Ito's Lemma

Itô's Lemma is the chain rule for stochastic processes. It's essential for differentiating a function $f(t,W_t)$ where $W_t$ is a Brownian motion, because classical calculus fails.

The core problem is that a Wiener process is "rough." Its quadratic variation is non-zero. The key Itô rule is:

This contrasts with classical calculus, where $(dx{)}^2$ would be considered 0.

Derivation from Taylor Expansion

To find $df$, we start with a Taylor expansion for $f(t,W_t)$:

We use the Itô multiplication rules to simplify the higher-order terms:

• $(d{W}_t{)}^2=dt$ (The crucial rule)
• $(dt{)}^2=0$
• $dt\cdot d{W}_t=0$

Substituting these rules, all terms except three vanish:

The Formula (1D)

By grouping the $dt$ terms from the derivation, we get the standard form of Itô's Lemma for a function $f(t,W_t)$:

• $\frac{\partial f}{\partial t}$: The standard "drift" from time.
• $\frac{\partial f}{\partial W}$: The standard sensitivity to the process $W_t$.
• $\frac{1}{2}\frac{{\partial }^{2}f}{\partial W^2}$: The Itô correction term. This is the crucial adjustment for stochasticity.

Example: Geometric Brownian Motion

Let’s find the SDE for $S(t)=f(t,W(t))$, which is the solution to Geometric Brownian Motion.

Let $A=\mu -\frac{{\sigma }^2}{2}$. Our function is $f=\exp (At+\sigma W_t)$.

We need the partial derivatives:

• $\frac{\partial f}{\partial t}=Af$
• $\frac{\partial f}{\partial W}=\sigma f$
• $\frac{{\partial }^{2}f}{\partial W^2}={\sigma }^{2}f$

Now plug these into the two components of the Itô formula:

1. The $dt$ term:

   $$(\frac{\partial f}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}f}{\partial W^2})=(Af+\frac{1}{2}{\sigma }^{2}f)=f(A+\frac{1}{2}{\sigma }^2)$$

   Substitute $A=\mu -\frac{{\sigma }^2}{2}$:

   $$f((\mu -\frac{{\sigma }^2}{2})+\frac{1}{2}{\sigma }^2)=f\left(\mu \right)=\mu f$$

2. The $d{W}_t$ term:

   $$\frac{\partial f}{\partial W}=\sigma f$$

Combining them gives the differential $dS$:

Replacing $f$ with $S(t)$, we get the familiar SDE: