Brownian Motion and Wiener Process

In this article, we will define Brownian Motion and explain some of its properties, which are very important in creating models for future asset price paths.

Previously, we introduced stochastic calculus in the context of quantitative finance, including the definitions and properties of Markov and Martingale processes, which are fundamental tools in modeling asset price paths.

This article will formally define Brownian Motion and explain the Wiener Process, which is the mathematical version of Brownian Motion, along with why standard Brownian Motion is not suitable for price modeling and the reasons for using Geometric Brownian Motion instead.

From the experiment of tossing a coin to a continuous random process

In the previous article, we considered tossing a coin an unlimited number of times in discrete-time intervals (discrete-time random walk). The current goal is to transition to a continuous-time random walk.

Define the real interval  [0,T][0, T]. [0,T]

Divide the given time period into nnn. Coin tossing, with each toss taking a duration of

Δt=T/n\Delta t = T/nΔt=T/n

The sequence of random variables for tossing a coin is XiX_iXi. It seems there is no text to translate. Please provide the text you'd like me to translate, and I'll be happy to help!

Define the partial sums (partial sums)

Sn=∑i=1nXiS_n = \sum_{i=1}^{n} X_iSn​=∑i=1n​Xi​

By adjusting the scale of XiX_iXi to achieve a special quadratic variation, which is:

QV(Sn)=∑i=1n(Si−Si−1)2=TQV(S_n) = \sum_{i=1}^n (S_i - S_{i-1}) ^2 = TQV(Sn) = ∑_{i=1}^{n} (Si - Si-1)^2 = T

This construction preserves the properties of Markov and Martingale, and as n→∞, it results in a non-diverging random walk.

Definition: Wiener Process or Standard Brownian Motion

A sequence of random variables {B(t), t≥0} {B(t),t≥0} is a Brownian Motion if:

• B(0)=0B(0) = 0B(0)=0
• For 0≤s<t0≤s<t0≤s<t, the difference B(t)−B(s)B(t)−B(s) It has a normal distribution. Has a mean of 0 and a variance of t−st - st−s
• The difference B(t)−B(s)B(t) - B(s)B(t)−B(s) is independent of the information before time sss.

The important properties of Brownian Motion

• Finite: The sum does not diverge even as the number of steps increases.
• Unbounded Variation: If you reverse the direction of the graph to the same direction, it can reach unbounded values at any time.
• Continuous but Nowhere Differentiable: The graph is continuous everywhere, but there are no points where it has a derivative (similar to a fractal structure).
• Markov Property: The future depends only on the present, not on the past.
• Martingale Property: The conditional expectation of the future is equal to the present value.
• Strong Normal Distribution: B(t)B(t)B(t) has a true normal distribution (mean = 0, variance = t)

Note: The strong normal distribution differs slightly from the normal distribution in terms of the strength of the conditions.

Brownian Motion is a key component of stochastic differential equations, which form the basis for the Black-Scholes equation used in contingent claims pricing.

Reference: Brownian Motion and the Wiener Process

From https://www.quantstart.com/articles/Brownian-Motion-and-the-Wiener-Process/