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Stochastic Calculus / Brownian Motion: Construction and Properties

Overview

The Black-Scholes model rests on a single stochastic process: Brownian motion (also called the Wiener process). Every derivative price, every stochastic volatility model, every interest-rate SDE is ultimately driven by Brownian motion — or by a transformation of it. Yet Brownian motion is not just a modelling convenience. It is the universal limiting object for sums of independent increments (the Donsker invariance principle), the unique continuous process with independent Gaussian increments, and the prototype of a process with non-zero quadratic variation — the property that makes ordinary calculus fail and forces us to invent Itô's formula.
This topic builds Brownian motion from first principles: starting from the CLT, constructing the process rigorously, exploring its four faces as a martingale / Markov process / Gaussian process / self-similar process, establishing the reflection principle and strong Markov property (which power barrier-option pricing), proving that its quadratic variation equals tt (which forces the extra 12σ2\frac{1}{2}\sigma^2 term in Itô's formula), and extending to the multidimensional and correlated setting needed for multi-asset derivatives.

By the end of this topic you should be able to:

  • Derive the CLT and explain why the normal distribution is the universal limit for i.i.d. increments.
  • State the defining properties of Brownian motion and explain why they uniquely determine its law.
  • Use the four characterisations of BM (martingale, Markov, Gaussian, self-similar) to compute expectations and transition probabilities.
  • State and apply the strong Markov property and the reflection principle; derive the distribution of the running maximum max⁡s≤tWs\max_{s \le t} W_s.
  • Prove that [W,W]t=t[W, W]_t = t in L2L^2 and explain why this forces the Itô correction term.
  • Set up correlated multi-dimensional BM using Cholesky factorisation and compute joint increments.
  • Derive the backward heat equation from the Markov property of BM.

In this lesson

  1. Contents
  2. The big picture

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