Chapter 2: Brownian Motion
After establishing the measure-theoretic foundations of conditional expectations and martingale theory, we formally introduce a classic continuous-time stochastic process—Brownian Motion (BM), also known as the Wiener Process. It serves as the foundation for constructing the integral theory of stochastic differential equations (SDEs), such as Itô integration.
1. Basic Definitions and Properties
Definition: Standard Brownian Motion
Let \((\Omega, \mathcal{F}, P)\) be a probability space, on which a real-valued stochastic process \(W = \{W(t), t \ge 0\}\) is defined. If \(W\) satisfies the following four conditions, it is called a Standard Brownian Motion:
1. Initial Zero Point: \(P(W(0) = 0) = 1\) a.s.
2. Independent Increments: For any time partition \(0 \le t_1 < t_2 < \dots < t_n\), the sequence of increments:
are mutually independent.
3. Stationary Gaussian Increments: For any \(0 \le s < t\), the increment follows a normal distribution with mean 0 and variance equal to the time difference:
4. Continuous Paths: Almost all sample paths \(t \mapsto W(t, \omega)\) are continuous (i.e., a.s. continuous).
From the above definition, we can immediately derive the low-order moments and covariance structure of Brownian motion, which is key to deriving the properties of white noise later.
Basic Properties: Moments and Covariance Structure
1. Mean and Variance: Since \(W(t) = W(t) - W(0) \sim \mathcal{N}(0, t)\), we obviously have:
2. Autocovariance Function: For any \(s, t \ge 0\), we have:
Derivation of the Autocovariance (click to expand)
Without loss of generality, assume \(0 \le s \le t\). We decompose \(W(t)\) into an incremental form:
Expanding gives:
Due to the independent increments property of Brownian motion, \(W(t) - W(s)\) and \(W(s) = W(s) - W(0)\) are independent. Furthermore, since the increments have mean 0, the first term is 0:
Since we assumed \(s \le t\), the general case can be written as \(s \wedge t\). \(\square\)
Joint Probability Density and Transition Density of Brownian Motion
The values \((W(t_1), \dots, W(t_n))\) of Brownian motion at time points \(0 < t_1 < t_2 < \dots < t_n\) follow a multivariate normal distribution. Due to the Markov property, its Joint Probability Density (Joint PDF) can be elegantly expressed as a product of Transition Density Functions:
Define the Gaussian transition density (from spatial point \(y\) to \(x\) over time \(t\)):
Then the joint probability density is:
2. Introducing White Noise and the Prototype of SDE
When transitioning from an ordinary differential equation (ODE) \(\frac{dX(t)}{dt} = b(X(t), t)\) to a stochastic differential equation (SDE), we need to add a noise term \(\xi(t)\). The ideal white noise \(\xi(t)\) should be completely uncorrelated at different times, meaning its covariance exhibits the property of the Dirac \(\delta\) function: \(E[\xi(s)\xi(t)] = \delta(s-t)\). Mathematically, this so-called white noise is precisely the "formal derivative" of Brownian motion \(\dot{W}(t)\).
Theorem: The Covariance Limit of the Increment Quotient of Brownian Motion is the \(\delta\) Function
Consider the difference quotient process of Brownian motion \(\xi_h(t) = \frac{W(t+h) - W(t)}{h}\) (\(h > 0\)). As \(h \to 0\), its autocovariance function converges to the Dirac \(\delta\) function in the sense of generalized functions (distributions).
Derivation of the Limit (Click to Expand)
We compute the covariance \(E[\xi_h(s) \xi_h(t)]\) of the difference quotient at different times \(s, t\). Using the covariance property of Brownian motion \(E[W(u)W(v)] = u \wedge v\):
Assuming \(s \le t\), we analyze the non-zero region of the above expression: 1. When the time difference \(|t-s| \ge h\), the four terms cancel each other out, resulting in \(0\). This shows that as long as the time interval is greater than \(h\), the difference quotient process is uncorrelated. 2. When the time difference \(|t-s| < h\), there is an overlapping interval. The calculation yields the covariance as:
This is an isosceles triangular function with a base width of \(2h\) and a height of \(\frac{1}{h}\). Clearly, the integral \(\int_{-\infty}^\infty \varphi_h(x) dx = 1\). As \(h \to 0\), this function tends to 0 at non-zero points, tends to infinity at 0, and its integral remains constant at 1. This is precisely the definition of the Dirac \(\delta\) function:
This explains why SDEs are usually written in differential form \(dX(t) = b(X,t)dt + \sigma(X,t)dW(t)\), because the true derivative of \(W(t)\) does not exist and can only be treated as a generalized function. \(\square\)
3. Multidimensional Brownian Motion and Core Properties
In quantitative finance and multi-particle systems, we often encounter high-dimensional stochastic phenomena.
Definition: Multidimensional Brownian Motion (n-dimensional BM)
An \(n\)-dimensional Brownian motion is defined as a vector process \(W(t) = (W^1(t), W^2(t), \dots, W^n(t))^T\), where: 1. Each component \(W^k(t)\) is a standard one-dimensional Brownian motion. 2. The components are mutually independent: for any \(k \neq l\), the \(\sigma\)-algebras \(\sigma(W^k(t), t \ge 0)\) and \(\sigma(W^l(t), t \ge 0)\) are independent.
The covariance structure between its components is:
(Here \(\delta_{kl}\) is the Kronecker delta, not the Dirac \(\delta\) function)
The classic nature of Brownian motion lies in its combination of Martingale and Markov Process characteristics.
Theorem: Brownian Motion is a Continuous Martingale
Let \(\{\mathcal{F}_t\}_{t \ge 0}\) be the natural filtration generated by the Brownian motion, \(\mathcal{F}_t = \sigma(W(u), 0 \le u \le t)\). Then \(W(t)\) is a martingale.
Proof of Martingale Property (click to expand)
For any \(s \le t\), we need to prove \(E[W(t) | \mathcal{F}_s] = W(s)\). Construct independence via increment decomposition:
By the linearity of conditional expectation:
Because Brownian motion has independent increments, the future increment \(W(t) - W(s)\) is completely independent of the historical information flow \(\mathcal{F}_s\), so independence implies irrelevance (equals the unconditional expectation); And \(W(s)\) itself is \(\mathcal{F}_s\)-measurable, so it is known as a constant (directly taken out):
Q.E.D. \(\square\)
Theorem: Brownian Motion is a Markov Process
Brownian motion satisfies the Markov property: i.e., "the future depends only on the present, not on the past". For any Borel set \(B \in \mathcal{B}(\mathbb{R}^n)\) and \(s \le t\):
Rigorous Measure-Theoretic Proof of Markov Property (click to expand)
Using the indicator function \(\chi_B\) (or denoted as \(I_B\)), we can write the probability as a conditional expectation:
Introduce the increment, and define the function \(f(x, y) = \chi_B(x + y)\). Decompose \(W(t)\) into \(W(s)\) and \(W(t) - W(s)\):
Here we apply a core lemma in measure theory concerning independence and conditional expectation (the freezing lemma/substitution theorem): Because \(W(s)\) is \(\mathcal{F}_s\)-measurable, and \(W(t) - W(s)\) is independent of \(\mathcal{F}_s\), we can treat \(W(s)\) as a "frozen" constant \(x\), take the unconditional expectation of the other part, and then substitute \(W(s)\) back:
Let the increment \(Z = W(t) - W(s) \sim \mathcal{N}(0, t-s)\), with density function \(g(z)\). The above expression equals:
Change variables by letting \(y = x + z\), then \(dz = dy\):
The result of this integral clearly depends only on the value of the random variable \(W(s)\), and not on any information in \(\mathcal{F}_s\) prior to time \(s\). By the definition of conditional expectation, this is exactly equal to \(E[\chi_B(W(t)) | W(s)]\), i.e., \(P(W(t) \in B | W(s))\). \(\square\)
4. Kolmogorov Continuity Theorem and Path Properties
The definition of Brownian motion directly assumes "continuous paths." However, mathematically we need to ask: given a consistent set of finite-dimensional distributions, does there necessarily exist a modification with continuous paths? This requires the extremely powerful Kolmogorov continuity theorem.
To quantify the "roughness" of continuity, we introduce Hölder spaces.
Definition: Hölder Continuous Space \(C^\gamma\)
A function \(f(t)\) is said to be Hölder continuous with exponent \(\gamma\) if there exists a constant \(K > 0\) such that:
It is denoted as \(f \in C^\gamma\). (Note: When \(\gamma = 1\), this is Lipschitz continuity. The paths of Brownian motion are extremely rough; they are nowhere differentiable, thus not Lipschitz continuous).
Theorem: Kolmogorov Continuity Theorem
Let \(X(t)\) be a stochastic process defined on an interval \([0, T]\). If there exist constants \(\alpha > 0, \beta > 0, C > 0\) such that for all \(s, t \in [0, T]\):
then \(X(t)\) has a modification with continuous sample paths. Furthermore, for any \(\gamma \in \left(0, \frac{\alpha}{\beta}\right)\), the sample paths of this modification are almost surely (a.s.) locally \(\gamma\)-Hölder continuous.
Core Proof Idea (Based on the Borel-Cantelli Lemma) (Click to expand)
The rigorous proof of the theorem is quite involved, but its core mechanism is elegant.
1. Examine dyadic rational points: Consider the dyadic grid points on the interval: \(t_i = \frac{i}{2^n}\). Define the event \(A_n\) where the increment between adjacent points is too large:
2. Bound using Chebyshev/Markov inequality:
Substituting the condition given in the theorem, \(E[|X_{i+1} - X_i|^\beta] \le C (2^{-n})^{1+\alpha}\), yields:
3. Apply the Borel-Cantelli Lemma: Since we chose \(\gamma < \frac{\alpha}{\beta}\), we have \(\alpha - \gamma\beta > 0\). This forms a geometric series with ratio less than 1, therefore:
By the Borel-Cantelli Lemma (the first lemma), \(P(\limsup A_n) = 0\). That is, almost surely, there exists \(N(\omega)\) such that for all \(n \ge N(\omega)\), the increments on the dyadic grid are tightly controlled within \(2^{-n\gamma}\). Through uniform convergence, this guarantees the Hölder continuity of the limiting process. \(\square\)
We then apply this theorem to the continuity analysis of Brownian motion, obtaining its important \(C^{\frac{1}{2}-}\) property.
Conclusion: The Hölder Roughness of Brownian Motion Paths is \(C^{\frac{1}{2}-}\)
The sample paths of Brownian motion \(W(t)\) are almost everywhere \(\gamma\)-Hölder continuous for any \(\gamma \in \left(0, \frac{1}{2}\right)\). (That is, it approaches \(1/2\)-Hölder continuity arbitrarily closely, but does not achieve \(1/2\)).
Derivation (Using Gaussian Moments) (Click to expand)
Since the increment of Brownian motion \(W(t) - W(s) \sim \mathcal{N}(0, |t-s|)\), we know the explicit formula for higher even moments of the normal distribution. For even powers \(2m\) (\(m \in \mathbb{N}\)):
This perfectly matches the condition of the Kolmogorov continuity theorem. Let: - \(\beta = 2m\) - \(1 + \alpha = m \implies \alpha = m - 1\)
According to the theorem, the Hölder exponent \(\gamma\) of the paths must satisfy:
Since this holds for all positive integers \(m\), we can let \(m \to \infty\), where \(\frac{1}{2m} \to 0\). Therefore, for any \(\gamma\) strictly less than \(\frac{1}{2}\), Brownian motion is \(\gamma\)-Hölder continuous. Consequently, we arrive at the conclusion that Brownian motion paths are \(C^{\frac{1}{2}-}\) continuous. \(\square\)