9.2 Wiener process

The Wiener process, also known as Brownian motion, is a cornerstone of stochastic processes. It models continuous-time random phenomena with Gaussian increments, independent increments, and continuous sample paths. This process serves as a building block for more complex stochastic models in physics, finance, and biology.

Understanding the Wiener process is crucial for analyzing random walks, stochastic integration, and differential equations. It provides a mathematical framework for modeling unpredictable behavior in various systems, from particle motion to stock prices. The process's unique properties make it invaluable in both theoretical and applied stochastic analysis.

Definition of Wiener process

• A Wiener process, also known as Brownian motion, is a continuous-time stochastic process that plays a fundamental role in the study of stochastic processes
• It serves as a building block for more complex stochastic models and is used to describe random phenomena in various fields, such as physics, finance, and biology
• The Wiener process is characterized by its Gaussian increments, independent increments, and continuous sample paths

Standard Wiener process

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• A standard Wiener process $W_t$ is a stochastic process indexed by time $t \geq 0$ that satisfies the following properties:
  • $W_0 = 0$ (starts at zero)
  • $W_t - W_s \sim N(0, t-s)$ for $0 \leq s < t$ (Gaussian increments)
  • $W_t - W_s$ is independent of $W_u - W_v$ for any non-overlapping time intervals $[s, t]$ and $[u, v]$ (independent increments)
  • $W_t$ has continuous sample paths (almost surely)
• The standard Wiener process has a mean of zero and a variance equal to the time interval over which it is defined

Brownian motion vs Wiener process

• Brownian motion and Wiener process are often used interchangeably, but there is a subtle difference between the two terms
• Brownian motion refers to the physical phenomenon of a particle undergoing random motion due to collisions with other particles (e.g., pollen grains in water)
• The Wiener process is the mathematical model that describes Brownian motion, providing a rigorous framework for studying and analyzing such random phenomena

Properties of Wiener process

• The Wiener process possesses several key properties that make it a fundamental object in the study of stochastic processes
• These properties include Gaussian increments, independent increments, continuous sample paths, and non-differentiable paths
• Understanding these properties is crucial for analyzing and modeling random phenomena using the Wiener process

Gaussian increments

• The increments of a Wiener process over any time interval follow a Gaussian (normal) distribution
• For any times $s$ and $t$ with $0 \leq s < t$, the increment $W_t - W_s$ is normally distributed with mean zero and variance $t-s$
• This property allows for the characterization of the probability distribution of the Wiener process at any given time

Independent increments

• The increments of a Wiener process over non-overlapping time intervals are independent of each other
• For any times $s$, $t$, $u$, and $v$ with $0 \leq s < t \leq u < v$, the increments $W_t - W_s$ and $W_v - W_u$ are independent random variables
• This property simplifies the analysis of the Wiener process and enables the application of various probabilistic techniques

Continuous sample paths

• The sample paths of a Wiener process are continuous functions of time (almost surely)
• This means that for almost every realization of the Wiener process, the resulting function $W_t$ is continuous in $t$
• Continuity is an important property that distinguishes the Wiener process from other stochastic processes with jumps or discontinuities

Non-differentiable paths

• Despite having continuous sample paths, the Wiener process is nowhere differentiable (almost surely)
• The sample paths of a Wiener process are highly irregular and exhibit fractal-like behavior
• This non-differentiability has significant implications for the study of stochastic calculus and the development of stochastic integration theories (e.g., Itô and Stratonovich integrals)

Wiener process as limit of random walk

• The Wiener process can be obtained as the limit of a properly scaled random walk as the number of steps tends to infinity
• This connection between discrete random walks and the continuous Wiener process is established through Donsker's theorem and the functional central limit theorem
• Understanding this limit behavior provides insights into the nature of the Wiener process and its role in modeling continuous-time random phenomena

Donsker's theorem

• Donsker's theorem, also known as the functional central limit theorem for random walks, states that a properly scaled random walk converges in distribution to a Wiener process as the number of steps tends to infinity
• Let $S_n = \sum_{i=1}^n X_i$ be a random walk with i.i.d. increments $X_i$ having mean zero and variance $\sigma^2$. Define the scaled random walk $W_n(t) = \frac{S_{\lfloor nt \rfloor}}{\sigma \sqrt{n}}$ for $t \in [0, 1]$. Then, as $n \to \infty$, $W_n(t)$ converges in distribution to a standard Wiener process $W_t$
• Donsker's theorem provides a rigorous justification for using the Wiener process as a continuous-time approximation of discrete random walks

Functional central limit theorem

• The functional central limit theorem is a generalization of the classical central limit theorem to function spaces
• It states that the sum of i.i.d. random functions, properly scaled, converges in distribution to a Gaussian process (e.g., Wiener process) as the number of functions tends to infinity
• In the context of random walks, the functional central limit theorem establishes the convergence of the scaled random walk to the Wiener process in the space of continuous functions

Quadratic variation of Wiener process

• The quadratic variation is a fundamental concept in the study of stochastic processes, particularly in the context of the Wiener process
• It measures the accumulated squared increments of a process over a given time interval and plays a crucial role in stochastic calculus and the theory of stochastic integration
• The quadratic variation of the Wiener process has several important properties and applications in finance, physics, and other fields

Definition of quadratic variation

• The quadratic variation of a stochastic process $X_t$ over the time interval $[0, T]$ is defined as the limit (in probability) of the sum of squared increments of the process over a partition of the interval, as the mesh size of the partition goes to zero
• For a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = T\}$ of $[0, T]$, the quadratic variation of $X_t$ is given by: $[X]_T = \lim_{|\Pi| \to 0} \sum_{i=1}^n (X_{t_i} - X_{t_{i-1}})^2$ where $|\Pi| = \max_{1 \leq i \leq n} (t_i - t_{i-1})$ denotes the mesh size of the partition
• The quadratic variation measures the accumulated squared fluctuations of the process over the given time interval

Quadratic variation over partitions

• For the Wiener process $W_t$, the quadratic variation over any finite partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = T\}$ of $[0, T]$ is given by: $[W]_T = \sum_{i=1}^n (W_{t_i} - W_{t_{i-1}})^2$
• Remarkably, as the mesh size of the partition goes to zero, the quadratic variation of the Wiener process converges to the length of the time interval: $\lim_{|\Pi| \to 0} \sum_{i=1}^n (W_{t_i} - W_{t_{i-1}})^2 = T$
• This property distinguishes the Wiener process from other stochastic processes and has significant implications for stochastic calculus and the development of stochastic integration theories

Stochastic integration with Wiener process

• Stochastic integration extends the concept of integration to stochastic processes, allowing for the integration of random functions with respect to stochastic processes, such as the Wiener process
• The two main approaches to stochastic integration with the Wiener process are the Itô integral and the Stratonovich integral
• Stochastic integration is a fundamental tool in the study of stochastic differential equations and has applications in various fields, including mathematical finance, physics, and engineering

Itô integral

• The Itô integral is a stochastic integral defined for adapted processes with respect to the Wiener process
• Let $f(t, \omega)$ be an adapted process (i.e., $f(t, \omega)$ is $\mathcal{F}_t$-measurable for each $t$, where $\mathcal{F}_t$ is the natural filtration of the Wiener process). The Itô integral of $f$ with respect to the Wiener process $W_t$ over the interval $[0, T]$ is defined as: $\int_0^T f(t, \omega) \, dW_t = \lim_{|\Pi| \to 0} \sum_{i=1}^n f(t_{i-1}, \omega) (W_{t_i} - W_{t_{i-1}})$ where $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = T\}$ is a partition of $[0, T]$ and the limit is taken in probability
• The Itô integral has several important properties, such as linearity, isometry, and martingale property, which make it a powerful tool in stochastic calculus

Stratonovich integral

• The Stratonovich integral is an alternative stochastic integral that differs from the Itô integral in the choice of the evaluation point for the integrand
• While the Itô integral uses the left endpoint of each subinterval in the partition, the Stratonovich integral uses the midpoint
• The Stratonovich integral of an adapted process $f$ with respect to the Wiener process $W_t$ over the interval $[0, T]$ is defined as: $\int_0^T f(t, \omega) \circ dW_t = \lim_{|\Pi| \to 0} \sum_{i=1}^n f\left(\frac{t_{i-1} + t_i}{2}, \omega\right) (W_{t_i} - W_{t_{i-1}})$ where $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = T\}$ is a partition of $[0, T]$ and the limit is taken in probability
• The Stratonovich integral satisfies the usual chain rule of calculus, making it more intuitive in some applications, but it lacks the martingale property of the Itô integral

Stochastic differential equations (SDEs)

• Stochastic differential equations (SDEs) are differential equations that incorporate random terms, typically in the form of a Wiener process or other stochastic processes
• SDEs are used to model the evolution of systems subject to random fluctuations and have applications in various fields, such as finance, physics, engineering, and biology
• The study of SDEs involves the existence and uniqueness of solutions, numerical methods for simulation, and the analysis of the properties of the solutions

SDEs driven by Wiener process

• An SDE driven by a Wiener process is an equation of the form: $dX_t = \mu(t, X_t) \, dt + \sigma(t, X_t) \, dW_t$ where $X_t$ is the stochastic process of interest, $\mu(t, X_t)$ is the drift coefficient, $\sigma(t, X_t)$ is the diffusion coefficient, and $W_t$ is a Wiener process
• The drift term $\mu(t, X_t) \, dt$ represents the deterministic part of the equation, while the diffusion term $\sigma(t, X_t) \, dW_t$ represents the random fluctuations driven by the Wiener process
• SDEs driven by Wiener processes are used to model various phenomena, such as stock prices in finance (e.g., Black-Scholes model), particle motion in physics (e.g., Langevin equation), and population dynamics in biology (e.g., stochastic Lotka-Volterra model)

Itô's lemma

• Itô's lemma, also known as the stochastic chain rule, is a fundamental result in stochastic calculus that allows for the computation of the differential of a function of a stochastic process
• Let $X_t$ be an Itô process satisfying the SDE $dX_t = \mu_t \, dt + \sigma_t \, dW_t$, and let $f(t, x)$ be a twice continuously differentiable function. Then, the differential of $f(t, X_t)$ is given by: $df(t, X_t) = \left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2} \sigma_t^2 \frac{\partial^2 f}{\partial x^2}\right) dt + \sigma_t \frac{\partial f}{\partial x} dW_t$
• Itô's lemma is essential for the analysis and solution of SDEs, as it allows for the transformation of SDEs into more tractable forms and the derivation of important results, such as the Black-Scholes formula in option pricing

Existence and uniqueness of solutions

• The existence and uniqueness of solutions to SDEs are important theoretical questions that ensure the well-posedness of the equations and the reliability of the models they represent
• Under certain conditions on the drift and diffusion coefficients (e.g., Lipschitz continuity and linear growth), the existence and uniqueness of strong solutions to SDEs can be guaranteed
• Strong solutions are stochastic processes that satisfy the SDE almost surely for a given initial condition and a specific Wiener process
• In some cases, weaker notions of solutions, such as weak solutions or martingale problem solutions, may be considered when strong solutions do not exist or are difficult to obtain

Simulation of Wiener process

• Simulating the Wiener process is essential for numerical studies, Monte Carlo simulations, and the analysis of stochastic models driven by Wiener processes
• There are two main approaches to simulating the Wiener process: discrete approximation methods and exact simulation methods
• The choice of the simulation method depends on the specific requirements of the problem, such as the desired accuracy, computational efficiency, and the nature of the stochastic model

Discrete approximation methods

• Discrete approximation methods simulate the Wiener process by discretizing the time interval and generating increments based on the properties of the Wiener process
• One common approach is the Euler-Maruyama method, which approximates the Wiener process $W_t$ at discrete time points $t_0, t_1, \ldots, t_n$ using the following iterative scheme: $W_{t_{i+1}} = W_{t_i} + \sqrt{t_{i+1} - t_i} Z_i$ where $Z_i$ are independent standard normal random variables
• The Euler-Maruyama method is simple to implement but has a strong convergence order of 0.5, meaning that the approximation error decreases with the square root of the step size
• Higher-order approximation methods, such as the Milstein scheme or the Runge-Kutta schemes, can provide better accuracy at the cost of increased computational complexity

Exact simulation methods

• Exact simulation methods generate samples of the Wiener process at specific time points without discretization error
• One approach is to use the Brownian bridge technique, which leverages the properties of the Wiener process to generate samples at intermediate time points given the values at the endpoints
• The Brownian bridge method exploits the fact that, given $W_0$ and $W_T$, the value of the Wiener process at an intermediate time point $t \in (0, T)$ follows a normal distribution with mean $W_0 + \frac{t}{T}(W_T - W_0)$ and variance $\frac{t(T-t)}{T}$
• By recursively applying the Brownian bridge technique, samples of the Wiener process can be generated at any desired set of time points without discretization error
• Exact simulation methods are particularly useful when precise samples of the Wiener process are required, such as in the study of path-dependent options or the analysis of hitting times

Applications of Wiener process

• The Wiener process finds applications in various fields, where it is used to model and analyze random phenomena and stochastic systems
• Some of the main areas where the Wiener process plays a crucial role include mathematical finance, physics and engineering, and biology and neuroscience
• In each of these fields, the Wiener process provides a mathematical framework for understanding and quantifying the effects of random fluctuations on the behavior of the systems under study