Applying Itô’s Lemma to Powers of a Wiener Process

Summary

The document explains why the differential of the fourth power of standard Brownian motion includes a time term as well as a stochastic term. It applies Itô’s lemma to a general Itô process, then specializes to standard Wiener motion, whose drift is zero and diffusion coefficient is one. For the fourth power, the second derivative contributes the additional term proportional to time.

The same differentiation gives a formula for the differential of the nth power: the stochastic component comes from the first derivative, while the time component comes from half the second derivative. This illustrates the quadratic-variation correction that distinguishes stochastic calculus from ordinary differentiation. The result assumes standard Brownian motion and a power function; the document does not explore other processes, interpretations of stochastic integration, or applications to trading models.

Key ideas

• Itô’s lemma adds a second-derivative term to the differential of a function of Brownian motion.
• For standard Wiener motion, the drift is zero and the diffusion coefficient is one.
• The differential of a power has both a stochastic term and a time term.
• The time correction depends on the second derivative of the power function.

Tags

# Differentiating Wiener process


# Differentiating Wiener process












I have come across an expression as below

$d\left({W_t}^4\right) = 4 {W_t}^3 d\left({W_t}\right) + 6{W_t}^2 dt$

where $W_t$ is standard Wiener process.

While I understand the first part of the RHS, I fail to understand the second part.

Could you please help to explain it? What is the expression for the general case i.e. $d\left({W_t}^n\right)$

## Answer by Kevin (score 5, accepted)

https://quant.stackexchange.com/a/79153

Let $\text dX_t=\mu_t\text dt+\sigma_t\text dW_t$ be an Itô process. Itô's Lemma tells us $$\text df(t,X_t)=\left(f_t+\mu_tf_x+\frac{1}{2}\sigma_t^2f_{xx}\right)\text dt+\sigma_tf_x\text dW_t.$$

You're interested in the special case of a Brownian motion with $\mu_t=0$ and $\sigma_t=1$ (ie, $X_t=W_t$). Thus, Itô's Lemma simplifies to $$\text df(t,W_t)=\left(f_t+\frac{1}{2}f_{xx}\right)\text dt+f_x\text dW_t.$$

In your case, $f(t,x)=x^4$ with $f_t=0$, $f_x=4x^3$, and $f_{xx}=12x^2$. Thus, $$\text d (W_t^4)=6W_t^2\text dt+4W_t^3\text dW_t.$$

As for the more general $f(t,x)=x^n$, we have $f_t=0$, $f_x=nx^{n-1}$, and $f_{xx}=n(n-1)x^{n-2}$. Thus, $$\text d (W_t^n)=\frac{n(n-1)}{2}W_t^{n-2}\text dt+nW_t^{n-1}\text dW_t.$$ Of course, if you set $n=4$, you recover the prior special case.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.