MHB Help solving/proofing an Integral Equation

[cdbsmith]

Need help showing that both sides of the following integral are equal. Any help would be greatly appreciated.

 

[Euge]

Need help showing that both sides of the following integral are equal. Any help would be greatly appreciated.

Since the integral equation is given (and assuming $X_0 = 0$), it suffices to show

$\displaystyle d(X_t^4 - 6tX_t^2 + 3t^2) = (4X_t^3 - 12tX_t)\, dX_t$.

To do this, set $f(t, x) = x^4 - 6tx^2 + 3t^2$ and apply Ito's lemma to get

$\displaystyle df(t, X_t) = \left(f_t + \frac{f_{xx}}{2}\right) dt + f_x \, dX_t$,

$\displaystyle df(t, X_t) = \left(-6X_t^2 + 6t + \frac{12X_t^2 - 12t}{2}\right) dt + (4X_t^3 - 12tX_t)\, dX_t$,

$\displaystyle df(t, X_t) = (-6X_t^2 + 6t + 6X_t^2 - 6t)\, dt + (4X_t^3 - 12tX_t)\, dX_t$,

$\displaystyle df(t, X_t) = 0\, dt + (4X_t^3 - 12tX_t)\, dX_t$,

$\displaystyle d(X_t^4 - 6tX_t^2 + 3t^2) = (4X_t^3 - 12tX_t)\, dX_t$.

 

[cdbsmith]

Since the integral equation is given (and assuming $X_0 = 0$), it suffices to show

$\displaystyle d(X_t^4 - 6tX_t^2 + 3t^2) = (4X_t^3 - 12tX_t)\, dX_t$.

To do this, set $f(t, x) = x^4 - 6tx^2 + 3t^2$ and apply Ito's lemma to get

$\displaystyle df(t, X_t) = \left(f_t + \frac{f_{xx}}{2}\right) dt + f_x \, dX_t$,

$\displaystyle df(t, X_t) = \left(-6X_t^2 + 6t + \frac{12X_t^2 - 12t}{2}\right) dt + (4X_t^3 - 12tX_t)\, dX_t$,

$\displaystyle df(t, X_t) = (-6X_t^2 + 6t + 6X_t^2 - 6t)\, dt + (4X_t^3 - 12tX_t)\, dX_t$,

$\displaystyle df(t, X_t) = 0\, dt + (4X_t^3 - 12tX_t)\, dX_t$,

$\displaystyle d(X_t^4 - 6tX_t^2 + 3t^2) = (4X_t^3 - 12tX_t)\, dX_t$.

Thanks, Euge!

I'm going to study your solution and try to understand it. I will let you know if I get stuck on something.

Thanks again!