🔢Numerical Analysis II Unit 11 – Stochastic Differential Equations: Methods

Key Concepts and Definitions

• Stochastic differential equations (SDEs) mathematical models that describe the evolution of a system subject to random perturbations
• Brownian motion a continuous-time stochastic process with independent, normally distributed increments (also known as a Wiener process)
• Itô calculus extends classical calculus to handle stochastic processes and SDEs
  • Itô's lemma a key result that allows for the computation of differentials of functions of stochastic processes
• Drift coefficient determines the deterministic part of an SDE and represents the average rate of change
• Diffusion coefficient determines the stochastic part of an SDE and represents the intensity of random fluctuations
• Itô integral a stochastic integral that defines the integration of a stochastic process with respect to Brownian motion
• Stratonovich integral an alternative stochastic integral with different properties than the Itô integral
• Numerical methods techniques used to approximate solutions to SDEs when analytical solutions are not available (Euler-Maruyama, Milstein)

Foundations of Probability Theory

• Probability space a mathematical construct consisting of a sample space, a set of events, and a probability measure
• Random variables functions that assign numerical values to outcomes in a sample space
  • Continuous random variables can take on any value within a specified range
  • Discrete random variables can only take on a countable set of values
• Probability distributions describe the likelihood of different outcomes for a random variable (normal, exponential, Poisson)
• Expected value the average value of a random variable over its entire range
• Variance a measure of the spread or dispersion of a random variable around its expected value
• Stochastic processes collections of random variables indexed by time (discrete-time or continuous-time)
• Markov property a property of stochastic processes where the future state depends only on the current state, not on the past states
• Martingales stochastic processes whose expected value at any future time, given the current information, is equal to its current value

Introduction to Stochastic Processes

• Stochastic processes model systems that evolve over time with random elements
• Markov processes a class of stochastic processes that satisfy the Markov property (Markov chains, Poisson processes)
• Brownian motion a continuous-time stochastic process with independent, normally distributed increments
  • Mathematical description involves the Wiener process, a specific type of Brownian motion
• Stochastic calculus extends classical calculus to handle stochastic processes
• Stochastic differential equations (SDEs) model the evolution of a system subject to random perturbations
  • Consist of a drift term (deterministic) and a diffusion term (stochastic)
• Itô calculus a framework for working with SDEs and stochastic processes
• Applications of stochastic processes include finance (stock prices), physics (particle motion), and biology (population dynamics)

Itô Calculus Basics

• Itô calculus extends classical calculus to handle stochastic processes and SDEs
• Brownian motion a key building block in Itô calculus, representing random fluctuations
• Itô integral defines the integration of a stochastic process with respect to Brownian motion
  • Constructed using a limit of Riemann sums with a specific choice of evaluation points
• Itô's lemma a key result that allows for the computation of differentials of functions of stochastic processes
  • Analogous to the chain rule in classical calculus, but with an additional term due to the quadratic variation of Brownian motion
• Quadratic variation a unique property of Brownian motion that measures the accumulated squared increments over time
• Stratonovich integral an alternative stochastic integral with different properties than the Itô integral
  • Follows the standard chain rule, but requires more restrictive conditions on the integrand
• Stochastic differential equations (SDEs) model systems subject to random perturbations using Itô calculus
  • Consist of a drift term (deterministic) and a diffusion term (stochastic)

Types of Stochastic Differential Equations

• Linear SDEs have coefficients that are linear functions of the state variable
  • Ornstein-Uhlenbeck process a mean-reverting linear SDE used to model interest rates and other financial quantities
• Nonlinear SDEs have coefficients that are nonlinear functions of the state variable
  • Geometric Brownian motion a nonlinear SDE used to model stock prices and other assets with exponential growth
• Scalar SDEs involve a single state variable and a single Brownian motion
• Multidimensional SDEs involve multiple state variables and possibly multiple Brownian motions
  • Stochastic volatility models (Heston model) use multidimensional SDEs to capture the dynamics of asset prices and their volatilities
• Itô SDEs interpret the stochastic integral using the Itô calculus
• Stratonovich SDEs interpret the stochastic integral using the Stratonovich calculus
  • Can be converted to Itô SDEs by adjusting the drift term
• Stochastic partial differential equations (SPDEs) extend SDEs to include spatial dimensions
  • Used to model phenomena such as fluid dynamics and heat transfer with random fluctuations

Numerical Methods for SDEs

• Analytical solutions to SDEs are rarely available, necessitating the use of numerical methods
• Euler-Maruyama method a simple and widely used numerical scheme for SDEs
  • Generalizes the Euler method for ordinary differential equations (ODEs) to the stochastic setting
  • Converges to the true solution as the time step tends to zero, but with a slower rate than in the deterministic case
• Milstein method a higher-order numerical scheme that includes an additional term to improve accuracy
  • Requires the computation of the derivative of the diffusion coefficient
• Runge-Kutta methods extend deterministic Runge-Kutta schemes to the stochastic setting
  • Provide better accuracy and stability than the Euler-Maruyama method, but at a higher computational cost
• Monte Carlo simulation a general technique for estimating quantities of interest by averaging over many realizations of the SDE
  • Useful for computing expectations, probabilities, and other statistics
• Multilevel Monte Carlo a variance reduction technique that combines simulations at different time step sizes to improve efficiency
• Quasi-Monte Carlo methods use deterministic sequences (low-discrepancy) instead of random numbers to improve convergence rates

Applications in Finance and Physics

• Financial mathematics SDEs are used to model asset prices, interest rates, and other financial quantities
  • Black-Scholes model an SDE that describes the dynamics of a stock price and forms the basis for option pricing theory
  • Stochastic volatility models (Heston model) capture the time-varying nature of asset price volatility
  • Interest rate models (Vasicek, Cox-Ingersoll-Ross) describe the evolution of interest rates over time
• Physics SDEs are used to model various physical phenomena subject to random fluctuations
  • Langevin equation an SDE that describes the motion of a particle subject to random forces (Brownian motion)
  • Stochastic differential equations in quantum mechanics (stochastic Schrödinger equation) incorporate random effects into quantum systems
  • Stochastic partial differential equations (SPDEs) model systems with both temporal and spatial randomness (fluid dynamics, heat transfer)
• Other applications include biology (population dynamics), engineering (signal processing), and social sciences (opinion dynamics)

Advanced Topics and Current Research

• Rough path theory an extension of Itô calculus to handle stochastic processes with less regularity (fractional Brownian motion)
  • Signature of a path a key concept in rough path theory that encodes the essential information of a stochastic process
• Malliavin calculus a variational calculus for stochastic processes that allows for the computation of sensitivities (Greeks)
  • Malliavin derivative measures the sensitivity of a random variable to perturbations in the underlying stochastic process
• Stochastic control theory studies the optimization of systems described by SDEs
  • Hamilton-Jacobi-Bellman (HJB) equation a partial differential equation that characterizes the optimal control policy
• Stochastic differential games extend stochastic control theory to multiple agents with conflicting objectives
• Machine learning and SDEs combining SDEs with machine learning techniques for model calibration, parameter estimation, and data-driven modeling
• Numerical methods for high-dimensional SDEs developing efficient algorithms for SDEs with many state variables (tensor approximations, sparse grids)
• Stochastic partial differential equations (SPDEs) analysis and numerical methods for SPDEs, which incorporate both temporal and spatial randomness