Determine the dynamics (SDE) of X_t^4

Question

Suppose X and Y evolve according to:

dXt= (2 + 5t + Xt)dt + 3 dz_1t

dYt= 4Ytdt + 8Ytdz_1t + 6dz_2t

where z_1t and z_2t are Brownian motions with (dz_1t )(dz_2t )=0.1dt

Can you give me some starting point how to transform the equations to the initial SDE in order to apply the Ito's lemma?

But you say you only want to look at $X_t^4$ so why do you state $Y_t$? Is there more to this question? — Chinny84, Nov 12 '14 at 14:19
because the is the first sub-question the additional questions are for a.) X^4t b.) e^Xt c.) XtYt d.)X^4tYt — Slavia Sofia, Nov 12 '14 at 14:25
I am good but not that good to know the other questions. I did realise that it was part of a larger question. Just in case you was going to add the other parts you either state in full or split the questions out. — Chinny84, Nov 12 '14 at 14:29

score 0 · Answer 1 · answered Nov 12 '14 at 15:53

$$ df = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial X_t}dX_t + \frac{1}{2}\frac{\partial^2 f}{\partial X_t^2}dX_t^2 $$ That's Ito's

thus $$ dX_t^4 = \frac{\partial }{\partial X_t}\left(X_t^4\right)dX_t + \frac{1}{2}\frac{\partial^2 }{\partial X_t^2}\left(X_t^4\right)dX_t^2 $$ Compute the partials first then use the fact that $\left(dz_t^1\right)^2 = dt$ to compute the sde. Then do a similar process for all the above, except you have to use the modified Ito for multivariable when you have to it for $f\left(X_t,Y_t\right)$.

Determine the dynamics (SDE) of X_t^4

1 Answers1