT+6amWXGwW9mlhkHv5lZZhz8ZmaZcfCbmWXGwW9mlhkHv5lZZhz8ZmaZcfCbmWXGwW9mlhkHv5lZZhz8ZmaZcfCbmWXGwW9mlhkHv5lZZioFv6QNkvZIavgELRWul9Qt6SFJZ5fq1kp6Mr3WNqvjZmY2MVWP+G8GVo5SfzGwLL06gc8CSJpH8cSucyiet3utpLkT7ayZmR2+SsEfEfcB+0Zpsgr4UhTuB06WtAC4CNgSEfsi4jlgC6P/AjEzs0nWrHP8C4FdpfmeVDZSuZmZTZFmBb8alMUo5cNXIHVK6pLUtXfv3iZ1y8zM6jUr+HuAxaX5RcDuUcqHiYj1EVGLiFp7e3uTumVmZvWaFfybgMvT1T3nAs9HxDPAZmCFpLnpn7orUpmZmU2RtiqNJG0ELgDmS+qhuFJnFkBE3AjcBVwCdAMvAe9KdfskXQdsTataFxGj/ZPYzMwmWaXgj4g1Y9QH8N4R6jYAG8bfNTMzmwz+5K6ZWWYc/GZmmXHwm5llxsFvZpYZB7+ZWWYc/GZmmXHwm5llxsFvZpYZB7+ZWWYc/GZmmXHwm5llxsFvZpYZB7+ZWWYc/GZmmXHwm5llxsFvZpaZSsEvaaWkHZK6JV3doP5Tkh5Mryck/bxU11eq29TMzpuZ2fiN+QQuSa3ADcCFFA9P3yppU0Q8OtAmIj5Qav/nwFmlVbwcEWc2r8tmZnY4qhzxLwe6I2JnRBwAbgFWjdJ+DbCxGZ0zM7PmqxL8C4FdpfmeVDaMpNOApcA9peJjJHVJul/SWyfcUzMza4oqD1tXg7IYoe1q4NaI6CuVdUTEbkmnA/dIejginhq2EakT6ATo6Oio0C0zM5uIKkf8PcDi0vwiYPcIbVdTd5onInanrzuB7zH0/H+53fqIqEVErb29vUK3zMxsIqoE/1ZgmaSlkmZThPuwq3MkvQqYC/xnqWyupDlpej5wPvBo/bJmZnbkjHmqJyJ6JV0FbAZagQ0RsV3SOqArIgZ+CawBbomI8mmgVwM3Seqn+CXz0fLVQGZmduRpaE4fHWq1WnR1dU11N8zMpg1J2yKiVqWtP7lrZpYZB7+ZWWYc/GZmmXHwm5llxsFvZpYZB7+ZWWYc/GZmmXHwm5llxsFvZpYZB7+ZWWYc/GZmmXHwm5llxsFvZpYZB7+ZWWYc/GZmmXHwm5llplLwS1opaYekbklXN6h/p6S9kh5MrytLdWslPZlea5vZeTMzG78xH70oqRW4AbiQ4sHrWyVtavAIxa9FxFV1y84DrgVqQADb0rLPNaX3ZmY2blWO+JcD3RGxMyIOALcAqyqu/yJgS0TsS2G/BVg5sa6amVkzVAn+hcCu0nxPKqv3B5IeknSrpMXjXBZJnZK6JHXt3bu3QrfMzGwiqgS/GpTVP6H9X4AlEfFa4F+BL45j2aIwYn1E1CKi1t7eXqFbZmY2EVWCvwdYXJpfBOwuN4iIn0XE/jT7OeD1VZc1M7Mjq0rwbwWWSVoqaTawGthUbiBpQWn2MuCxNL0ZWCFprqS5wIpUZmZmU2TMq3oiolfSVRSB3QpsiIjtktYBXRGxCfgLSZcBvcA+4J1p2X2SrqP45QGwLiL2TcI4zMysIkU0POU+pWq1WnR1dU11N8zMpg1J2yKiVqWtP7lrZpYZB7+ZWWYc/GZmmXHwm5llxsFvZpYZB7+ZWWYc/GZmmXHwm5llxsFvZpYZB7+ZWWYc/GZmmXHwm5llxsFvZpYZB7+ZWWYc/GZmmakU/JJWStohqVvS1Q3qPyjp0fSw9e9KOq1U1yfpwfTaVL+smZkdWWM+gUtSK3ADcCHFM3S3StoUEY+Wmv0IqEXES5LeA3wc+KNU93JEnNnkfpuZ2QRVOeJfDnRHxM6IOADcAqwqN4iIeyPipTR7P8VD1c3M7ChUJfgXArtK8z2pbCRXAHeX5o+R1CXpfklvnUAfzcysicY81QOoQVnDB/VK+mOgBryxVNwREbslnQ7cI+nhiHiqwbKdQCdAR0dHhW6ZmdlEVDni7wEWl+YXAbvrG0l6C/Ah4LKI2D9QHhG709edwPeAsxptJCLWR0QtImrt7e2VB2BmZuNTJfi3AsskLZU0G1gNDLk6R9JZwE0Uob+nVD5X0pw0PR84Hyj/U9jMzI6wMU/1RESvpKuAzUArsCEitktaB3RFxCbgE8ArgG9IAvhpRFwGvBq4SVI/xS+Zj9ZdDWRmZkeYIhqerp9StVoturq6probZmbThqRtEVGr0taf3DUzy4yD38wsMw5+M7PMOPjNzDLj4Dczy4yD38wsMw5+M7PMOPjNzDLj4Dczy4yD38wsMw5+M7PMOPjNzDLj4Dczy4yD38wsMw5+M7PMOPjNzDJTKfglrZS0Q1K3pKsb1M+R9LVU/4CkJaW6a1L5DkkXNa/rZmY2EWMGv6RW4AbgYuAMYI2kM+qaXQE8FxG/CXwK+Fha9gyKZ/T+FrAS+Exan5mZTZEqR/zLge6I2BkRB4BbgFV1bVYBX0zTtwK/q+Lhu6uAWyJif0Q8DXSn9ZmZ2RSpEvwLgV2l+Z5U1rBNRPQCzwOnVFzWzMyOoLYKbdSgrP4J7SO1qbJssQKpE+hMs7+UtKNC3xqZDzw7wWWnK4955sttvOAxj9dpVRtWCf4eYHFpfhGwe4Q2PZLagJOAfRWXBSAi1gPrq3V7ZJK6qj5pfqbwmGe+3MYLHvNkqnKqZyuwTNJSSbMp/lm7qa7NJmBtmn4bcE9ERCpfna76WQosA/6rOV03M7OJGPOIPyJ6JV0FbAZagQ0RsV3SOqArIjYBXwC+LKmb4kh/dVp2u6SvA48CvcB7I6JvksZiZmYVVDnVQ0TcBdxVV/bh0vSvgD8cYdmPAB85jD6O12GfLpqGPOaZL7fxgsc8aVSckTEzs1z4lg1mZpmZMcE/1m0lphNJiyXdK+kxSdslvS+Vz5O0RdKT6evcVC5J16exPyTp7NK61qb2T0paO9I2jwaSWiX9SNKdaX5pugXIk+mWILNT+Yy5RYikkyXdKunxtL/Pm8n7WdIH0vf0I5I2SjpmJu5nSRsk7ZH0SKmsaftV0uslPZyWuV5So0vnRxYR0/5F8U/np4DTgdnAj4EzprpfhzGeBcDZafoE4AmK22V8HLg6lV8NfCxNXwLcTfG5iXOBB1L5PGBn+jo3Tc+d6vGNMu4PAl8F7kzzXwdWp+kbgfek6T8DbkzTq4Gvpekz0r6fAyxN3xOtUz2uMcb8ReDKND0bOHmm7meKD28+DRxb2r/vnIn7Gfgd4GzgkVJZ0/YrxdWR56Vl7gYuHlf/pvoNatKbfB6wuTR/DXDNVPerieP7FnAhsANYkMoWADvS9E3AmlL7Hal+DXBTqXxIu6PpRfEZj+8CbwbuTN/QzwJt9fuY4gqz89J0W2qn+v1ebnc0voATUxCqrnxG7mcGP8k/L+23O4GLZup+BpbUBX9T9muqe7xUPqRdlddMOdUzY28Nkf68PQt4ADg1Ip4BSF9fmZqNNP7p9L78A/A3QH+aPwX4eRS3AIGhfZ8ptwg5HdgL/HM6xfV5ScczQ/dzRPwP8PfAT4FnKPbbNmb+fh7QrP26ME3Xl1c2U4K/8q0hphNJrwBuA94fEb8YrWmDsnHdMmMqSboU2BMR28rFDZrGGHXTYrwlbRSnAz4bEWcBL1KcAhjJtB53Oqe9iuL0zK8Dx1Pc9bfeTNvPYxnvOA97/DMl+CvfGmK6kDSLIvS/EhG3p+L/k7Qg1S8A9qTykcY/Xd6X84HLJP2E4u6vb6b4C+BkFbcAgaF9PzQuTfAWIUeJHqAnIh5I87dS/CKYqfv5LcDTEbE3Ig4CtwO/zczfzwOatV970nR9eWUzJfir3FZi2kj/of8C8FhEfLJUVb41xlqKc/8D5ZenqwPOBZ5Pf0puBlZImpuOtlaksqNKRFwTEYsiYgnFvrsnIt4O3EtxCxAYPt5pf4uQiPhfYJekV6Wi36X4lPuM3M8Up3jOlXRc+h4fGO+M3s8lTdmvqe4FSeem9/Hy0rqqmep/gDTxHymXUFz98hTwoanuz2GO5Q0Uf7o9BDyYXpdQnN/8LvBk+jovtRfFw3KeAh4GaqV1/QnFcxC6gXdN9dgqjP0CBq/qOZ3iB7ob+AYwJ5Ufk+a7U/3ppeU/lN6HHYzzSocpGu+ZQFfa19+kuHpjxu5n4O+Ax4FHgC9TXJkz4/YzsJHi/xgHKY7Qr2jmfgVq6T18Cvg0dRcIjPXyJ3fNzDIzU071mJlZRQ5+M7PMOPjNzDLj4Dczy4yD38wsMw5+M7PMOPjNzDLj4Dczy8z/A3y3Tt2Ggk7XAAAAAElFTkSuQmCC\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Pas de variance, mais la condition initiale \n", "# a vraiment beaucoup de mal a etre oubliee !\n", "A_FAIRE\n", "\n", "x_0=1.5;\n", "\n", "X=algorithme_rm(x_0,n,C,Beta,F,np.random.normal(size=n));\n", "plt.plot(X);\n", "plt.plot(1.96*np.ones(n));plt.ylim(0,+2.2);plt.show()" ] }, { "cell_type": "code", "execution_count": 169, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Cas beta=1\n", "A_FAIRE\n", "\n", "Beta=1;\n", "x_0=1.5;\n", "n=1000;\n", "\n", "X=algorithme_rm(x_0,n,C,Beta,F,np.random.normal(size=n));\n", "plt.plot(X);\n", "plt.plot(1.96*np.ones(n));plt.ylim(0,+2.2);plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Question 3.__ Montrer que la pente de la fonction $f$ en $x_p$ est donnée par \n", "$$\n", " c^*= {2e^{-\\frac{x_p^2}{2}}}/{\\sqrt{2\\pi}}.\n", "$$\n", "Pour quelle valeur de $C$ peut on s'attendre à obtenir un TCL ?\n", "\n", "Verifier le théorème de la limite centrale pour cet algorithme sous\n", "cette condition." ] }, { "cell_type": "code", "execution_count": 172, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "C= 8.555645458894494 , Beta= 1.0 , n= 1000\n" ] }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# La valeur pour avoir le TCL 2c^* alpha > 1\n", "def alpha_star(x_star):\n", " c_star=2*math.exp(-x_star**2/2)/math.sqrt(2*math.pi);\n", " return 1/(2*c_star);\n", "\n", "n=1000;\n", "Beta=1.0;\n", "C=2.0*alpha_star(1.96);\n", "x_0=1.0;\n", "\n", "print('C=',C,',','Beta=',Beta,',','n=',n);\n", "\n", "G=np.random.normal(size=n);\n", "X=algorithme_rm(x_0,n,C,Beta,F,G);\n", "plt.plot(X);\n", "plt.plot(1.96*np.ones(n));plt.ylim(0,+5);plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Optimisation du paramètre d'une fonction d'importance importance" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On veut calculer $\\E(f(G))$. Comme \n", "$$\n", " \\E\\left(e^{-\\lambda G - \\frac{\\lambda^2}{2}} f(G+\\lambda)\\right)\n", " = \\E(f(G)),\n", "$$\n", "On cherche a minimiser en $\\lambda$ la variance de \n", "$\n", " X_\\lambda = e^{-\\lambda G - \\frac{\\lambda^2}{2}} f(G+\\lambda).\n", "$\n", "Notez que l'on a (cours)\n", "$$\n", "\\Var\\left(X_\\lambda\\right) \n", " = \\E\\left(e^{-\\lambda G + \\frac{\\lambda^2}{2}} f^2(G)\\right) - \\E(f(G))^2.\n", "$$\n", "On traitera le cas du call dans le modèle de Black et Scholes avec les paramètres suivants\n", "$r=2\\%$, $\\sigma=30\\%/\\mbox{an}$, $S_0=x=100$, $T=1$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Question 1.1.__ Tracer (une approximation Monte-Carlo de) la courbe $\\lambda \\to \\Var(X_\\lambda)$ lorsque $K=100$." ] }, { "cell_type": "code", "execution_count": 176, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "r=0.02;\n", "sigma=0.3;\n", "T=1;\n", "x=100;\n", "\n", "def f(g):\n", " d=np.size(g);\n", " S_T=np.zeros(d);\n", " S_T= x * np.exp((r - sigma*sigma/2)*T*np.ones(d)+sigma*np.sqrt(T)*g);\n", " return math.exp(-r*T)*np.maximum(S_T-K*np.ones(np.size(S_T)),np.zeros(np.size(S_T)));\n", "\n", "\n", "def approx_variance(Lambda,G):\n", " # ATTENTION: G est un vecteur echantillon de gaussiennes\n", " A_FAIRE\n", "\n", "K=100;\n", "n=1000;\n", "Lambda = np.linspace(0,3,100)\n", "z=np.zeros(np.size(Lambda));\n", "G=np.random.normal(size=n);\n", "for i in range(np.size(Lambda)):\n", " z[i]=approx_variance(Lambda[i],G);\n", " #print(Lambda[i],z[i])\n", "plt.plot(Lambda,z);\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Question 1.2.__ Reprendre la simulation précédente avec $K=200$." ] }, { "cell_type": "code", "execution_count": 177, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "K=200;\n", "Lambda = np.linspace(0,6,100)\n", "\n", "A_FAIRE\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Question 2.__ Montrer que la dérivée peut s'écrire sous les deux formes suivantes\n", "$$\n", " \\frac{\\partial}{\\partial \\lambda}\\Var\\left(X_\\lambda\\right)\n", " = \\E\\left((\\lambda - G)e^{-\\lambda G + \\frac{\\lambda^2}{2}} f^2(G)\\right)\n", " = - \\E\\left( G e^{-2\\lambda G - \\lambda^2} f^2(G+\\lambda)\\right).\n", "$$\n", "Vérifier ce fait par simulation en tracant les deux approximations\n", "Monte-Carlo de la dérivées. Laquelle de ces deux représentations vous\n", "parait elle préférable du point de vue de la simulation ?" ] }, { "cell_type": "code", "execution_count": 178, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "r=0.02;\n", "sigma=0.3;\n", "T=1;\n", "x=100;\n", "\n", "def approx_1_derivee_variance(Lambda):\n", " A_FAIRE\n", " return np.mean(Y);\n", "\n", "def approx_2_derivee_variance(Lambda):\n", " A_FAIRE\n", " return np.mean(Y)\n", "\n", "n=10000;\n", "\n", "K=100;\n", "Lambda=np.linspace(0,3,100)\n", "z1=np.zeros(np.size(Lambda));\n", "z2=np.zeros(np.size(Lambda));\n", "G=np.random.normal(size=n);\n", "for i in range(np.size(Lambda)):\n", " z1[i]=approx_1_derivee_variance(Lambda[i]);\n", " z2[i]=approx_2_derivee_variance(Lambda[i]);\n", "plt.plot(Lambda,z1);\n", "plt.plot(Lambda,z2);" ] }, { "cell_type": "code", "execution_count": 182, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n=10000\n", "K=200\n", "Lambda=np.linspace(0,6,100)\n", "\n", "A_FAIRE\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# __La méthode de projection de Chen__" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On chercher à résoudre $\\frac{\\partial}{\\partial\n", " \\lambda}\\Var\\left(X_\\lambda\\right)=0$ en utilisant un algorithme de type\n", "Robbins et Monro utilisant la première représentation\n", "de la dérivée. \n", "\n", "On prendra $K=200$.\n", "\n", "Pour cela, on pose $X_0=0$, puis\n", "$$\n", " X_{n+1}=X_n \n", " -\\gamma_n (\\lambda - G_{n+1})e^{-\\lambda G_{n+1} + \\frac{\\lambda^2}{2}} f^2(G_{n+1}).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Question 3.__ Implémenter cet algorithme de façon directe (sans borner l'algorithme).\n", "Constater l'instabilité de cet algorithme. Quelle hypothèse permettant\n", "de prouver la convergence n'est pas vérifiée ? \n", "\n", "On essaye quand même, mais ne pas s'étonner si l'on rencontre quelques problèmes !" ] }, { "cell_type": "code", "execution_count": 183, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/home/bl/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:11: RuntimeWarning: overflow encountered in exp\n", " # This is added back by InteractiveShellApp.init_path()\n", "/home/bl/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:11: RuntimeWarning: invalid value encountered in multiply\n", " # This is added back by InteractiveShellApp.init_path()\n" ] }, { "data": { "text/plain": [ "[]" ] }, "execution_count": 183, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def gamma(n,C,beta):\n", " return C / ((n+1)**beta);\n", "\n", "r=0.02;\n", "sigma=0.3;\n", "T=1;\n", "x=100;\n", "K=100;\n", "\n", "def F(Lambda,g):\n", " return A_FAIRE\n", "\n", "C=0.5;\n", "beta=1.0;\n", "n=10000;\n", "G=np.random.normal(size=n);\n", "x_0=0;\n", "\n", "X=algorithme_rm(x_0,n,C,beta,F,G);\n", "plt.plot(X)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Question 4.__ Implémenter cet algorithme en ramenant $X_n$ à une valeur initiale commune lorsque $X_n$ dépasse la\n", "valeur de $5$ (procédure dite de ``projection de Chen''). Vérifier que\n", "l'algorithme converge (mieux!)." ] }, { "cell_type": "code", "execution_count": 184, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def algorithme_rm_chen(x_0,MAX,n,C,beta,F,G):\n", " X=np.zeros(n);\n", " X[0] = x_0;\n", " for k in range(n-1):\n", " X[k+1]=X[k]-gamma(k,C,beta)*F(X[k],G[k]);\n", " if(abs(X[k+1]) >= MAX): \n", " A_FAIRE\n", " return X\n", " \n", "r=0.02;\n", "sigma=0.3;\n", "T=1;\n", "x=100;\n", "K=100;\n", "\n", "C=0.5;\n", "beta=1.0;\n", "Lambda0=0;\n", "#Lambda0=(math.log(K/x)-(r-sigma**2/2)*T)/(sigma*math.sqrt(T));\n", "MAX=5;\n", "n=10000;\n", "\n", "G=np.random.normal(size=n)\n", "Y=algorithme_rm_chen(Lambda0,MAX,n,C,beta,F,G);\n", "plt.plot(Y);plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Question 5.__ Proposer un algorithme basée sur la deuxième représentation de\n", " la dérivée. Lequel de ces deux algorithmes vous parait le plus\n", " pertinent." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.4" } }, "nbformat": 4, "nbformat_minor": 2 }