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Can someone help me apply Ito's lemma to the function $f(t,x,k)$ where t is the time and x,k dimensions where x and k refer to dynamics

$dX(t)=\mu(t)dt+\sigma(t)dB(t)$

$dK(t)=\nu(t)dt+\theta(t)dW(t)$

What i have done so far $$ \begin{align}df(t,X(t),K(t))=\left(\frac{\partial f}{\partial t} + \mu(t)\frac{\partial f}{\partial x}+\nu(t)\frac{\partial f}{\partial k}+\frac{1}{2}\sigma(t)^{2}\frac{\partial ^{2}f}{\partial x^{2}}+\frac{1}{2}\theta(t)^{2}\frac{\partial ^{2}f}{\partial k^{2}}\right)dt + \sigma(t)\frac{\partial f}{\partial x}dB(t) + \theta(t)\frac{\partial f}{\partial k}dW(t) + \frac{\partial^2 f}{\partial x\partial k}dB(t)dW(t) \end{align} $$ Can someone correct me please ?

thanks for your time

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    To clarify, you are trying to find the stochastic differential equation governing $f(t,X(t),K(t))$ where $X$ and $K$ satisfy the SDE's above, and $B$ and $W$ are independent brownian motions?2012-02-03
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    I've made some correction which are just notational, the most crucial one - you missed $2$ in $\partial^2 f$ for a mixed derivative. Seems to be correct for me. @Aaron: independence is not used as I see.2012-02-03
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    @Ilya Independence might not be necessary, but it would eliminate the $dB(t)dW(t)$ term, and knowing how they relate would simplify the term in any event. However, knowing for sure that they are brownian motions (and not some other process) is necessary to know that you only need up to quadratic terms, and that $dB^2=dt$, etc. The question confuses me, though, because I'm not really sure what it is asking, other than "is this right?" Without looking carefully at the individual terms, I have seen statements of Ito's lemma that look like what is written, so it doesn't seem an application.2012-02-03
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    First of all, thank you for you immediate responce.2012-02-03
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    @Ilya, i you are correct about the square :P2012-02-03
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    @Aaron, firstly i agree that the dependence of the Brownian motions B(t) and W(t) is not an issue. Maybe they are correlated in some sense or they not (in my case however are, so the term does not vanish). Secondly, i see the above as an application of Ito's lemma, since you apply Ito's lemma to find the dynamics of the above stochastic dynamical system.2012-02-03
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    I just needed some kind of verification , since i missed the square and lead myself to dead end :p Now, after two coffees, i see that clearly :p2012-02-03

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