# 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?

• 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. Oct 3, 2019 at 2:07
• @AlexC I see. What about the meaning of $f'(X_t,t)?$ Which derivative is the author referring to? Oct 3, 2019 at 3:26
• The "prime" is equivalent to differentiation with respect to x. Oct 3, 2019 at 12:05

$$f'(X_t, t)$$ 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.