Applying Itô’s Lemma to Arithmetic Brownian Motion

Summary

The document clarifies how to apply Itô’s lemma to arithmetic Brownian motion written as a deterministic time term plus a scaled Brownian process. The key point is to define a function of two arguments, time and the current process value, then take the partial derivatives with respect to those arguments separately. For this function, the time derivative is the drift, the derivative with respect to the process value is the volatility coefficient, and the second derivative with respect to that value is zero.

Substituting these derivatives into Itô’s formula gives the differential as a drift term proportional to time increment plus a volatility term proportional to the Brownian increment. The explanation emphasizes that one does not take an ordinary partial derivative of the Brownian process with respect to time inside the formula. This resolves the apparent conflict between Brownian motion varying over time and holding the process argument fixed for the partial derivative. The answer is limited to this simple linear process and does not discuss more general stochastic processes.

Key ideas

• Itô’s lemma treats time and the current process value as separate arguments of a function.
• The time partial derivative holds the process-value argument fixed.
• For the stated linear function, the time derivative is the drift and the process derivative is the volatility coefficient.
• The second derivative with respect to the process value vanishes in this example.
• The Brownian increment enters through the stochastic differential, not an ordinary time derivative.

Tags

# Answer by user16651 (score 1)


# Why does the partial derivative, $X_t$, of an ABM $X(t)$ not involve standard Brownian motion $Z(t)$, even though $Z(t)$ varies with $t$?












Consider the arithmetic Brownian motion $X(t) = \alpha t + \sigma Z(t)$ and evaluating $dX(t)$ using Ito's lemma.

We have $\frac{\partial X}{\partial t} = \alpha$, which does not involve $Z(t)$, even though $Z(t)$ varies with $t$.

When using Ito's lemma, why is $Z(t)$ treated as an independent random variable from $t$ and therefore the partial with respect to $t$ treats $Z(t)$ as a constant?

Edit: I'm not sure why this is getting downvoted. What I have above is coming directly from my textbook "Models for Financial Economics" by Abraham Weishaus.

I'm looking for clarification on what the author writes in italics: "When using Ito's lemma, $Z(t)$ treated as an independent random variable from $t$ and therefore the partial with respect to $t$ treats $Z(t)$ as a constant"

## Answer by user16651 (score 1)

https://quant.stackexchange.com/a/31522

Your imagine from Ito's lemma is false. Although $Z(t)$ is a time-dependent process, we shouldn't apply $\frac{\partial Z(t)}{\partial t}$ in Ito's lemma. It is meaningless.

Let $f(t,x)=\alpha t+\sigma x\in \mathbb{C}^{1,2}\left([0,+\infty)\times \mathbb{R}\right)$. By application of Ito's lemma, we have $$df(t, Z(t))=\frac{\partial f}{\partial t}(t, Z(t))dt+\frac{\partial f}{\partial x}(t, Z(t))dZ(t)+\frac 12\frac{\partial^2 f}{\partial x^2}(t, Z(t))d[Z(t),Z(t)]\tag 1$$ thus $$dX(t)=\alpha dt+\sigma dZ_t\tag 2$$

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