Brownian Motion

Brownian Motion (or Wiener Process) is a fundamental concept in probability theory and stochastic processes. It describes the random motion of particles suspended in a fluid, but it also has applications in various fields such as physics, finance, and biology.

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Properties

The best way to describe Brownian motion is through its defining properties:

1. (Start at zero) $B_0=0$ almost surely.
2. Independent increments: For $0\le t_0<t_1<\cdots <t_n$, the increments $B_{t_1}-B_{t_0},\dots ,B_{t_n}-B_{t_{n-1}}$ are independent.
3. Stationary Gaussian increments: For $0\le s<t$, $B_t-B_s\sim \mathcal{N}(0,t-s)$.
4. Continuous paths: With probability $1$, $t\mapsto B_t$ is continuous (nowhere differentiable a.s.).
5. Martingale: $(B_t)$ is a martingale; also $(B_t^2-t)$ is a martingale.
6. Scaling (self-similarity): For any $c>0$, $(B_{ct})\stackrel{d}{=}(\sqrt{c}{B}_t)$.
7. Quadratic variation: $[B{]}_t=t$; sums of $(\Delta B{)}^2$ over partitions converge to $t$.
8. Markov property: Conditional on $B_s$, the future increments depend only on $s$.
9. Time homogeneity: Distributions depend only on time differences $t-s$.

Definition

Brownian motion can be formally defined as a stochastic process $(B_t{)}_{t\ge 0}$ that satisfies the properties listed above. It can be constructed in various ways, such as through the limit of random walks or using the Wiener measure.

As a function:

$$B_t=\underset{n\to \infty }{\lim }\frac{1}{\sqrt{n}}\sum _{i=1}^{\lfloor nt\rfloor }{X}_i$$