10.1 Brownian motion

Brownian motion, named after botanist Robert Brown, describes the random movement of particles in fluids. This mathematical concept has far-reaching applications in physics, finance, and other fields, serving as a foundation for modeling diffusion processes and stochastic phenomena.

The Wiener process, a continuous-time stochastic process, forms the mathematical basis for Brownian motion. It's characterized by independent, stationary increments and plays a crucial role in various applications, from option pricing to polymer dynamics.

Origins of Brownian motion

• Brownian motion named after botanist Robert Brown who observed random motion of pollen grains suspended in water in 1827
• Brown's observations sparked interest in understanding the underlying mechanisms of this seemingly erratic motion
• Further investigations by physicists and mathematicians in the late 19th and early 20th centuries laid the foundation for the mathematical formulation of Brownian motion and its applications in various fields

Mathematical formulation

Wiener process

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• Wiener process, also known as standard Brownian motion, is a continuous-time stochastic process with independent and stationary increments
• Characterized by three key properties:
  • $W(0) = 0$ almost surely
  • For $0 \leq s < t$, the increment $W(t) - W(s)$ is normally distributed with mean 0 and variance $t-s$
  • For any non-overlapping intervals $[s_1, t_1]$ and $[s_2, t_2]$, the increments $W(t_1) - W(s_1)$ and $W(t_2) - W(s_2)$ are independent
• Wiener process serves as the foundation for modeling and analyzing Brownian motion in various contexts

Lévy characterization

• Lévy characterization provides an alternative definition of Brownian motion based on its characteristic function
• States that a continuous, adapted process $X_t$ is a Brownian motion if and only if:
  • $X_0 = 0$ almost surely
  • $X_t$ has independent increments
  • $X_t - X_s \sim N(0, t-s)$ for $0 \leq s < t$
• Lévy characterization emphasizes the independence and normality of increments, which are essential properties of Brownian motion

Martingale property

• Brownian motion possesses the martingale property, which means that the conditional expectation of future values given the past values is equal to the present value
• Mathematically, for a Brownian motion $W_t$ and $s < t$, $\mathbb{E}[W_t | \mathcal{F}_s] = W_s$, where $\mathcal{F}_s$ represents the filtration (information) up to time $s$
• Martingale property is crucial in financial applications, such as option pricing, where the fair price of an asset is determined by the expectation of its future payoff under a risk-neutral measure

Physical interpretation

Diffusion processes

• Brownian motion serves as a fundamental model for diffusion processes, which describe the random motion of particles in a medium
• Diffusion processes are characterized by the mean-squared displacement of particles growing linearly with time, $\langle x^2 \rangle \propto t$
• Examples of diffusion processes include heat conduction, molecular diffusion in fluids, and charge carrier transport in semiconductors

Einstein's theory

• In 1905, Albert Einstein provided a theoretical explanation for Brownian motion based on the kinetic theory of gases
• Einstein's theory related the mean-squared displacement of a particle to the diffusion coefficient $D$ and time $t$: $\langle x^2 \rangle = 2Dt$
• Einstein's work established a link between the microscopic motion of molecules and the macroscopic observable of diffusion, providing strong evidence for the atomic nature of matter

Properties of Brownian motion

Continuity vs non-differentiability

• Brownian motion paths are almost surely continuous functions of time, meaning they have no jumps or discontinuities
• However, Brownian motion paths are also almost surely nowhere differentiable, implying that they are extremely irregular and do not possess a well-defined velocity at any point
• The non-differentiability of Brownian motion paths is a consequence of their fractal-like nature, with self-similar structures appearing at all scales

Self-similarity

• Brownian motion exhibits self-similarity, meaning that the statistical properties of the process remain unchanged when the time scale is rescaled
• Mathematically, for any $a > 0$, the process $\{a^{-1/2}W(at), t \geq 0\}$ is also a Brownian motion
• Self-similarity is a key feature of fractal objects and is closely related to the concept of scale invariance

Markov property

• Brownian motion satisfies the Markov property, which states that the future evolution of the process depends only on its current state and not on its past history
• Formally, for any times $t_0 < t_1 < \cdots < t_n$ and any Borel set $A$, $\mathbb{P}(W_{t_n} \in A | W_{t_0}, \ldots, W_{t_{n-1}}) = \mathbb{P}(W_{t_n} \in A | W_{t_{n-1}})$
• The Markov property simplifies the analysis and simulation of Brownian motion, as it allows for the construction of the process incrementally based on its current state

Variations and generalizations

Fractional Brownian motion

• Fractional Brownian motion (fBm) is a generalization of Brownian motion that allows for long-range dependence and self-similarity with a parameter $H \in (0, 1)$
• For $H = 1/2$, fBm reduces to standard Brownian motion, while for $H \neq 1/2$, the increments of fBm are correlated (positively for $H > 1/2$ and negatively for $H < 1/2$)
• fBm is used to model phenomena exhibiting long-memory effects, such as network traffic, financial time series, and geophysical processes

Geometric Brownian motion

• Geometric Brownian motion (GBM) is a continuous-time stochastic process used to model the exponential growth or decay of a quantity subject to random fluctuations
• GBM is defined as the exponential of a Brownian motion with drift: $S_t = S_0 \exp(\mu t + \sigma W_t)$, where $\mu$ is the drift parameter and $\sigma$ is the volatility
• GBM is widely used in financial modeling, particularly in the Black-Scholes model for option pricing, where it represents the dynamics of the underlying asset price

Brownian bridge

• A Brownian bridge is a conditional Brownian motion that starts and ends at specified values over a given time interval
• Mathematically, a Brownian bridge from $a$ to $b$ over the interval $[0, T]$ is defined as $X_t = a(1-t/T) + b(t/T) + W_t - (t/T)W_T$, where $W_t$ is a standard Brownian motion
• Brownian bridges are used in various applications, such as interpolation, simulation of conditioned diffusion processes, and the construction of Gaussian processes

Applications in physics

Diffusion-limited aggregation

• Diffusion-limited aggregation (DLA) is a process by which particles undergoing Brownian motion cluster together to form complex, fractal-like structures
• In DLA, particles are released one at a time from a distant source and diffuse until they encounter an existing cluster, at which point they stick irreversibly
• DLA models are used to describe the formation of various natural structures, such as mineral deposits, bacterial colonies, and lightning patterns

Polymer dynamics

• Brownian motion plays a crucial role in the dynamics of polymers, which are long chain-like molecules composed of repeating subunits
• The conformational changes and diffusive motion of polymer chains in solution can be modeled using Brownian motion and its variations, such as the Rouse model and the Zimm model
• Understanding polymer dynamics is essential for designing and optimizing materials with desired mechanical, thermal, and rheological properties

Applications in finance

Black-Scholes model

• The Black-Scholes model is a mathematical framework for pricing financial derivatives, such as options, based on the assumption that the underlying asset price follows a geometric Brownian motion
• The model uses the concept of risk-neutral valuation, where the expected return of the asset is replaced by the risk-free rate, and the option price is determined by the expectation of its discounted payoff under this measure
• The Black-Scholes formula provides a closed-form solution for the price of European-style options, which has revolutionized the field of quantitative finance

Option pricing

• Option pricing theory relies heavily on the properties of Brownian motion and its generalizations, such as jump-diffusion processes and stochastic volatility models
• Exotic options, such as barrier options, Asian options, and lookback options, require more sophisticated mathematical tools based on Brownian motion and stochastic calculus
• Numerical methods, such as Monte Carlo simulation and finite difference schemes, are used to price options when closed-form solutions are not available

Simulation and visualization

Discrete approximations

• Brownian motion can be approximated using discrete-time random walks, where the particle takes independent, identically distributed steps at regular time intervals
• The simplest approximation is the symmetric random walk, where the particle moves either up or down by a fixed amount with equal probability at each step
• More refined approximations, such as the Euler-Maruyama scheme, use the properties of the Wiener process to simulate Brownian motion paths with better accuracy

Stochastic differential equations

• Stochastic differential equations (SDEs) provide a framework for modeling and simulating Brownian motion and its generalizations in the presence of deterministic drift and diffusion terms
• The most common SDE is the Itô diffusion, which describes the evolution of a process $X_t$ as $dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t$, where $\mu$ and $\sigma$ are the drift and diffusion coefficients, respectively
• Numerical methods for solving SDEs, such as the Euler-Maruyama and Milstein schemes, are used to simulate and visualize the sample paths of Brownian motion and related processes

Connections to other fields

Central limit theorem

• The central limit theorem (CLT) states that the sum of a large number of independent, identically distributed random variables with finite mean and variance converges in distribution to a Gaussian random variable
• Brownian motion can be seen as a continuous-time analog of the CLT, where the random walk converges to a Wiener process as the time step tends to zero and the number of steps tends to infinity
• The connection between Brownian motion and the CLT highlights the universality of Gaussian distributions in modeling random phenomena

Stochastic calculus

• Stochastic calculus extends the concepts of ordinary calculus to stochastic processes, such as Brownian motion, which are not differentiable in the classical sense
• The two main branches of stochastic calculus are the Itô calculus and the Stratonovich calculus, which differ in their interpretation of stochastic integrals and the resulting rules for stochastic differentiation
• Stochastic calculus provides a rigorous framework for defining and manipulating stochastic differential equations, which are used to model a wide range of phenomena in physics, biology, and finance

Potential theory

• Potential theory studies the properties of harmonic functions, which are solutions to Laplace's equation and related partial differential equations
• Brownian motion is intimately connected to potential theory through the Dirichlet problem, which asks for a harmonic function with prescribed boundary values on a given domain
• The probability distribution of the first exit time and location of a Brownian motion from a domain is determined by the solution to the corresponding Dirichlet problem, establishing a deep link between stochastic processes and elliptic PDEs