🧮Computational Mathematics Unit 10 – Numerical Methods for Stochastic DEs

Key Concepts and Terminology

• Stochastic differential equations (SDEs) model systems with random fluctuations or noise
• Brownian motion represents the random movement of particles suspended in a fluid
• Wiener process is a continuous-time stochastic process with independent increments
• Itô calculus extends calculus to stochastic processes and is used to manipulate SDEs
• Stratonovich calculus is an alternative to Itô calculus with different integration rules
• Strong convergence measures the pathwise approximation of the numerical solution to the exact solution
• Weak convergence measures the approximation of the probability distribution of the numerical solution to the exact solution
• Numerical stability ensures that small errors in the input do not lead to large errors in the output

Stochastic Differential Equations Basics

• SDEs consist of a deterministic drift term and a stochastic diffusion term
• The drift term represents the expected change in the system over time
• The diffusion term represents the random fluctuations or noise in the system
• SDEs are written in differential form as $dX_t = a(t, X_t)dt + b(t, X_t)dW_t$
• $X_t$ represents the stochastic process, $a(t, X_t)$ is the drift term, $b(t, X_t)$ is the diffusion term, and $W_t$ is the Wiener process
• Itô's lemma is used to find the differential of a function of a stochastic process
• The solution to an SDE is a stochastic process that satisfies the equation
• SDEs can model various phenomena such as financial markets (stock prices), population dynamics, and physical systems (Brownian motion)

Numerical Methods Overview

• Numerical methods approximate the solution of SDEs since analytical solutions are often unavailable
• Time discretization techniques divide the time interval into smaller steps and approximate the solution at each step
• Stochastic numerical schemes combine time discretization with a method for approximating the stochastic integral
• The Euler-Maruyama method is a simple and widely used stochastic numerical scheme
• The Milstein method is a higher-order scheme that includes the Itô-Taylor expansion terms
• The stochastic Runge-Kutta methods generalize deterministic Runge-Kutta methods to SDEs
• The implicit methods (backward Euler, Crank-Nicolson) can handle stiff systems and provide better stability
• Adaptive time-stepping adjusts the step size based on the local error estimate to improve efficiency and accuracy

Time Discretization Techniques

• The time interval $[0, T]$ is divided into $N$ subintervals of size $\Delta t = T/N$
• The discrete time points are denoted as $t_n = n\Delta t$ for $n = 0, 1, \ldots, N$
• The numerical solution is approximated at each time point $t_n$
• The Itô integral $\int_{t_n}^{t_{n+1}} b(t, X_t)dW_t$ is approximated using a stochastic numerical scheme
• The Riemann-Stieltjes sum approximates the Itô integral as $\sum_{i=1}^m b(t_{n,i}, X_{t_{n,i}})(W_{t_{n,i+1}} - W_{t_{n,i}})$
• The left-point rule evaluates the integrand at the left endpoint of each subinterval
• The midpoint rule evaluates the integrand at the midpoint of each subinterval
• The trapezoidal rule approximates the integral using a trapezoidal approximation

Stochastic Numerical Schemes

• The Euler-Maruyama method approximates the Itô integral using the left-point rule
  • $X_{n+1} = X_n + a(t_n, X_n)\Delta t + b(t_n, X_n)\Delta W_n$
  • $\Delta W_n = W_{t_{n+1}} - W_{t_n}$ is the Brownian motion increment
• The Milstein method includes the Itô-Taylor expansion term
  • $X_{n+1} = X_n + a(t_n, X_n)\Delta t + b(t_n, X_n)\Delta W_n + \frac{1}{2}b(t_n, X_n)\frac{\partial b}{\partial x}(t_n, X_n)((\Delta W_n)^2 - \Delta t)$
• The stochastic Runge-Kutta methods use intermediate stages to improve accuracy
  • The stochastic Heun method is a two-stage scheme that uses the Euler-Maruyama method for the intermediate stage
  • The stochastic Runge-Kutta method of order 1.5 uses two intermediate stages and achieves strong order 1.5
• The implicit methods solve a nonlinear equation at each time step
  • The backward Euler method solves $X_{n+1} = X_n + a(t_{n+1}, X_{n+1})\Delta t + b(t_{n+1}, X_{n+1})\Delta W_n$
  • The Crank-Nicolson method uses a trapezoidal rule for the drift term and a midpoint rule for the diffusion term

Stability and Convergence Analysis

• Stability analysis studies the behavior of numerical methods when applied to test equations (linear scalar SDEs)
• Mean-square stability requires the expectation and variance of the numerical solution to remain bounded
• Asymptotic stability requires the numerical solution to converge to the equilibrium solution as time tends to infinity
• Convergence analysis investigates the rate at which the numerical solution approaches the exact solution as the step size decreases
• Strong convergence of order $\gamma$ means $E[|X_N - X(T)|] \leq C\Delta t^\gamma$ for some constant $C$
• Weak convergence of order $\beta$ means $|E[g(X_N)] - E[g(X(T))]| \leq C\Delta t^\beta$ for some smooth function $g$ and constant $C$
• The Euler-Maruyama method has strong order 0.5 and weak order 1
• The Milstein method has strong order 1 and weak order 1

Implementation and Algorithms

• Pseudocode outlines the steps of the numerical method without specifying the programming language
• Initialization sets the initial conditions and parameters of the SDE
• The main loop iterates over the time steps and updates the numerical solution at each step
• Random number generation simulates the Brownian motion increments $\Delta W_n$
  • The Box-Muller transform generates normally distributed random numbers from uniformly distributed random numbers
  • The Mersenne Twister is a widely used pseudorandom number generator
• Postprocessing computes statistics, visualizes the results, and saves the output
• Parallel computing can be used to simulate multiple paths of the SDE simultaneously
• GPU acceleration can significantly speed up the computation of large-scale SDEs

Applications and Case Studies

• Finance: SDEs model stock prices (geometric Brownian motion), interest rates (Cox-Ingersoll-Ross model), and option pricing (Black-Scholes model)
• Physics: SDEs describe Brownian motion, diffusion processes, and stochastic resonance
• Biology: SDEs model population dynamics (stochastic Lotka-Volterra equations), gene expression, and epidemiology (SIR model with noise)
• Engineering: SDEs are used in control theory, signal processing, and stochastic optimization
• Neuroscience: SDEs model the firing of neurons and the propagation of signals in neural networks
• Climate science: SDEs describe the variability and uncertainty in climate models
• Case study: The Ornstein-Uhlenbeck process is a mean-reverting SDE used to model interest rates and commodity prices
  • $dX_t = \theta(\mu - X_t)dt + \sigma dW_t$, where $\theta$ is the mean reversion speed, $\mu$ is the long-term mean, and $\sigma$ is the volatility
• Case study: The stochastic Lorenz system is a chaotic system with additive noise
  • $dX_t = \sigma(Y_t - X_t)dt + \alpha_1 dW_{1,t}$
  • $dY_t = (X_t(\rho - Z_t) - Y_t)dt + \alpha_2 dW_{2,t}$
  • $dZ_t = (X_tY_t - \beta Z_t)dt + \alpha_3 dW_{3,t}$, where $\sigma$, $\rho$, and $\beta$ are parameters, and $\alpha_1$, $\alpha_2$, and $\alpha_3$ are noise intensities