Exercise of applying Ito's Lemma

Question

Let ${(X_{t})}_{t\geq 0}$ a stochastic process that evolves with dynamic $\frac{dX_{t}}{X_{t}}=(t+1)dt+\frac{1}{2}dW_{t}$. Determine the stochastic differential of process $Y_{t}=X_{t}e^{-\frac{1}{2}[(t+1)^2]}$.

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Trivially, i have to calculate $\frac{\partial Y_{t}}{\partial t}$, $\frac{\partial Y_{t}}{\partial X_{t}}$ and $\frac{\partial ^{2}Y_{t}}{\partial X_{t}^{2}}$ and replace these partial derivatives in $dY_{t}=\frac{\partial Y_{t}}{\partial t}dt+\frac{\partial Y_{t}}{\partial X_{t}}dX_{t}+\frac{1}{2}\frac{\partial ^{2}Y_{t}}{\partial X_{t}^{2}}\left \langle dX_{t},dX_{t} \right \rangle$, right?

Thanks for any possible confirmations.

Answer 1

Let $Y_{t}=X_{t}e^{-\frac{1}{2}[(t+1)]^{2}}$. So:

• $\frac{\partial Y_{t}}{\partial t}=-e^{-\frac{1}{2}[(t+1)]^{2}}(t+1)$
• $\frac{\partial Y_{t}}{\partial X_{t}}=e^{-\frac{1}{2}[(t+1)]^{2}}$
• $\frac{\partial^2 Y_{t}}{\partial X_{t}^2}=0$

Replacing in Ito's SDE:

$dY_{t}=\frac{\partial Y_{t}}{\partial t}dt+\frac{\partial Y_{t}}{\partial X_{t}}dX_{t}+\frac{1}{2}\frac{\partial^2 Y_{t}}{\partial X_{t}^2}\left \langle dX_t,dX_t \right \rangle=e^{-\frac{1}{2}[(t+1)]^2}[(t+1)+(t+1)X_tdt+\frac{1}{2}X_tdW_t]$.

I wait for confirmations!

Thanks in advance!