Reconciling Two Notations for Itô’s Lemma
Summary
The document compares two presentations of Itô’s lemma for a function of an Itô process and time. The question focuses on whether a book’s notation is correct, what the symbols for the first and second derivatives mean when the function has two arguments, and how that presentation relates to a more familiar partial-derivative formula.
The answer clarifies that the prime notation denotes differentiation with respect to the state variable, holding time as the other argument. With that interpretation and a noted typographical correction, the two formulas agree. The response also says the book’s notation is nonstandard, which explains the reader’s confusion. This is a concise notation clarification rather than a derivation of Itô’s lemma; readers still need to understand the assumptions on the process and the differential rules used in the theorem.
Key ideas
- In this setting, the prime on the function denotes a partial derivative with respect to its state variable.
- The derivative notation can be reconciled with the version written explicitly using partial derivatives.
- The formulas match after interpreting the notation and correcting a typographical issue.
- The book’s notation is described as nonstandard, and the answer does not derive the theorem.
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Full text
# Are the Ito's Lemma given in Mark Joshi's Concept and Practice in Mathematical Finance same as what I learn?
# Are the Ito's Lemma given in Mark Joshi's Concept and Practice in Mathematical Finance same as what I learn?
In Joshi's Concepts and Practice in Mathematical Finance, page $110,$ he stated the Ito's Lemma:
> Theorem $5.1$ (Ito's Lemma) Let $X_t$ be an Ito process satisfying $$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t,$$ and let $f(x,t)$ be twice differentiable function; then we have that $f(X_t,t)$ is an Ito process, and that $$d(f(X_t,t)) = \frac{\partial f}{\partial t}(X_t,t)dX_t + f'(X_t,t)dX + \frac{1}{2}f''(X_t,t) dX_t^2$$ where $dX_t^2$ is defined by $$dt^2 = 0, \quad dtdW_t = 0\quad dW_t^2=dt.$$
I have some doubt on the Ito's Lemma above. Is it stated correctly? What is the meaning of $f'(X_t,t)$ and $f''(X_t,t)$ as we have multivariable function? Also, the Ito's Lemma that I know is defined by $$d(f(X_t,t)) = \frac{\partial f}{\partial t}(X_t,t) dt + \frac{\partial f}{\partial x}(x,t) dX_t + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}(x,t) (dX_t)^2$$ where $(dX_t)^2 = \sigma^2(X_t,t)dt.$
Are the two Ito's Lemma above equivelent?
## Answer by Dhruv Gupta (score 2, accepted)
https://quant.stackexchange.com/a/49046
$f'(X_t, t)$ refers to $\frac{\partial f}{\partial x} (X_t, t)$. If you make this change in notation, along with correcting the typo pointed out by @Alex C, the two versions of the Ito's lemma will match.
Also, the notation used in Mark Joshi's book is not standard; your confusion in this scenario is natural.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.