{ "cells": [ { "cell_type": "markdown", "metadata": { "tags": [ "hide-cell" ] }, "source": [ "# Numerical simulation of SDEs\n", "\n", "\n", "Star\n", "Issue\n", "Watch\n", "Follow\n", "\n", "This is a reproduction of certain scripts found in Higham, *An algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations.*{cite}`higham`\n", "This paper is an accessible introduction to SDEs, which is centered around ten scripts.\n", "Below are reproductions of these scripts (excluding two on linear stability) and some supplementary notes.\n", "\n", "## Why stochastic differential equations\n", "\n", "We are often interested in modelling a system whose state takes values in a continuous range, and over a continuous time domain.\n", "Whereas ordinary differential equations (ODEs) describe variables which change according to a deterministic rule, SDEs describe variables whose change is governed partly by a deterministic component and partly by a stochastic component.\n", "SDEs are therefore an appropriate model for systems whose dynamics involve some true randomness, or some fine grained complexity which we cannot afford to or do not wish to model." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "tags": [ "remove-cell" ] }, "outputs": [], "source": [ "import numpy as np\n", "\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "from matplotlib_inline.backend_inline import set_matplotlib_formats\n", "\n", "set_matplotlib_formats('pdf', 'svg')\n", "\n", "matplotlib.rcParams['mathtext.fontset'] = 'stix'\n", "matplotlib.rcParams['font.family'] = 'STIXGeneral'" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Wiener process\n", "\n", "In order to define the stochastic component of the transition rule of a stochastic system, we must define an appropriate noise model. The Wiener process is a stochastic process that is commonly used for this purpose.\n", "\n", ":::{prf:definition} Wiener process\n", "\n", "A standard Wiener process over [0, T] is a random variable $W(t)$ that depends continuously on $t \\in [0, T]$ and satisfies:\n", "\n", "1. $W(0) = 0,$ with probability 1.\n", "2. For $0 \\leq t_1 < t_2 \\leq T$ the random variable $W(t_2) - W(t_1)$ has distribution $\\mathcal{N}(0, t_2 - t_1)$.\n", "3. For $0 \\leq t_1 < t_2 < t_3 < t_4 \\leq T,$ then $W(t_2) - W(t_1) \\perp W(t_4) - W(t_3)$ are independent.\n", " \n", ":::\n", "\n", "\n", "We can imagine the Wiener process as the path followed by a particle that experiences infinitely many, infinitesimal kicks. The size of these kicks, $W(t_2) - W(t_1)$, diminishes as the interval between them, $t_2 - t_1$, diminishes. The kicks are also independent from each other, so the future path of the particle is independent of its past path given its present position.\n", "\n", "## Sampling from a Wiener process\n", "\n", "Before using the Wiener process to define an SDE, let's look at the process itself. How can we draw a sample $W(t)$ from it? Since a sample from the Wiener process takes a random value for every $t \\in [0, \\infty)$, the best we can do on a computer is to sample the process at a finite subset of time instances. We specify the times $t_1 < t_2 < ... < t_N$ at which to sample $W(t)$, and then use the definition of the Wiener process to see that we should sample as follows:\n", "\n", "$$\\begin{align}\n", "W(t_{n+1}) = W(t_{n}) + \\Delta W(t_{n}), \\text{ where } \\Delta W(t_{n}) \\sim \\mathcal{N}(0, t_{n+1} - t_n),\n", "\\end{align}$$\n", "\n", "where $W(t_0) = W(0) = 0$. We can therefore sample all the $\\Delta W(t_n)$ independently, and take their cumulative sum to compute $W(t_n)$, as shown below." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "tags": [ "hide-input" ] }, "outputs": [], "source": [ "# Set random seed\n", "np.random.seed(0)\n", "\n", "# Integration parameters\n", "T = 1\n", "N = 500\n", "dt = T / N\n", "\n", "# Times to sample at\n", "t = np.linspace(0, T, N)[:, None]\n", "\n", "# Sample dW's and compute cumulative sum\n", "dW = dt ** 0.5 * np.random.normal(size=(N - 1, 1))\n", "W = np.concatenate([[0], np.cumsum(dW)])" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "tags": [ "center-output", "remove-input" ] }, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(10, 4))\n", "plt.plot(t, W, color='k')\n", "\n", "plt.xticks(np.linspace(0, T, 6), fontsize=16)\n", "plt.yticks(np.linspace(-2, 2, 5), fontsize=16)\n", "plt.xlim([0, 1])\n", "plt.ylim([-2, 2])\n", "\n", "plt.title('Sample from a Wiener process', fontsize=24)\n", "plt.xlabel(r'$t$', fontsize=22)\n", "plt.ylabel(r'$W(t)$', fontsize=22)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So even though we can't represent the entirety of the path, we can sample it to arbitrary precision.\n", "\n", "## Function of a Wiener process\n", "\n", "Suppose we are interested in a stochastic process which is a function of a Wiener process, say\n", "\n", "$$\\begin{align}\n", "X(t) = \\exp\\left(t + \\frac{1}{2}W(t)\\right).\n", "\\end{align}$$\n", "\n", "In this case we can sample $X(t)$ by first sampling $W(t)$, and then computing the corresponding values of $X(t)$. We'll draw `S = 1000` samples of $X(t)$, compute their mean, standard deviation and show three of these samples." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "tags": [ "hide-input" ] }, "outputs": [], "source": [ "# Set random seed \n", "np.random.seed(0)\n", "\n", "# Time to simulate for, discretisation level and number of paths\n", "T = 1\n", "N = 500\n", "S = 1000\n", "dt = T / N\n", "\n", "# Times to sample at\n", "t = np.linspace(0, T, N)\n", "\n", "# Sample dW's and compute cumulative sum\n", "dW = dt ** 0.5 * np.random.normal(size=(N - 1, S))\n", "W = np.concatenate(\n", " [np.zeros(shape=(1, S)), np.cumsum(dW, axis=0)],\n", " axis=0,\n", ")\n", "\n", "# Compute values \n", "X = np.exp(t[:, None] + 0.5 * W)\n", "X_mean = np.mean(X, axis=1)\n", "X_stdev = np.var(X, axis=1) ** 0.5" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "tags": [ "center-output", "remove-input" ] }, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(10, 4))\n", "plt.plot(t, X[:, 0], color='r', label='Sample 1')\n", "plt.plot(t, X[:, 1], color='g', label='Sample 2')\n", "plt.plot(t, X[:, 2], color='b', label='Sample 3')\n", "\n", "plt.plot(t, X_mean, color='k', label='Mean')\n", "plt.fill_between(\n", " t,\n", " X_mean - 2 * X_stdev,\n", " X_mean + 2 * X_stdev,\n", " color='gray',\n", " alpha=0.3,\n", " label='Std. dev.',\n", ")\n", "\n", "plt.xticks(np.linspace(0, T, 6), fontsize=16)\n", "plt.yticks(np.linspace(0, 6, 4), fontsize=16)\n", "plt.xlim([0, 1])\n", "plt.ylim([0, 6])\n", "\n", "plt.title(\n", " r'Samples from $\\exp\\left(t + \\frac{1}{2} W(t)\\right)$',\n", " fontsize=24,\n", ")\n", "plt.xlabel(r'$t$', fontsize=22)\n", "plt.ylabel(r'$X(t)$', fontsize=22)\n", "plt.legend(fontsize=18)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Evaluating a stochastic integral\n", "\n", "The next thing that we look at is the evaluation of a stochastic integral. Let's consider the integral\n", "\n", "$$\\begin{align}\n", "Y = \\int^1_0 W(t)~dW(t),\n", "\\end{align}$$\n", "\n", "where $W(t)$ is a Wiener process. To evaluate it, we first need to define what the $\\int$ means in the stochastic case. In the deterministic case we use Riemann integral, which is the limit of a discretised sum:\n", "\n", "$$\\begin{align}\n", "R = \\int^b_a f(t)~dt = \\lim_{N \\to \\infty} \\sum_{n = 0}^{N - 1} f\\left(a + n\\delta t\\right) \\delta t\n", "\\end{align}$$\n", "\n", "where $\\delta t = \\frac{b - a}{N}$. We have chosen to evaluate $f$ on the left side of the discretisation bins. In the deterministic case, it does not matter where we evaluate $f$ within a discretisation bin, meaning that the integral\n", "\n", "$$\\begin{align}\n", "R_\\lambda = \\lim_{N \\to \\infty} \\sum_{n = 0}^{N - 1} f\\left(a + \\left(n + \\lambda\\right) \\delta t\\right) \\delta t,\n", "\\end{align}$$\n", "\n", "does not depend on the choice of $\\lambda \\in [0, 1]$ when we take $\\delta t \\to 0$ (provided $f$ is sufficiently well behaved). For stochastic integrals, this does not hold: the choice of where to evaluate the integrand affects the value of the integral, even in the limit $\\delta t \\to 0$ - we will see an example from Higham shortly. We therefore have to make a choice in defining the integral. Two widespread choices are the Ito and the Stratonovich integrals:\n", "\n", "$$\\begin{align}\n", "&\\int^b_a h(t) dW(t) = \\lim_{N \\to \\infty} \\sum_{n = 0}^{N - 1} h(t_n) \\left(W(t_{n + 1}) - W(t_n)\\right), \\text{ Ito}.\\\\\n", "\\\\\n", "&\\int^b_a h(t) dW(t) = \\lim_{N \\to \\infty} \\sum_{n = 0}^{N - 1} h\\left(\\frac{t_n + t_{n+1}}{2}\\right) \\left(W(t_{n + 1}) - W(t_n)\\right), \\text{ Stratonovich}.\n", "\\end{align}$$\n", "\n", "where we have defined $t_n = a + n \\delta t$. While Ito evaluates the integrand on the left side of the discretisation bin ($\\lambda = 0$), Stratonovich evaluates it at the midpoint of the bin ($\\lambda = 1/2$). Our stochastic integral of interest\n", "\n", "$$\\begin{align}\n", "Y = \\int^1_0 W_t~dW_t,\n", "\\end{align}$$\n", "\n", "is equal to the following values under the Ito and Stratonovich definitions:\n", "\n", "$$\\begin{align}\n", "\\int^1_0 W_t~dW_t = \\begin{cases}\n", "\\frac{1}{2}W(T)^2 - \\frac{1}{2}T^2 & \\text{ under Ito,}\\\\\n", "\\frac{1}{2}W(T)^2 & \\text{ under Stratonovich.}\n", "\\end{cases}\n", "\\end{align}$$\n", "\n", "\n", ":::{dropdown} Proof: Evaluating $~\\int W(t)~dW(t)~$ under the Ito and Stratonovich integrals\n", " \n", "If we use the Ito integral, we have\n", " \n", "$$\\begin{align}\n", "\\sum_{n = 0}^{N - 1} W(t_n) \\left(W(t_{n + 1}) - W(t_n)\\right) &= \\frac{1}{2}\\sum_{n = 0}^{N - 1} \\left( W(t_{n + 1})^2 - W(t_n)^2 - (W(t_{n + 1}) - W(t_n))^2 \\right)\\\\\n", "&= \\frac{1}{2}\\left( W(T)^2 - W(0)^2 - \\sum_{n = 0}^{N - 1} (W(t_{n + 1}) - W(t_n))^2 \\right).\n", "\\end{align}$$\n", " \n", "The distribution of $(W(t_{n + 1}) - W(t_n))^2$ has mean equal to the second moment of $\\Delta W_{t_n}$ and variance equal to the fourth moment of $\\Delta W_{t_n}$, which are $\\delta t$ and $3 \\delta t^2$ respectively. Therefore, the sum above is a random variable with mean $T$ and variance $\\mathcal{O}(\\delta t)$ - where we have used the fact that the summands are independent, so the variance of the sum is the sum of the variances. So in the limit of $\\delta t \\to 0$, the integral converges to $\\frac{1}{2}W(T)^2 - \\frac{1}{2}T^2$ under the Ito definition.\n", " \n", "By contrast, if we use the Stratonovich integral, we have\n", " \n", "$$\\begin{align}\n", "\\sum_{n = 0}^{N - 1} W\\left(\\frac{t_{n+1} + t_n}{2}\\right) \\left(W(t_{n + 1}\\right) - W(t_n)) &= \\sum_{n = 0}^{N - 1} \\left(\\frac{W(t_{n + 1}) + W(t_n)}{2} + \\Delta Z_n\\right) \\left(W(t_{n + 1}\\right) - W(t_n))\\\\\n", "&= \\sum_{n = 0}^{N - 1} \\frac{1}{2} W(t_{n + 1})^2 - \\frac{1}{2} W(t_n)^2 + \\Delta Z_n \\left(W(t_{n + 1}\\right) - W(t_n))\\\\\n", "&= \\frac{1}{2} W(T)^2 - \\frac{1}{2} W(0)^2 + \\sum_{n = 0}^{N - 1} \\Delta Z_n \\left(W(t_{n + 1}\\right) - W(t_n)),\n", "\\end{align}$$\n", " \n", "where $\\Delta Z_n \\sim \\mathcal{N}(0, \\delta t / 4)$. To obtain the first equality above, we used the fact that\n", " \n", "$$\\begin{align}\n", "p\\left(W\\left(\\frac{t_{n+1} + t_n}{2}\\right)\\big |~ W(t_{n+1}), W(t_{n})\\right) = \\frac{p\\left( W(t_{n+1}) | W\\left(\\frac{t_{n+1} + t_n}{2}\\right)\\right) p\\left( W\\left(\\frac{t_{n+1} + t_n}{2}\\right) \\big | W(t_{n}) \\right) p(W(t_n))}{p(W(t_{n+1}), W(t_{n}))},\\\\\n", "\\end{align}$$\n", " \n", "and observing that the distribution above has the form of a product of normal distributions over $W\\left(\\frac{t_{n+1} + t_n}{2}\\right)$, we arrive at the result:\n", " \n", "$$\\begin{align}\n", "p\\left(W\\left(\\frac{t_{n+1} + t_n}{2}\\right)\\big |~ W(t_{n+1}), W(t_{n})\\right) = \\mathcal{N}\\left(W\\left(\\frac{t_{n+1} + t_n}{2}\\right); \\frac{W(t_{n+1}) + W(t_{n})}{2}, \\frac{\\delta t}{4}\\right).\n", "\\end{align}$$\n", " \n", "Since $\\Delta Z_n$ is independent of $W_t$, the term $\\Delta Z_n \\left(W(t_{n + 1}\\right) - W(t_n))$ has mean 0, and variance $\\delta t^2 / 4$. Therefore, the sum term has mean 0 and variance $\\mathcal{O}(\\delta t)$, so in the limit of $\\delta t \\to 0$ the integral converges to $\\frac{1}{2}W(T)^2$ under the Stratonovich definition.\n", "\n", "::: " ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "tags": [ "remove-input" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Ito integral approximation (exact): 1.419 (1.415)\n", "Stratonovich approximation (exact): 1.917 (1.915)\n" ] } ], "source": [ "# Set random seed\n", "np.random.seed(3)\n", "\n", "# Time to simulate for and discretisation level\n", "T = 1\n", "N = int(1e6)\n", "dt = T / N\n", "\n", "# Sample dW's and compute cumulative sum\n", "dW = dt ** 0.5 * np.random.normal(size=(N,))\n", "W = np.concatenate([[0], np.cumsum(dW)])\n", "\n", "# Evaluate the Ito integral\n", "ito_approx = np.sum(W[:-1] * dW)\n", "ito_exact = 0.5 * W[-2] ** 2 - 0.5 * T\n", "\n", "# The Wiener process must be evaluated at the midpoints\n", "# for the Stratonovich integral\n", "W_midpoint = (W[:-1] + W[1:]) / 2\n", "W_midpoint = W_midpoint + np.random.normal(0, (dt / 4) ** 0.5, size=(N,))\n", "\n", "# Evaluate the Stratonovich integral\n", "strat_approx = np.sum(W_midpoint * dW)\n", "strat_exact = 0.5 * W[-2] ** 2\n", "\n", "print(f'Ito integral approximation (exact): {ito_approx:.3f} ({ito_exact:.3f})')\n", "print(f'Stratonovich approximation (exact): {strat_approx:.3f} ({strat_exact:.3f})')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The difference between the Ito and Stratonovich integrals does not vanish as $dt \\to 0$, which you can verify by experimenting with $dt$ above. The choice of definition has implications about the resulting integral (itself a stochastic process), which may be more or less appropriate for different applications. From here onwards we will work with the Ito integral exclusively, although much of the discussion is ammenable to the Stratonovich integral too.\n", "\n", "## Euler-Maruyama method\n", "\n", "The Euler-Maruyama method is the analoge of the Euler method for deterministic integrals, applied to the stochastic case.\n", "\n", ":::{prf:definition} Euler-Maruyama method\n", "\n", "Given a scalar SDE with drift and diffusion functions $f$ and $g$\n", " \n", "$$\\begin{align}\n", "dX(t) = f(X(t))dt + g(X(t)) dW(t),\n", "\\end{align}$$\n", " \n", "the Euler-Maruyama method approximates $X$ by\n", "\n", "$$\\begin{align}\n", "X_{n + 1} = X_n + f(X_n) \\Delta t + g(X_n) \\Delta W_n,\n", "\\end{align}$$\n", "\n", "where $\\delta t > 0$ is the time step, $X_n = X(t_n), W_n = W(t_n)$ and $t_n = n\\delta t$.\n", "\n", ":::\n", "\n", "Let's look at an implementation of the Euler-Maruyama (EM) method. \n", "The `euler_maruyama` below function takes the drift and diffusion functions $f$ and $g,$ and applies the EM algorithm, from the specified initial conditions.\n", "Note that we can sample the `dW` in advance." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "def euler_maruyama(seed, X0, T, N, f, g):\n", " \n", " # Set random seed\n", " np.random.seed(seed)\n", " \n", " # Set discretisation, initial value and times at which to evaluate\n", " dt = T / N\n", " X = [X0]\n", " t = np.linspace(0, T, N + 1)\n", " \n", " # Sample Wiener process dW's\n", " dW = dt ** 0.5 * np.random.normal(size=(N,))\n", " \n", " for i in range(N):\n", " \n", " # Apply Euler-Maruyama at each point in time\n", " dX = f(X[-1], t[i]) * dt + g(X[-1], t[i]) * dW[i]\n", " \n", " # Store the new X\n", " X.append(X[-1] + dX)\n", " \n", " # Compute W to return it at the end\n", " W = np.concatenate([[0], np.cumsum(dW)])\n", " \n", " return t, X, W" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Below is the definition of `f` and `g` that we will be integrating, namely\n", "\n", "$$\\begin{align}\n", "f(x, t) &= \\lambda x,\\\\\n", "g(x, t) &= \\mu x,\\\\\n", "\\end{align}$$\n", "\n", "known as the Black-Scholes model. This is implemented as a closure, i.e. `f_g_black_scholes` takes in the appropriate `lambda` and `mu` and returns the corresponding `f` and `g`." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "def f_g_black_scholes(lamda, mu):\n", " \n", " def f(X, t):\n", " return lamda * X\n", " \n", " def g(X, t, grad=False):\n", " return mu if grad else mu * X\n", " \n", " return f, g" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We choose these drift and diffusion terms because the associated SDE has a closed form solution, with which we can compare our numerical solution. The analytic solution to the Black-Scholes model is\n", "\n", "$$\\begin{align}\n", "X_t = X_0 \\exp \\left[\\Big(\\lambda - \\frac{1}{2} \\mu^2\\Big)~t + \\mu W(t)\\right],\n", "\\end{align}$$\n", "\n", "which we implement in `exact_black_scholes` below. Unlike ODEs, whose solution is a unique function, the solution of an SDE depends on the random noise sample $W(t)$. It's important to remember to share the same $W(t)$ sample between the exact solution and its numerical approximation." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "def exact_black_scholes(X0, t, W, lamda, mu):\n", " return X0 * np.exp((lamda - 0.5 * mu ** 2) * t + mu * W)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We are can now run the EM method and compare it aginst the exact solution below." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "tags": [ "remove-cell" ] }, "outputs": [], "source": [ "# Black-Scholes parameters\n", "lamda = 2\n", "mu = 1\n", "\n", "# Seed and integration parameters\n", "seed = 0\n", "X0 = 1\n", "T = 1\n", "N = int(1e2)\n", "\n", "# Get drift and diffusion functions of the Black-Scholes model\n", "f, g = f_g_black_scholes(lamda=lamda, mu=mu)\n", "\n", "# Solve approximately via the EM method\n", "t, X, W = euler_maruyama(seed=seed, X0=X0, T=T, N=N, f=f, g=g)\n", "\n", "# Get the exact solution for the same W sample\n", "X_exact = exact_black_scholes(X0=X0, t=t, W=W, lamda=lamda, mu=mu)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "tags": [ "center-output", "remove-input" ] }, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(10, 4))\n", "plt.scatter(t, X_exact, s=20, marker=\"x\", color='k', zorder=1, label='Exact solution')\n", "plt.scatter(t, X, s=20, marker='x', color='red', zorder=2, label='Euler-Maruyama')\n", "\n", "plt.xlim([0, T])\n", "plt.xticks(np.linspace(0, 1, 6), fontsize=16)\n", "plt.yticks(np.linspace(0, 8, 5), fontsize=16)\n", "plt.title('EM and exact solutions of Black-Scholes', fontsize=24)\n", "plt.xlabel('$t$', fontsize=22)\n", "plt.ylabel('$X(t)$', fontsize=22)\n", "plt.legend(loc=\"upper left\", fontsize=18)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can change the accuracy of the solution by adjusting $N$. As a fun aside, what if we try out a different drift term? One nice choice is\n", "\n", "$$\\begin{align}\n", "f(x, t) &= \\omega~\\text{cos}(\\omega t),\\\\\n", "g(x, t) &= \\mu x.\\\\\n", "\\end{align}$$\n", "\n", "In the case $\\mu = 0$, the solution is the deterministic function $X_t = \\text{sin}(\\omega t)$. When $\\mu \\neq 0$, the solution will be perturbed by the gaussian noise." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "def f_g_sine(omega, mu):\n", " \n", " def f(X, t):\n", " return omega * np.cos(omega * t)\n", " \n", " def g(X, t):\n", " return mu * X\n", " \n", " return f, g" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "tags": [ "remove-cell" ] }, "outputs": [], "source": [ "# SDE parameters\n", "omega = 2. * np.pi\n", "mu = 0.3\n", "\n", "# Seed and integration parameters\n", "seed = 0\n", "X0 = 1\n", "T = 2\n", "N = int(1e3)\n", "\n", "# Get drift and diffusion functions of our sine model\n", "f, g = f_g_sine(omega=omega, mu=mu)\n", "\n", "# Solve approximately via the EM method\n", "t, X, W = euler_maruyama(seed=seed, X0=X0, T=T, N=N, f=f, g=g)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "tags": [ "center-output", "remove-input" ] }, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(10, 4))\n", "plt.scatter(t, X, s=1, marker='x', color='red', zorder=2)\n", "\n", "plt.xlim([0, T])\n", "plt.xticks(np.linspace(0, T, 6), fontsize=16)\n", "plt.yticks(np.linspace(-3, 3, 5), fontsize=16)\n", "plt.title('EM solution of sine model', fontsize=24)\n", "plt.xlabel('t', fontsize=22)\n", "plt.ylabel('$X(t)$', fontsize=22)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Strong and weak convergence\n", "\n", "Since the choice of the number of bins $N$ of the discretisation affects the accuracy of our method, we are interested in how quickly the approximation converges to the exact solution as a function of $N$. To do so, we must first define *what convergence means* in the stochastic case, which leads us to two disctinct notions of convergence, the strong sence and the weak sense.\n", "\n", ":::{prf:definition} Strong convergence\n", "\n", "A method for approximating a stochastic process $X(t)$ is said to have strong order of convergence $\\gamma$ if there exists a constant such that\n", "\n", "$$\\begin{align}\\mathbb{E}|X_n - X(\\tau_n)| \\leq C\\Delta t^\\gamma\n", "\\end{align}$$\n", "\n", "for any fixed $\\tau_n = n\\Delta t \\in [0, T]$ and $\\Delta t$ sufficiently small.\n", "\n", ":::\n", "\n", "Strong convergence refers to the rate of convergence of the approximation $X_n$ to the exact solution $X(\\tau_n)$ as $\\Delta t \\to 0$, in expectation. A weaker condition for convergence is rate at which the expected value of the approximation converges to the true expected value, as $\\Delta t \\to 0$, as given below.\n", "\n", "\n", ":::{prf:definition} Weak convergence\n", "\n", "A method for approximating a stochastic process $X(t)$ is said to have weak order of convergence $\\gamma$ if there exists a constant such that\n", " \n", "$$\\begin{align}|\\mathbb{E}[X_n] - \\mathbb{E}[X(\\tau_n)]| \\leq C\\Delta t^\\gamma\n", "\\end{align}$$\n", " \n", "for any fixed $\\tau_n = n\\Delta t \\in [0, T]$ and $\\Delta t$ sufficiently small.\n", "\n", ":::\n", "\n", "The paper states without proof that, under conditions on $f$ and $g$, Euler-Maruyama has strong order of convergence $\\frac{1}{2}$ and weak order of convergence $1$.\n", "We do not provide a proof for any of the above statements, but instead evaluate the rate of convergence empirically." ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "tags": [ "remove-cell" ] }, "outputs": [], "source": [ "def parallel_euler_maruyama(seed, num_paths, X0, T, N, f, g):\n", " \n", " # Set the random seed\n", " np.random.seed(seed)\n", " \n", " # Time increment\n", " dt = T / N\n", " \n", " # Set initial X values\n", " X = X0 * np.ones(shape=(num_paths, N + 1))\n", " \n", " # Times at which to evaluate the integral\n", " t = np.linspace(0, T, N + 1)\n", " \n", " # Wiener process samples\n", " dW = dt ** 0.5 * np.random.normal(size=(num_paths, N))\n", " \n", " for i in range(N):\n", " \n", " # Calculate new X according to EM rule\n", " X[:, i+1] = X[:, i] + f(X[:, i], t[i]) * dt + g(X[:, i], t[i]) * dW[:, i]\n", " \n", " W = np.concatenate(\n", " [np.zeros(shape=(num_paths, 1)),\n", " np.cumsum(dW, axis=1)],\n", " axis=1,\n", " )\n", " \n", " return t, X, W" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "tags": [ "center-output", "remove-input" ] }, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Black-Scholes parameters\n", "lamda = 2\n", "mu = 1\n", "\n", "# Seed and integration parameters\n", "seed = 0\n", "X0 = 1\n", "T = 1\n", "Ns = (10 ** np.linspace(2, 4, 4)).astype(dtype=np.int32)\n", "num_paths = int(1e4)\n", "\n", "# Get drift and diffusion functions of the Black-Scholes model\n", "f, g = f_g_black_scholes(lamda=lamda, mu=mu)\n", "\n", "dts = []\n", "X_approx = []\n", "X_exacts = []\n", "\n", "for N in Ns:\n", " \n", " t, X, W = parallel_euler_maruyama(\n", " seed=seed,\n", " num_paths=num_paths,\n", " X0=X0,\n", " T=T,\n", " N=N,\n", " f=f,\n", " g=g,\n", " )\n", "\n", " X_exact = exact_black_scholes(\n", " X0=X0,\n", " t=t[None, :],\n", " W=W,\n", " lamda=lamda,\n", " mu=mu,\n", " )\n", " \n", " dts.append(T / N)\n", " X_approx.append(X[:, -1])\n", " X_exacts.append(X_exact[:, -1])\n", " \n", "X_approx = np.stack(X_approx, axis=0)\n", "X_exacts = np.stack(X_exacts, axis=0)\n", "\n", "X_abs_diffs = np.abs(X_approx - X_exacts)\n", "em_strong_errors = np.mean(X_abs_diffs, axis=1)\n", "\n", "X_approx_means = np.mean(X_approx, axis=-1)\n", "X_exacts_means = np.mean(X_exacts, axis=-1)\n", "em_weak_errors = np.abs(X_approx_means - X_exacts_means)\n", "\n", "plt.figure(figsize=(10, 4))\n", "\n", "plt.subplot(121)\n", "plt.plot(dts, em_strong_errors, color='k')\n", "\n", "plt.loglog()\n", "plt.xticks(fontsize=18)\n", "plt.yticks([1e-2, 1e-1, 1e0], fontsize=18)\n", "plt.xlabel(r'$\\delta t$', fontsize=20)\n", "plt.ylabel(r'$\\mathbb{E}|X_n - X(t_n)|$', fontsize=20)\n", "plt.title('Euler-Maruyama\\nstrong convergence', fontsize=24)\n", "\n", "plt.subplot(122)\n", "plt.plot(dts, em_weak_errors, color='k')\n", "\n", "plt.loglog()\n", "plt.xticks(fontsize=18)\n", "plt.yticks([1e-3, 1e-2, 1e-1, 1e0], fontsize=18)\n", "plt.xlabel(r'$\\delta t$', fontsize=20)\n", "plt.ylabel(r'$|\\mathbb{E}X_n - \\mathbb{E}X(t_n)|$', fontsize=20)\n", "plt.title('Euler-Maruyama\\nweak convergence', fontsize=24)\n", "\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Milstein's higher order method\n", "\n", "Just as higher order methods for ODEs exist for obtaining refined estimates of the solution, so do methods for SDEs, such as Milstein's higher order method.\n", "\n", ":::{prf:definition} Milstein's method\n", "\n", "Given a scalar SDE with drift and diffusion functions $f$ and $g$\n", "\n", "$$\\begin{align}\n", "dX(t) = f(X(t))dt + g(X(t)) dW(t),\n", "\\end{align}$$\n", "\n", "the Milstein method approximates $X$ by\n", " \n", "$$\\begin{align} X_{j + 1} = X_j + f(X_j) \\Delta t + g(X_j) \\Delta W_j + \\frac{1}{2}g(X_j)g'(X_j) (\\Delta W_j^2 - \\Delta t),\n", "\\end{align}$$\n", " \n", "where $\\Delta t > 0$ is the time step, $X_j = X(\\tau_j), W_j = W(\\tau_j)$ and $\\tau_j = j\\Delta t$.\n", " \n", ":::\n", "\n", " Milstein's method achieves a strong convergence rate of $1$ and a weak convergence rate of $1$." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "tags": [ "remove-cell" ] }, "outputs": [], "source": [ "def parallel_milstein(seed, num_paths, X0, T, N, f, g):\n", " \n", " # Set the random seed\n", " np.random.seed(seed)\n", " \n", " # Time increment\n", " dt = T / N\n", " \n", " # Set initial X values\n", " X = X0 * np.ones(shape=(num_paths, N + 1))\n", " \n", " # Times at which to evaluate the integral\n", " t = np.linspace(0, T, N + 1)\n", " \n", " # Wiener process samples\n", " dW = dt ** 0.5 * np.random.normal(size=(num_paths, N))\n", " \n", " for i in range(N):\n", " \n", " # Compute the EM term and the higher order correction term\n", " dX = f(X[:, i], t[i]) * dt + g(X[:, i], t[i]) * dW[:, i]\n", " dX = dX + 0.5 * g(X[:, i], t[i]) * g(X[:, i], t[i], grad=True) * (dW[:, i] ** 2 - dt)\n", " \n", " X[:, i+1] = X[:, i] + dX\n", " \n", " W = np.concatenate([np.zeros(shape=(num_paths, 1)), np.cumsum(dW, axis=1)], axis=1)\n", " \n", " return t, X, W" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's first use Milstein's method to get a single solution of the Black-Scholes model, as we did for EM." ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "tags": [ "remove-cell" ] }, "outputs": [], "source": [ "# Black-Scholes parameters\n", "lamda = 2\n", "mu = 1\n", "\n", "# Seed and integration parameters\n", "seed = 0\n", "X0 = 1\n", "T = 1\n", "N = int(1e2)\n", "num_paths = 1\n", "\n", "# Get drift and diffusion functions of the Black-Scholes model\n", "f, g = f_g_black_scholes(lamda=lamda, mu=mu)\n", "\n", "# Solve using milstein's method\n", "t, X, W = parallel_milstein(\n", " seed=seed,\n", " num_paths=num_paths,\n", " X0=X0,\n", " T=T,\n", " N=N,\n", " f=f,\n", " g=g,\n", ")\n", "\n", "X_exact = exact_black_scholes(\n", " X0=X0,\n", " t=t[None, :],\n", " W=W,\n", " lamda=lamda,\n", " mu=mu,\n", ")" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "tags": [ "center-output", "remove-input" ] }, "outputs": 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+yRypp0Zg11uTolsTpd7eUWYK2QgjZLJ+FdPHVM5QtbUWiXFN0uZMtNSa3L3dq/213zeCG1/yCmVuZHN0cmVhbQplbmRvYmoKMjEgMCBvYmoKPDwgL0xlbmd0aCA0MDIgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicNVJN0t0wCNvnFFygM+bX+Dyv800Xr/ffVsLpIkGxA5KAiJAlvuXXkQiTqCW/9SGskr8DvOX7hJYUYPFPT2P4PK5BYMdEU6xbynFsvQm+j6E8LvRMmsabproIOsENGm2cBaoXlZTyfYLlN3LwyxLdSF9bTJX5Dk4H50FMPOCjhsvsKPW9aJcYMv20WKWEjrpUCeupEOC0dSaibl+kmzcK6fi2YQxm4wTMsdZEB8PV4q0j1HeMSa+5gXlP2s7p2EQ8t5p73hxDBJfr5BiyV4uCy1cIDDJ8HroB2Huu67z/1y2Q6Bx0M+TCMUGbbIfYnsaFrxHbEsl21oza7I4cTXlPHOOKEbpgmGIpPZaOFc7fMVnfbLYX+RonPXYmjloiVW4AEdVuEEM7/AR1hxpBGES7vOv2ef7M2NQ2Vm+GWtw9g0ML7olDccxYbNn44SmazyZm3mjTYrxBxs42/GIUdyboTKICOwOJ2LLg8hm2O7gRSTkGx0UbBlEYqe0UUqPQu2SogCUYrTDyXzP1//wDoa+dzAplbmRzdHJlYW0KZW5kb2JqCjIyIDAgb2JqCjw8IC9MZW5ndGggMjkzIC9GaWx0ZXIgL0ZsYXRlRGVjb2RlID4+CnN0cmVhbQp4nD1RS3LFMAjb+xS6wJvhH/s86XS6eL3/toK0XcQoNgghUgQCx+sgaqO240OXphMqvgf5VXgvLUeIQnkfpjBJhF+4lxmL02B5kURge0+8l4cMctKGM0sv+OmaSIVnTlMXm2jRNY20ChEJlZqeD1kWNJgtrdY8OtxLz6hXkr0MSmWqvN5FPtRhvz5Ze6+JPUuDIyAu6Q+lzvdkgVEx/1lrT2CHHEAeoxk8rdWgyPxeThC7/Wmdrb0bbqHUx8773833+nr81P3r7BVdSJA9c7sw9nqLNc5pHII9rb0VTF/bp4FTHdM8SJ0tpy6cpNEyAiaO0Y2M1J3gnNKjjR8/tfO4yH1mKcYVPssxy+eGi3G+6ma8ZpRm4ppnDD/4G+fmaJ8/bCdtYAplbmRzdHJlYW0KZW5kb2JqCjIzIDAgb2JqCjw8IC9MZW5ndGggMjY4IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlID4+CnN0cmVhbQp4nDWRSW5EMQhE9z4FF4hkBmP7PD9qZdG5/zYPWllYlBiKKhwRMsWWfF0JNYmc8q3D7xFdKb+jkhpb3sOPy1Vx0svEfYqbPMNuSsCwi8K0mZ6h6xZQDZ7sI8vJhsmt3mYkmDYDABI6PFk2UTI/HFMiXPSu1mVorcjEyUZuKI6LJqLVjB/r6TLgZ30i2p625FEZ9jjxBPNd2SBlJqlgwJd2pEK9M3how7M0arumwkHeIGWjirEtsGt7o6V7EidWRlHrdPjpiDOmC2niCH7122xUYKj9OjkxPipa6y+kt06afE19ympMZbsczoPI/HRnnxVBANcpiXXWnUoXO9cu02pT/n/6GT/j9QcnnmGHCmVuZHN0cmVhbQplbmRvYmoKMjQgMCBvYmoKPDwgL0xlbmd0aCAzMTUgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicNVE5cgMxDOv3FfyAZ8RT1Huc8aRw/t8G4NoVIYoHAOZassS2PI7EUal2+dGLMEz+Lo/g1xtgyaM+IDfBalEXq2SFqTE8L80iULx1yW7JhSzG9OEeDWWwlcgS7JTY4p7SLd5Mq7UEErq3RIEeMCMbdJCvkogUkgnd+Llp+ZfgcYzcUoujIUuDNZZShm5wKwuxExMxt29k2FgGUdsmciOQjwMOlNjZmFewS8/YBXHoLGTfgxS7bnRAzkqokL3NUm+QksgxMZBLsg9s4XufCXA/p/o+w/v6HVW67pNAgdtHqaPQ7YCVwRWwQg31AJnNcbjbvCfiBwcn0lEAj2k+uNP1gJ+4EI7qSydazD2AFEoV/qlDIdgf0lZI4324F0ty2HK7ETjMTwR07bGfN75vdkK+gp4Q9/oHHj94TAplbmRzdHJlYW0KZW5kb2JqCjI1IDAgb2JqCjw8IC9MZW5ndGggMjQ3IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlID4+CnN0cmVhbQp4nC2RS24EMQhE9z4FF4hkPjb4PBNFWczcf5uHOwsLGiiqio4ImWJLvo6EhcSe8q0jZonukM/ooq6U9/DqtrhGD5tezGvo2p2oTp7klmVUw+RY79XIDqZJtZMqgdPZVCZ+elhtSniK5rwCzO1GAOfJ3OtOeCoYpeOHKgpjUrFNZHc2ycmbvMdBpaNguhhPJyqrsYp2XXyBU+cFXWzRYT9Ywz59W9EBGbU7aesbEUDKW8MqgcQxolfCc7TX+G1uh+8zHObO3ngBupQLEAuXSefUPQvHR5PhsO9teOnYep6KLmeCiB8/t6P8iqQyn2P+Ezb3zx+nd1dTCmVuZHN0cmVhbQplbmRvYmoKMjYgMCBvYmoKPDwgL0xlbmd0aCA0MTkgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicNZNLjhxBCET3dQouYCn55O88bY288Nx/6xeUZ1GChgQiAnqOYcPmsl+eVjVt7WW//ZHrc9r36w2SsW2F5XXLsM+TZPk5TpdGvfbz+I32vBZ59/E+36VqBvnwNnWJyjlkysLLTlikwvdaEPMkfhl/lmV38cTbNB9hNZaRqLf9sJoNsIhpUgzCPXLbGbYGfZxUNsgZtpabM0mMAyCy0BrVXhZWfLdqikxBdTGzomzufG13k1eXDByKcQlmyUa3rfdJN1S7BzbIsb3lw0tFR3Msj+ZMN7GGVAuvrv9X8Xn+tOia+f16mkV2Up8BTn8ZCPG2EGvWEixC9vMEM+T5Yc8I7QsbQuPaPEuWvnPf1/aG2nNqeFFgVU0toVGXKtD7Bj3HgJ55VNN8xrX0y3rEE7VmH03xllNo9BzCDx9xC3SJHXATzuT7ywxUOUxfrzqOalGNOGEmXCjlwiU7T2d0dnyVtjelEOjbOboqCw6DPcW8MgLrctKlFFAPpHo/R2eZ65ouHuPRcXlTCcpQOHXmQwwSphFsgkvT3+GHj7h9/QN6A6H+CmVuZHN0cmVhbQplbmRvYmoKMjcgMCBvYmoKPDwgL0xlbmd0aCA5MSAvRmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeJxNjDsOwDAIQ/ecgiMACaDep6o6NPdfy0dpuuAn21gQAYHYz7AOagontUBSg9m6YdKThK588KeV0A+4U3RZc6q+NlUa65vq9c69cGeS2HBX1uAKo3i9EGIioAplbmRzdHJlYW0KZW5kb2JqCjI4IDAgb2JqCjw8IC9MZW5ndGggNTYgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicMzY2VjBQMLZUMLQ0UTCyMFUwMjVXSDHkAjFBQrlcULkcEAMkmcMFU5YDV5XDlcGVBgAcng6iCmVuZHN0cmVhbQplbmRvYmoKMjkgMCBvYmoKPDwgL0xlbmd0aCAyMzYgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicNVA7ksUgDOs5hY6AjQ34PNnZ2SJ7//ZJJK9BCjb6xNdGh00engNzD/xYs90xh+P/YbZgKzF7cjORlbiaJdnmTTjRYL4PcmLx3HQ/G+Xvk+1Hg5aSFNDjaiIxQeuZG0WYWzKdjDOzPLnM6yAn8SRVhLPBUO8bxaTIiU3Nb5Gr/TX166qksreAwndL1mcYhw8w9JBBddCipKrTeugyOTS+EMlQrbE69sagC9sNksIKjApMP3A1phAZ+qZJP0tXI4kuf0WOeRKtQuR8mb1Tlubvp3gErBhQolYFD7if8Gpm+sFPxZtlfz+8HFiUCmVuZHN0cmVhbQplbmRvYmoKMzAgMCBvYmoKPDwgL0xlbmd0aCAxNTQgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicLU9JDgMxCLvnFX5CgEDgPVNVPUz/f62ZVIqwA2ax7sSEFIP6RqThJaPpxHeoEO4GcaIrZGMbdIFCW7hGGdZCJeKJpZ1LeAgbmqSjHDEFWQgR1tkbOrGV4JQ+cI11/koZF0iDMd3EitOSdWfBOC6Kp5x7qejKbkmz3H/i1GRgtb+A9WadnK983k7apvTm4/cen/H+ATCkMysKZW5kc3RyZWFtCmVuZG9iagozMSAwIG9iago8PCAvTGVuZ3RoIDQyMSAvRmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeJw9UztyBTEI6/cUHMHm7/Mkk0nxcv82Es5Ls9KCMQhwVcuSnfhUhXgu+dwP6ZKfJ6KAr8EdJB2yS3KpqEnuJaby8eRu8S2pCtfAWTSD6EnEkZklIlIsHadcrBkaB55zJMrEV0t4DMKj+1oWEV7ktuPweB5Eu3gYbjvi1mJe9IAdFkqSiHFlYR5IsicSFeG/N/UE6gcgFQh1auc/jl7bo9fMGWZOPThvAeEwh0vkAAJoxrFTjAOzQgTaQw3a50+V1rqWWKNKcaEdFqGwWqPRB6oK2FRHzy7cFqwEaf38YY8Hc2PFJLDMSUcoCmE/eJcu1L45Y0VTRi/JROUbRq07kS3CregYZ9hsPuXF4beZFchhvoZg4H0Lrlv/x8NYtIH9x6LEHkDzLzEKR+OpezSgNZMee+GoemMYnNuOnn3EiUyMsKZSZEQ6sn6zw+WDYEW8hSg2wT3evUbf9NzNNo5+bjRuW7IQbo9OXivO0Otawu6JwCbfmMwaVitHAFGbHrJieSTIV4vapLjse/oxz6VwJR5KzQRpxisb4fe5vZ7v5+sXpASqEAplbmRzdHJlYW0KZW5kb2JqCjMyIDAgb2JqCjw8IC9MZW5ndGggMjg1IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlID4+CnN0cmVhbQp4nDVSuXXFMAzrNQVH4KljHuflpfjZvw1AOY0A8wRJl6qo2MSTuySnypcNUpXf4WsBP8AtViChLrYk7IinhINOeUYEUrdEqixrMC3awUKTiWT0oUzC53M1PsNLryUSaOKOyBP0wBpQYnsilyrREm2eQbYcZUkKkiaaQyLkhtKPDHdxDap1P1SPgiAcx/Y/9FDZUZjcQ1iIvdcUmLcKGuA9CRswYiGBZJqwGxazAr2pirlnSR3M4ZLVgCVYE+hJ7VXHuTNIdvsJh0OKbklKqegzMAKZtXpMbG6dZv4yV/SBVkcE40PnmxeZ1wJlmbO3mU5PWrxf2GTVReUFyXYvA6R2K8LJEucNDpGo6VcYdsX/ozd6f5TP+Bnff02QbOEKZW5kc3RyZWFtCmVuZG9iagozMyAwIG9iago8PCAvTGVuZ3RoIDIyNSAvRmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeJw1UTtuQCEM2zmFL1CJfIBwnldVXXr/tXZeO9kicWKHNScm/OLjIs9E7olPG6Iehp9mtg2ZFzYDOQ9W4RmxFywQZtJ6LsEzrELEwlW99ja32LXH1hW461nkHrAaq5vjbs2gOKNgtduQr2zkUntfYp/uyFl/muwh2SH8vmGU4BnfI4p8JtM0K+npPKgI8nJNVjZb8MOs3O+xGmmSJxKzQ1d2mIA+q7PSf/A2ZkT3Rj+hCpkVe2P3q7Rbie0mIhk/WYdvF+gQSsshR9XIAgPqxrqbLPITcuM/iEJ9/QL+YlIzCmVuZHN0cmVhbQplbmRvYmoKMzQgMCBvYmoKPDwgL0xlbmd0aCAxNDYgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicPY9RDsMwCEP/cwofAYfQhPN0mvbR3f93Jq0mRfiJCGPCDAaSqp4DxzzwYis0fBu5pNdWRkEMcOoluoPp8MDZuh0YE50uhy2524IYQ3Ob5N1NqimuW8/G+XRiIVaCXsr6UaqY3OtFq/b3tJ2xKEX+0L+lfG4BmTiVMeXj3pVUQWfdUKexwt03Xu3T3j8cmTFkCmVuZHN0cmVhbQplbmRvYmoKMzUgMCBvYmoKPDwgL0xlbmd0aCAzNDIgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicTVJJbsQwDLv7FfpAAGvxovekKHpI/38tKU+AXkaMLVMkNaN36TLkMp0Sq0vMLl/adKTE2PJbyLfK06yrhLtYZLV596p386HnJFEDJIabzpsYIY4bUnvPqhaLN0C6Av0pquxPmUGyiYMhrilXiuExyt3Mz7epXFOsLyG/pkO9riXQX1KdpwCXQsTzgVChIL84UBfs6q6+TZgGOnblKm0AmwMJGcvTxj+gmQBOspwynLXLNHIansEWxrjUlDFPdWoFcl94TABby8QB50K+VDMVAWIZzBGpRRUaL8BIYDbr925ZgUHRqLSVu0DyOt6dvBt82s/ZIQjPNvf+5AKVCrloYoxRw2DTkLIzaYTvLDjO+nbbGC8OiYsiHBa0Uz/1aVVuBjdEE5JpUq2eeO29OJCUIWT+Rwy2z3/FEE6dKK2DORkQV0GVDi3lI7a8fm54+/4DKHSDbAplbmRzdHJlYW0KZW5kb2JqCjM2IDAgb2JqCjw8IC9MZW5ndGggMTQzIC9GaWx0ZXIgL0ZsYXRlRGVjb2RlID4+CnN0cmVhbQp4nD2Puw0DMQxDe0/BBQ4w9bNvnguCFMn+bagLkEoEJT1K7o6J2Di4Cj4DpfrgaHmwJj7DTvu138PYrhuYEosgDYlr1IltzWFFF0uTKxFeoE/kknnmjb9uegkuovo1BTw1E6F97eytGUZqWx0PuOK6WnZYK6a4mhCOCuft96HyzJeuVMT/iWu8xvMLQYcr9QplbmRzdHJlYW0KZW5kb2JqCjM3IDAgb2JqCjw8IC9MZW5ndGggMTM5IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlID4+CnN0cmVhbQp4nC1POQ7DMAzb9Qp+IICuyPZ7UgQd0v+vpdxOlMQLiggofOGwUfB5oogvE56qFB9J26dHzAYqFC08a8FdkctxiVcgLbc9GNjoOZrpQC7qOAbJ+eu5ZDcy/5E1OGnAMnDQZNQvKmyOttpam2r0zCY4RZKhIL22IWe3te7MCUaWFv4vXPKW+wu1WSuQCmVuZHN0cmVhbQplbmRvYmoKMzggMCBvYmoKPDwgL0xlbmd0aCAxMjUgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicLU5BDgQhCLv7in5gElBQfM9MNnvZ/1+3MCZGSiml3QWCJbhUoKFQETzaEtrAr0AP6HIo3zQ47qY+cLG1s9i9KgeyC+2Z85ilfnd5hEb8zZIT+AYnq6csBrm9sA2ahpKmsypN7TC8mgKGKH2GWlqxPXAi3+3bPn9vXSdUCmVuZHN0cmVhbQplbmRvYmoKMzkgMCBvYmoKPDwgL0xlbmd0aCAyNjUgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicNVE7lgMxCOt9Co5gPgZ8nuzblyJ7/3YlkhRjGARIgLvLloPP/Ujklh9dChOn5W88z5TX0ga8TWynhKWYuURseSwLZrvYudPAeo8Fcms8V2YccYONBOIOz/NNik5jdw2CvMuIirehJsQriGiNFt8NCyU3x4Kn/R0pqm0oyamlNnYDQi6oMfDSAtFPZCPSJdoxtQ/MjP7RMrNDxVg9g2y5m8uAk0SoXrQgjGza4AaZEhJyTxkdVn1fBF/LQa0CoVaSoGZiGdtVCnaPtzl1cUmBAjpIL4xaKtTY3HHiv68ctLjBchouaxy9Mz70XSbjHJvHDFwK0G0cAaOf8znY9/Cv9Vy//0beZD0KZW5kc3RyZWFtCmVuZG9iago0MCAwIG9iago8PCAvTGVuZ3RoIDM0NyAvRmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeJwtkjtuBDEMQ3ufQhdYwPr4d54NghTJ/ds8erYZamRbIinlPtZtuL2OZW2rcezLm49p2d3+Wsxukdt+W5y08G3p0/xwPY75Dnu3LDefdQt4xYM+7sm27ZZZNsj3abFJx6KMRU21jb4F7+br/js3X7TIATOyvmyYQ4ZSx/R8LWAeK5vDXklGTxKOoA8Fc38COnkvO9wVqTKaUhRCkZCGOE/OEMDLj4JIAGJjwPzS7VbLxBqVgj0/KhwzZJFjQcS4FaWEusdpzycQgsGxS5CppwoWlNIKchvf49JyEtfEdSfhCBaK2TObKKjAKxiQbnKCrsL0wJXiVH2EnKiuMkeZxH691Um63+i3aZiJK0Q4/ERB11SvhX4JGNTplwUqK9Zdg8oFvwdhziRuRswT9RhX8cwPDV1O5a12cZ6P2hzrepilEbBCGXc8LItmoWXqWolnHd/tp33/A7hDgCcKZW5kc3RyZWFtCmVuZG9iago0MSAwIG9iago8PCAvTGVuZ3RoIDI3NiAvRmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeJw1kUFuxUAIQ/c5hS9QacDAkPP8quqi//7bmon+IgIBY55JroUFBr6MiGpUB77tiqiTvpWpqMbfxbpRmeAiyglPQ27H67K7kW6wSkQ2LB1sns5CSNopeW/4vk98XeQ+2czFUtU2eNRCuhTS4Hg/WG73dGq0VdGErXXe5DS4N0yv/DHiYpio/btOZpr0hTb0HqEbFjG+ffUJ+8io1I0t5V6aIFJ+paJL5C0F11VcURMV06GFrsNDXtr1udvr+hWUaCX3PhnHn67Adng88XW5nE9m+rjFWfobz07qjW5k2kdBTvSq6SgbXaNciMi0X8YGU6MlSunIUo3JYVwxCWlwbUuxzzR1zHvIRynxoR3yn39b9mPQCmVuZHN0cmVhbQplbmRvYmoKNDIgMCBvYmoKPDwgL0xlbmd0aCAxNyAvRmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeJwzMjVQQMAUQ640AB2YA00KZW5kc3RyZWFtCmVuZG9iago0MyAwIG9iago8PCAvTGVuZ3RoIDE5MyAvRmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeJw9kMsNwzAMQ++eggsUsGR97HlSFD20+19LOWkOCV/0iShpTnTIwGNBc8H5PKVpBDLxbRUjf5paQgLao0rFZ8nR1irNjjHqLZIMEkwmm/h1gcF0zzHVU5yFBYPhSQmQzIxhdxgHTHYvg2jCXapcJ2mwX6j0wyaPURljJOjLnCs4NfcqzLCzSNyodio3qgxNea/d9sCbTseVvUmG/qn3vX3WOQLndP7XOKIM6z4R5WiqG9TocuA66dHe7fUDAUlF8wplbmRzdHJlYW0KZW5kb2JqCjQ0IDAgb2JqCjw8IC9MZW5ndGggMjI5IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlID4+CnN0cmVhbQp4nDVRu3HFMAzrNQVG4F/iPC+XS+Hs3waUL4UNWCQhgE4RCKz5ih2oXfjSNVR943eFTf1Zt+OC2qBsmOiwdFg07PCkA86Ke+Kz3Mh28msjdL9YOhWyFFYskCGcEWTNjLE/D9VCUWLQPhc/S8veE3dkE+kjafOzWpGZ2ATjowwyAz6JNh1SKr1RvIseR9B8BJ2CeaC7b2zzc5Em2i/zOclmCHbaVELqXh5UyJQXaZ4VsmAcP4wTxlUkLK9xGqHYs5wOXnJqSIiD3aEFJh6JRnNFwd1rclUCjXH6/zee9bO+/wBt11EWCmVuZHN0cmVhbQplbmRvYmoKNDUgMCBvYmoKPDwgL0xlbmd0aCAyNTcgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicTZFLbsQwDEP3PgUvMID1sSyfJ0XRRXr/bSl1gnYTPUexSCprTkwcvA48J3xNfMgoJH03WOAerglVWCo2zA2vxDXMEq/FdsGOhvgFPQ45UI+aLceqXEPW6rMuKGQaCNdIoTq2YJ96itRwgsXhqALej4ALfSVcJ/tOy3pgBjdnBKf0NU5nuFvoj0T3Q5O+QpDEvbBrkuQBWzqjbKh2uYaWFM8ZJWrU3hXB6C65BKWosyRnSi+Dzic/pCOhw662u2OMkr0e0vtWhTGuyDutzYoTVKskwr1UtbbxZHLZ/0jmflNaryRqM4koM7WMxddL6kf69irX82Pv8TU+fwBzdF6QCmVuZHN0cmVhbQplbmRvYmoKNDYgMCBvYmoKPDwgL0xlbmd0aCAzMzUgL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicTVK5cQQxDMu3CpbAV5Tqscfj4Nx/aoA6zzhZUHxBcEtVVKzxyT6SpfJpD02Vn8d7A1+DVjROii0JLbEtYSpu8vEEgp54w6cXzrg3OuK9pAzgUg23o8ZOo52lSycN31Ih5s7KgUAmDSZHcbwtJ8ANg7SG9sCQQwkm5RJXKRBaSF1ghRnjPiqLPtcUD2MlFohmLfaKs2WbpKU0AKNRDlmC0gCKnAjokGO8xqhRCP3+WTeoDguCbcjgfdFmr02FDoiGxAIGaeZ09hHKMZIqe1LSiVBC3VgL6CG8SSCKCFhHYacNimDsWDTWKM21GrlYLXDTwLVjk0GAW+w7I87tHKfmaDrcyCKZBcapN4JtmQe10+yN8b5/Ojopdye3+lOK1uhC/z9rFOIcvqIxyzCzfNigjqq0TU7LnCixdNxuHId15me4/+rr+X6+fgEKBYVgCmVuZHN0cmVhbQplbmRvYmoKNDcgMCBvYmoKPDwgL0xlbmd0aCAzMTggL0ZpbHRlciAvRmxhdGVEZWNvZGUgPj4Kc3RyZWFtCnicTZJLbgQhDET3nMIXGMk/aDjPRFEWnftv8ww9UhY9LqiyKdvTVUXFUl5ul+TVJbvKl7UP/D0outwtR4D4Op8vyRxEl3fLmJJUyTBJdW67xEwYJ/vlSbbnktdKcTJelim2BuAKVHYpKhKNQtuKqR7wbtdzNawiD6zSKkxUgbkktAr1E/sqphD2gs4M81lv+oDpxTzVzEJ4f8d3W36Q6ToSo4sny6JkUaMqFSfrdgDk6M/d3EWnuAJmGXHjPay5UbQPuqebqJl5uUeaJueXPG4BXjIc8tpaYlZ1Btuh6N06k3blwLBd9sy3wdzrYszI/6G9OFtkMwqbfrIu1D639VohzICplivqYQphAUVMmJrvqH14ndDerVYcKzZChdmY2NN+IrtiE4XoJi7fbiJPhCm0nX7+bHf7ad9/STd6DgplbmRzdHJlYW0KZW5kb2JqCjQ4IDAgb2JqCjw8IC9MZW5ndGggMjc5IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlID4+CnN0cmVhbQp4nDVQS04FMQzbzyl8AaTm2/Y8DyE23H+LnQfSjOKmbmK71sKCJz4skLvR/D/tEYxY+Bnkp3hp5JHUAV+B15OVMJ4yDeYbyTE3dbE2eiFOIviJVmyH1vDxubPNa019Pb7eHdshgqUg+2aO2DgHXdgOo9jXUwXj2CEZp8PNNGTPRO7w6yoRrTbBPQhDlhyiRns7iufDMyWbB9pnZV0uu3ToE4VTvCrV2x0UFNLJ/ZaosyaHQllPfBmO/+xez/cTZIQXcxRKdqMP2azZqO6/XNqp+Rp3y8KZqmB6kO0ahlHy+43RkKaYa6pNDb+6IXJviGH0qzdXljUk5Iw9n6xYuINpKecJalQNWzL1XLKdBv6NyNTXL3oYaJ4KZW5kc3RyZWFtCmVuZG9iagoxNyAwIG9iago8PCAvVHlwZSAvRm9udCAvQmFzZUZvbnQgL0NaUkJMUitTVElYR2VuZXJhbC1SZWd1bGFyIC9GaXJzdENoYXIgMAovTGFzdENoYXIgMjU1IC9Gb250RGVzY3JpcHRvciAxNiAwIFIgL1N1YnR5cGUgL1R5cGUzCi9OYW1lIC9DWlJCTFIrU1RJWEdlbmVyYWwtUmVndWxhciAvRm9udEJCb3ggWyAtOTcwIC00NDMgMjAwMCAxMDIzIF0KL0ZvbnRNYXRyaXggWyAwLjAwMSAwIDAgMC4wMDEgMCAwIF0gL0NoYXJQcm9jcyAxOCAwIFIKL0VuY29kaW5nIDw8IC9UeXBlIC9FbmNvZGluZwovRGlmZmVyZW5jZXMgWyAzMiAvc3BhY2UgNDAgL3BhcmVubGVmdCAvcGFyZW5yaWdodCA0NSAvaHlwaGVuIC9wZXJpb2QgNDggL3plcm8gL29uZQovdHdvIDUyIC9mb3VyIDU0IC9zaXggNTYgL2VpZ2h0IDY5IC9FIDc3IC9NIDk3IC9hIC9iIC9jIC9kIC9lIDEwNSAvaSAxMDggL2wKL20gL24gL28gL3AgMTE0IC9yIC9zIC90IC91IDEyMCAveCAveSBdCj4+Ci9XaWR0aHMgMTUgMCBSID4+CmVuZG9iagoxNiAwIG9iago8PCAvVHlwZSAvRm9udERlc2NyaXB0b3IgL0ZvbnROYW1lIC9DWlJCTFIrU1RJWEdlbmVyYWwtUmVndWxhciAvRmxhZ3MgMzIKL0ZvbnRCQm94IFsgLTk3MCAtNDQzIDIwMDAgMTAyMyBdIC9Bc2NlbnQgMTA1NSAvRGVzY2VudCAtNDU1IC9DYXBIZWlnaHQgMAovWEhlaWdodCAwIC9JdGFsaWNBbmdsZSAwIC9TdGVtViAwIC9NYXhXaWR0aCAxMTA5ID4+CmVuZG9iagoxNSAwIG9iagpbIDI1MCAyNTAgMjUwIDI1MCAyNTAgMjUwIDI1MCAyNTAgMjUwIDI1MCAyNTAgMjUwIDI1MCAyNTAgMjUwIDI1MCAyNTAgMjUwCjI1MCAyNTAgMjUwIDI1MCAyNTAgMjUwIDI1MCAyNTAgMjUwIDI1MCAyNTAgMjUwIDI1MCAyNTAgMjUwIDMzMyA0MDggNTAwIDUwMAo3NDcgNzc4IDE4MCAzMzMgMzMzIDUwMCA2ODUgMjUwIDMzMyAyNTAgMjc4IDUwMCA1MDAgNTAwIDUwMCA1MDAgNTAwIDUwMCA1MDAKNTAwIDUwMCAyNzggMjc4IDY4NSA2ODUgNjg1IDQ0NCA5MjEgNzIyIDY2NyA2NjcgNzIyIDYxMSA1NTYgNzIyIDcyMiAzMzMgMzczCjcyMiA2MTEgODg5IDcyMiA3MjIgNTU3IDcyMiA2NjcgNTU2IDYxMSA3MjIgNzIyIDk0NCA3MjIgNzIyIDYxMiAzMzMgMjc4IDMzMwo0NjkgNTAwIDMzMyA0NDQgNTAwIDQ0NCA1MDAgNDQ0IDMzMyA1MDAgNTAwIDI3OCAyNzggNTAwIDI3OCA3NzggNTAwIDUwMCA1MDAKNTAwIDMzMyAzODkgMjc4IDUwMCA1MDAgNzIyIDUwMCA1MDAgNDQ0IDQ4MCAyMDAgNDgwIDU0MSAyNTAgNTAwIDI1MCAzMzMgNDM0CjQ0NCAxMDAwIDUwMCA1MDAgMzMzIDExMDkgNTU2IDMzMyA4ODkgMjUwIDYxMiAyNTAgMjUwIDMzMyAzMzMgNDQ0IDQ0NCA1MjMKNTAwIDEwMDAgMzMzIDk4MCAzODkgMzMzIDcyMiAyNTAgNDQ0IDcyMiAyNTAgMzMwIDUwMCA1MDAgNTAwIDUwMCAyMDAgNTAwCjMzMyA3NjAgMjc2IDUwMCA2MDAgMzMzIDc2MCAzMzMgNDAwIDY4NSAzMDAgMzAwIDMzMyA1MDAgNTkyIDI1MCAzMzMgMzAwIDMxMAo1MDAgNzUwIDc1MCA3NTAgNDQ0IDcyMiA3MjIgNzIyIDcyMiA3MjIgNzIyIDg4OSA2NjcgNjExIDYxMSA2MTEgNjExIDMzMyAzMzMKMzMzIDMzMyA3MjIgNzIyIDcyMiA3MjIgNzIyIDcyMiA3MjIgNjQwIDcyMiA3MjIgNzIyIDcyMiA3MjIgNzIyIDU1NiA1MDAgNDQ0CjQ0NCA0NDQgNDQ0IDQ0NCA0NDQgNjY3IDQ0NCA0NDQgNDQ0IDQ0NCA0NDQgMjc4IDI3OCAyNzggMjc4IDUwMCA1MDAgNTAwIDUwMAo1MDAgNTAwIDUwMCA1NjQgNTAwIDUwMCA1MDAgNTAwIDUwMCA1MDAgNTAwIDUwMCBdCmVuZG9iagoxOCAwIG9iago8PCAvRSAxOSAwIFIgL00gMjAgMCBSIC9hIDIxIDAgUiAvYiAyMiAwIFIgL2MgMjMgMCBSIC9kIDI0IDAgUiAvZSAyNSAwIFIKL2VpZ2h0IDI2IDAgUiAvZm91ciAyNyAwIFIgL2h5cGhlbiAyOCAwIFIgL2kgMjkgMCBSIC9sIDMwIDAgUiAvbSAzMSAwIFIKL24gMzIgMCBSIC9vIDMzIDAgUiAvb25lIDM0IDAgUiAvcCAzNSAwIFIgL3BhcmVubGVmdCAzNiAwIFIKL3BhcmVucmlnaHQgMzcgMCBSIC9wZXJpb2QgMzggMCBSIC9yIDM5IDAgUiAvcyA0MCAwIFIgL3NpeCA0MSAwIFIKL3NwYWNlIDQyIDAgUiAvdCA0MyAwIFIgL3R3byA0NCAwIFIgL3UgNDUgMCBSIC94IDQ2IDAgUiAveSA0NyAwIFIKL3plcm8gNDggMCBSID4+CmVuZG9iago1MyAwIG9iago8PCAvTGVuZ3RoIDMyNyAvRmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeJxNUjluBDEM6+cV/MAC1mnrPRsEKTb/b0NpskEKjzi2JVKUUwQLDz38ZgSX4UOuN/y+kW28rjRDaiFFuQpRjlyB5xU7+SeIcMReCONy5YmlwPww21NR1mAzzeAn4Qx1YFNDFpRlSKLMzg3J3iZYzNLKv8iD12X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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(10, 4))\n", "\n", "plt.scatter(\n", " t,\n", " X_exact[0, :],\n", " s=20,\n", " marker='x',\n", " color='k',\n", " zorder=1,\n", " label='Exact solution',\n", ")\n", "\n", "plt.scatter(\n", " t,\n", " X[0, :],\n", " s=20,\n", " marker='x',\n", " color='red',\n", " zorder=2,\n", " label='Euler-Maruyama (subsampled)',\n", ")\n", "plt.xlim([0, T])\n", "plt.xticks(np.linspace(0, 1, 6), fontsize=18)\n", "plt.yticks(np.linspace(0, 8, 5), fontsize=18)\n", "plt.title('Milstein solution', fontsize=20)\n", "plt.xlabel(r'$t$', fontsize=20)\n", "plt.ylabel(r'$X(t)$', fontsize=20)\n", "plt.legend(fontsize=18)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also look at the strong and weak rates of convergence for Milstein's method.\n", "Milstein's method achieves a strong convergence rate of $1$ as oposed to the $1/2$ strong rate of Euler Maruyama." ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "tags": [ "center-output", "remove-input" ] }, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Black-Scholes parameters\n", "lamda = 2\n", "mu = 1\n", "\n", "# Seed and integration parameters\n", "seed = 0\n", "X0 = 1\n", "T = 1\n", "Ns = (10 ** np.linspace(2, 4, 4)).astype(dtype=np.int32)\n", "num_paths = int(1e4)\n", "\n", "# Get drift and diffusion functions of the Black-Scholes model\n", "f, g = f_g_black_scholes(lamda=lamda, mu=mu)\n", "\n", "dts = []\n", "X_approx = []\n", "X_exacts = []\n", "\n", "for N in Ns:\n", " \n", " t, X, W = parallel_milstein(seed=seed, num_paths=num_paths, X0=X0, T=T, N=N, f=f, g=g)\n", " X_exact = exact_black_scholes(X0=X0, t=t[None, :], W=W, lamda=lamda, mu=mu)\n", " \n", " dts.append(T / N)\n", " X_approx.append(X[:, -1])\n", " X_exacts.append(X_exact[:, -1])\n", " \n", "X_approx = np.stack(X_approx, axis=0)\n", "X_exacts = np.stack(X_exacts, axis=0)\n", "\n", "X_abs_diffs = np.abs(X_approx - X_exacts)\n", "milstein_strong_errors = np.mean(X_abs_diffs, axis=1)\n", "\n", "X_approx_means = np.mean(X_approx, axis=-1)\n", "X_exacts_means = np.mean(X_exacts, axis=-1)\n", "milstein_weak_errors = np.abs(X_approx_means - X_exacts_means)\n", "\n", "plt.figure(figsize=(10, 4))\n", "\n", "plt.subplot(121)\n", "plt.plot(dts, milstein_strong_errors, color='k')\n", "\n", "plt.loglog()\n", "plt.xticks(fontsize=16)\n", "plt.yticks([1e-2, 1e-1, 1e0], fontsize=16)\n", "plt.xlabel(r'$\\delta t$', fontsize=20)\n", "plt.ylabel(r'$\\mathbb{E}|X_n - X(t_n)|$', fontsize=20)\n", "plt.title('Milstein\\nstrong convergence', fontsize=20)\n", "\n", "plt.subplot(122)\n", "plt.plot(dts, milstein_weak_errors, color='k')\n", "\n", "plt.loglog()\n", "plt.xticks(fontsize=16)\n", "plt.yticks([1e-3, 1e-2, 1e-1, 1e0], fontsize=16)\n", "plt.xlabel(r'$\\delta t$', fontsize=20)\n", "plt.ylabel(r'$|\\mathbb{E}X_n - \\mathbb{E}X(t_n)|$', fontsize=20)\n", "plt.title('Milstein\\nweak convergence', fontsize=20)\n", "\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Stochastic chain rule\n", "\n", "Suppose we want to evaluate a function $V(\\cdot)$ at various $X(t)$, i.e. $V(X(t))$. If $X(t)$ were a deterministic quantity, such as the solution to an ODE, we could solve for $X(t)$ and plug it into $V$. Alternatively, we could express the evolution of $V$ itself as a differential equation using the chain rule:\n", "\n", "$$\\begin{align}\n", "dV &= \\frac{dV}{dX}dX = \\frac{dV}{dX} f(t) dt, \\text{ where } dX = f(t) dt,\\\\\n", "\\end{align}$$\n", "\n", "This way, we could solve the following ODE directly for $V$\n", "\n", "$$\\begin{align}\n", "\\frac{dV}{dt} &= \\frac{dV}{dX} f(t).\n", "\\end{align}$$\n", "\n", "For an autonomous SDE however the chain rule takes a different form, which under the Ito definition is as follows.\n", "\n", ":::{prf:theorem} Ito's result for one dimension\n", "\n", "Let $X_t$ be an Ito process given by\n", "\n", "$$\\begin{align}\n", "dX_t = U_t dt + H_t dW_t.\n", "\\end{align}$$\n", " \n", "where $U_t, H_t$ are square-integrable processes, and let $V(X, t)$ be a twice continuously differentiable function.\n", "Then $Y_t = V(X_t, t)$ is again an Ito process and\n", " \n", "$$\\begin{align}\n", "dY_t = \\frac{\\partial V}{\\partial t} dt + \\frac{\\partial V}{\\partial X} dX_t + \\frac{1}{2}\\frac{\\partial^2 V}{\\partial X^2} V_t^2 dt. \n", "\\end{align}$$\n", " \n", "If $V$ does not depend on $t$, we have\n", " \n", "$$\\begin{align}\n", "dY_t = \\left(\\frac{\\partial V}{\\partial X} U_t + \\frac{1}{2}\\frac{\\partial^2 V}{\\partial X^2} H_t^2 \\right) dt + \\frac{\\partial V}{\\partial X}H_t dW_t. \n", "\\end{align}$$\n", " \n", ":::\n", "\n", "For a more formal definition and proof of Ito's result see {cite}`oksendal` (Theorem 4.1.8 and pages 44-48).\n", "Below is a short sketch proof, which highlights why the additional term appears in the formula.\n", "\n", "\n", ":::{dropdown} Informal argument: Ito's result for one dimension\n", " \n", "Writing the infinitesimal difference in $Y_t$ as a Taylor expansion we get\n", " \n", "$$\\begin{align}\n", "dY_t = \\frac{\\partial V}{\\partial t} dt + \\frac{\\partial V}{\\partial X} dX_t + \\frac{1}{2} \\left[\\frac{\\partial^2 V}{\\partial X^2} dX_t^2 + 2 \\frac{\\partial^2 V}{\\partial X \\partial t} dt dX_t + \\frac{\\partial^2 V}{\\partial t^2} dt^2 \\right] + o(dt^2),\n", "\\end{align}$$\n", " \n", "where the $o(dt^n)$ notation means that the ratio of the term being ommited, up to and including the infinitesimal $dt^n$ goes to 0 as $dt \\to 0$.\n", "Now since $dX_t = U_t dt + H_t dW_t$ and $dW_t$ is of order $dt^{1/2}$, the last two terms in the square brackets are $o(dt^{3/2})$ and we can neglect them.\n", "The reference provides a formal argument for neglecting these terms, showing that their contribution to the Ito integral has zero mean and a variance that tends to 0 as $dt \\to 0$ - these contributions converge to $0$ in mean square.\n", "However, the first term in the brackets is of order $dt$ and does not vanish. In particular\n", " \n", "$$\\begin{align}\n", "dX_t^2 = U_t^2 dt^2 + 2 U_tH_t dW_t dt + H_t^2 dW_t^2.\n", "\\end{align}$$\n", " \n", "In the above expression, we can neglect the first two terms which are of order $dt^2$ and $dt^{3/2}.$\n", "Again, here the formal argument is that their associated contributions to the Ito integral converge to 0 in mean square.\n", "This yields the expression\n", " \n", "$$\\begin{align}\n", "dY_t = \\frac{\\partial V}{\\partial X} U_t dt + \\frac{1}{2}\\frac{\\partial^2 V}{\\partial X^2} H_t^2 dW_t^2 + \\frac{\\partial V}{\\partial X}H_t dW_t. \n", "\\end{align}$$\n", " \n", "Now consider the contribution to the Ito integral of the last term, i.e. the sum\n", " \n", "$$\\begin{align}\n", "\\sum_{n = 1}^N \\frac{\\partial^2 V}{\\partial X^2}\\Bigg \\vert_{X_{t_n}} H_{t_n}^2 dW_{t_n}^2 = \\sum_{n = 1}^N a_n dW_{t_n}^2,\n", "\\end{align}$$\n", " \n", "where $t_n = n~dt$ and $N = T / dt$. This sum has expectation $\\sum_{n = 1}^N a_n dt$ and it can be shown that its variance goes to 0 as $dt \\to 0$, so the contribution converges to $\\sum_{n = 1}^N a_n dt$ in mean square and we can write\n", " \n", "$$\\begin{align}\n", "dY_t = \\left(\\frac{\\partial V}{\\partial X} U_t + \\frac{1}{2}\\frac{\\partial^2 V}{\\partial X^2} H_t^2 \\right) dt + \\frac{\\partial V}{\\partial X}H_t dW_t. \n", "\\end{align}$$\n", "\n", ":::\n", "\n", "With this corrected rule, we can directly integrate the stochastic process for $V.$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## References\n", "\n", "```{bibliography}\n", ":filter: docname in docnames\n", "```" ] } ], "metadata": { "kernelspec": { "display_name": "venv-rw", "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": "3.10.14" } }, "nbformat": 4, "nbformat_minor": 4 }