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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?

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    $\begingroup$ There seems to be an obvious typo in the text you quote above: $dX_t$ should be replaced by $dt$. Then the two "versions" agree. $\endgroup$ – Alex C Oct 3 at 2:07
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    $\begingroup$ @AlexC I see. What about the meaning of $f'(X_t,t)?$ Which derivative is the author referring to? $\endgroup$ – Idonknow Oct 3 at 3:26
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    $\begingroup$ The "prime" is equivalent to differentiation with respect to x. $\endgroup$ – Alex C Oct 3 at 12:05
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$f'(X_t, t)$ 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.

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  • $\begingroup$ Thanks for pointing out. Do you happen to have an errata to the book? $\endgroup$ – Idonknow Oct 3 at 12:10
  • $\begingroup$ No, I dont. Sorry. $\endgroup$ – Dhruv Gupta Oct 4 at 5:29

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