Applying Itô's Lemma to a Power of a Geometric Brownian Motion

Summary

This exchange resolves a discrepancy in the stochastic differential for a power of a geometric Brownian motion. Applying Itô's lemma to the power function produces a diffusion term proportional to the power and the Brownian increment, plus a drift with two contributions: the original process drift after differentiation and the second-derivative correction from Itô's lemma. Combining those terms gives the drift coefficient involving the power minus one.

The responses confirm the textbook expression and identify how the correction arises, clarifying why the alternative calculation has an erroneous drift. The document is a focused derivation check rather than a discussion of trading strategy, calibration, or empirical evidence. Its result applies to the stated diffusion process and power transformation; using it in another model requires checking that model's dynamics and notation.

Key ideas

• Differentiate the power function and apply Itô's lemma to account for stochastic curvature.
• The drift combines the differentiated process drift with the second-derivative correction.
• The resulting drift coefficient contains the power minus one factor.
• Check the underlying process dynamics and notation before transferring the result to another model.

Tags

# Shreve book II Question 4.6 Error?


# Shreve book II Question 4.6 Error?












I'm working through Shreve II, and on question 4.6, you are asked to compute

$d(S_t^p)$ where $S_t$ = $S_0e^{\sigma W_t + (\alpha - \frac{1}{2}\sigma^2)t}$

I get the answer $pS_t^p[\sigma dW_t + (\alpha + \frac{1}{2}\sigma^2p - \sigma^2)dt]$

whereas the online solutions manual gets $pS_t^p[\sigma dW_t + (\alpha + \frac{1}{2}\sigma^2(p - 1)dt]$

Clarifying that either I or the author of the solution has made an error. Going through their solution, I believe they drop an expression containing $\sigma$ where they should not drop it.

## Answer by fni (score 1, accepted)

https://quant.stackexchange.com/a/15577

$$d(S^p) = pS^p (\alpha +\sigma dW) + \frac{1}{2}p(p-1)S^p\sigma^2 dt $$

$$ = pS^p \left[ \left(\alpha +\frac{1}{2}\sigma^2(p-1)\right)dt + \sigma dW \right]$$

## Answer by Mr. Rodriguez (score 1)

https://quant.stackexchange.com/a/15576

Shreve's answer is the correct one:

The drift term of $\frac{dS^p}{S^p}$ has two parts:

- $p \left(\alpha - \frac{1}{2} \sigma^2 \right)$ from regular differentiation

- $\frac{1}{2} p^2 \sigma^2 $ is the Ito term.

When you sum them up you get $p \left(\alpha + \frac{1}{2} (p-1) \sigma^2 \right)$

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