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Created by Knut M. Synstadfrom the Noun Project

Brownian Motion

Definition (Brownian motion)

A process (Bt)t≥0(B_{t})_{t\ge{0}} on (Ω,F,P)(\Omega,\mathcal{F},P) is called a standard Brownian motion if

  1. B0=0B_{0}=0
  2. Increments are normally distributed: ∀0≤s<t:Bt−Bs∼N(0,t−s)\forall 0\le s<t:B_{t}-B_{s}\sim \mathcal{N}(0,t-s)
  3. Independent Increments: ∀0≤s<t:Bt−Bs⊥ ⁣ ⁣ ⁣⊥FsB\forall 0\le s<t:B_{t}-B_{s}\perp\!\!\!\perp\mathcal{F}_{s}^{B}where FsB=σ(Bu:0≤u<s)\mathcal{F}_{s}^{B}=\sigma(B_{u}:0\le u<s)and FtB\mathcal{F}_{t}^{B} is the natural filtration of (Bt)t≥0(B_{t})_{t\ge 0}
  4. Continuity: ∀ω∈Ω:t↦Bt(ω)\forall\omega \in\Omega:t\mapsto B_{t}(\omega) is continuous in R+\mathbb{R}^{+}

Theorem (Wiener’s theorem)

Brownian Motion does exist.

Theorem (Brownian motion is a martingale)

Let (Bt)t≥0(B_{t})_{t\ge 0} be standard BM on (Ω,F,P)(\Omega,\mathcal{F},P) and let (FtB)t≥0(\mathcal{F}_{t}^{B})_{t\ge0} (FtB=σ(Bu:0≤u<t)\mathcal{F}_{t}^{B}=\sigma(B_{u}:0\le u<t)) be natural filtration of (Bt)t≥0(B_{t})_{t\ge 0}. Then (Bt)t≥0(B_{t})_{t\ge 0} is a (FtB)t≥0(\mathcal{F}_{t}^{B})_{t\ge 0}-martingale.

Linked from

Itô Isometry

Itô Representation Theorem

Brownian Motion