Linear equation with additive noise

Question

The following is an excerpt from Mikosch's Elementary Stochastic Calculus:

I'm trying to understand the calculation in the red box where the version of Ito's lemma mentioned is as follows:

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Question: How is (2.30) used in the red box?

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I tried to apply (2.30) directly by using $f(t,x)=y(t)x$ and $$ A_s^{(1)}=c_1(s)X_s+c_2(s),\quad A_s^{(2)}=\sigma_2(s). $$ But I ended up with something very different from the desired formula in the red box. (For instance, in my calculation, the term $c_1'(t)$ pops up. )

Answer 1

I didn't get the desired formula due to wrong calculation of the partial derivatives. For $f(t,x)=y(t)x,$ one should have $$ f_1(t,x)=y'(t)x=-c_1(t)y(t)x\\ f_2(t,x)=y(t)\\ f_{22}(t,x)=0 $$ A direct substitution to (2.30) would then solve the problem.

Answer 2

Here you are using a multivariate Ito formula, which in this context is also known as stochastic integration by parts $$d(XZ) = XdZ+ZdX+d[X,Z].$$ In your case $Z\equiv y$ and $dZ\equiv dy = -c_1 y dt$ so the quadratic covariation part is zero. After substituting for $dX$ and simplifying you will end up with $$ d(Xy)=-c_1 Xy dt + y dX = y(c_2 dt +\sigma_2 dB).$$