Interpreting the Time Partial Derivative in Itô’s Lemma

Summary

The document clarifies what the time partial derivative means when applying Itô’s lemma to a function of a stochastic process. For a function written as f(S_t, t), the partial derivative with respect to t treats the process value S_t as the first, separate argument. It does not differentiate the time dependence already present in S_t as though applying an ordinary chain rule to a single-variable function.

This distinction explains why, for a function such as the logarithm of the process with no explicit second-argument dependence, the partial derivative with respect to that independent time argument is zero. The process’s stochastic time evolution is accounted for by the other terms in Itô’s lemma. The answer is a conceptual clarification rather than a worked derivation, and it notes that Itô’s lemma differs from ordinary calculus; readers may need a fuller stochastic calculus treatment for the underlying justification.

Key ideas

• In Itô’s lemma, the time partial derivative holds the process value fixed.
• Time dependence inside the stochastic process is handled through separate terms in the lemma.
• A function with no explicit dependence on the independent time argument has a zero time partial derivative.
• Ordinary chain rule intuition alone does not capture Itô’s lemma’s setup.

Tags

# Is there a better, more rigorous explanation for why this partial derivative is 0 using Ito's Lemma?


# Is there a better, more rigorous explanation for why this partial derivative is 0 using Ito's Lemma?












I encountered the following slide in a lecture on Ito's Lemma.

The lecturer explained that $$\frac{\partial V}{\partial t} = 0$$ because the first two derivatives on the slide already took into account time into the change of the value of V.

I'm not convinced. If $V = \log S(t)$ is a function of time, why wouldn't we have to use the chain rule for the third derivative on the slide?

$$\frac{\partial V}{\partial t} = \frac{\partial V}{\partial S(t)} \cdot \frac{\partial S(t)}{\partial t} = S^{-1} \cdot \frac{\partial S(t)}{\partial t} = ...$$

I'm not sure where to go from here to show that it is in fact 0.

## Answer by Gordon (score 3, accepted)

https://quant.stackexchange.com/a/28083

A process indeed depends on time $t$. However, in Ito's lemma, only derivatives with respect to independent time variable $t$ is considered. That is, for a process of the form $f(S_t, t)$, $\frac{\partial f}{\partial t}$ is the derivative with respect to the second, that is, the independent, $t$ variable, however, the parameter $t$ in the process $S_t$ is not considered. Ito's lemma takes a particular form, which can not be understood in the normal calculus sense.

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