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Question :

Let $$ dY_t=c_tdt+d_tdW^1_t+e_tdW^2_t $$ Where $W^1_t,~~W^2_t$ are standard independent brownian motions.

I am trying to apply Ito's formula to this, say for example trying to find $d(\frac {1}{Y_t})$

I'm not too sure about how to treat the variance term in Ito's formula.

Any hints would be appreciated,

Cheers

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I'll assume that $c,d,e$ depend only on $t$. In integral form, $$ Y_t=Y_0+\int_0^tc_s\,\mathrm ds+\int_0^td_s\,\mathrm dW_s^1+\int_0^te_s\,\mathrm dW_s^2. $$ Then, for any $C^2$ function $f$ Itô's formula (in differential form) states $$ \mathrm d\left(f(Y_t)\right)=f'(Y_t)\,\mathrm dY_t+\frac12f''(Y_t)\,\mathrm d\langle Y\rangle_t. $$ I think your problem is finding $\mathrm d\langle Y\rangle_t$. To do this, use bilinearity and symmetry of $\langle \cdot,\cdot\rangle$: \begin{multline*} \langle Y\rangle_t=\langle \int_0^\cdot c_s\,\mathrm ds\rangle_t+\langle \int_0^\cdot d_s\,\mathrm dW_s^1\rangle_t+\langle \int_0^\cdot e_s\,\mathrm dW_s^2\rangle_t +2\langle \int_0^\cdot c_s\,\mathrm ds,\int_0^\cdot d_s\,\mathrm dW_s^1\rangle_t\\ +2\langle \int_0^\cdot c_s\,\mathrm ds,\int_0^\cdot e_s\,\mathrm dW_s^2\rangle_t+2\langle\int_0^\cdot e_s\,\mathrm dW_s^2,\int_0^\cdot d_s\,\mathrm dW_s^1\rangle_t. \end{multline*} Now, $\langle \int_0^\cdot c_s\,\mathrm ds\rangle_t=0$ and $\langle \int_0^\cdot c_s\,\mathrm ds,\int_0^\cdot \varphi_s\,\mathrm dW_s^i\rangle_t=0$ because $\int_0^\cdot c_s\,\mathrm ds$ has finite variation. Also, $\langle\int_0^\cdot e_s\,\mathrm dW_s^2,\int_0^\cdot d_s\,\mathrm dW_s^1\rangle_t=0$ because the two Brownian motions are independent.

The only terms that remain are $$ \langle Y\rangle_t=\langle \int_0^\cdot d_s\,\mathrm dW_s^1\rangle_t+\langle \int_0^\cdot e_s\,\mathrm dW_s^2\rangle_t =\int_0^t d_s^2\,\mathrm ds+\int_0^t e_s^2\,\mathrm ds=\int_0^t \left(d_s^2+e_s^2\right)\,\mathrm ds. $$ Thus, $$ \mathrm d\left(f(Y_t)\right)=f'(Y_t)\left(c_t\mathrm dt+d_t\mathrm dW^1_t+e_t\mathrm dW^2_t\right)+\frac12f''(Y_t)\left(d_t^2+e_t^2\right)\,\mathrm dt. $$ Lastly, if you look around, you'll see that a general form of Itô's formula can be applied directly to $f(Y_t)=f\circ F(\int c_s\,\mathrm ds,\int d_s\,\mathrm dW_s^1,\int e_s\,\mathrm dW_s^2)$, making directly appear the quadratic variations of $W_t^1$ and $W_t^2$. The two approaches coincide.

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  • $\begingroup$ Thanks again Ian, this really helps. Are there any books you would recommend? I am currently using Baxter & Rennie but they seem to assume a lot of this without any explanation. $\endgroup$ – dimebucker May 25 '14 at 4:41
  • $\begingroup$ @dimebucker91, sorry but I learnt stochastic calculus during lectures, and do not know of any good book on the subject. You could try asking for references in another question or searching MSE for good references, but I'm afraid I can't help. $\endgroup$ – Ian May 25 '14 at 4:46
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$$ d(1/Y_{t}) = -Y_{t}^{-2}dY_{t} + Y_{t}^{-3}dY_{t}dY_{t} $$ Substituting $dY_{t}$ into this equation.

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  • $\begingroup$ why $dY_t dY_t$? $\endgroup$ – dimebucker May 24 '14 at 11:05
  • $\begingroup$ This is the Ito formula, we also can write as $d[Y]_{t}$. $\endgroup$ – Eric May 24 '14 at 12:46
  • $\begingroup$ Sorry but we better AVOID writing things such as $dY_tdY_t$ (and it seems you forgot a factor 1/2 in Itô). $\endgroup$ – Did Jun 21 '14 at 13:22
  • $\begingroup$ Yes, I am so sorry. $\endgroup$ – Eric Jun 21 '14 at 14:31

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