I'm trying to teach myself stochastic processes and I'm going through Kurt Jacobs' Stochastic Processes for physicists and am having difficulty following the math for calculating the second derivative of an example of a stochastic differential equation following Ito's rule of $(dW)^{2}=dt$.
In the following differential equation, where $dW$ is an increment of a Wiener process and $\gamma$ and $\beta$ are positive constants.
$dx=-\gamma x dt\, + \, \beta xdW$
To calculate the second moment, you start by calculating the differential for $x^2$. The solution is given as
$d(x^2)=-2\gamma x^{2} dt + 2 \beta x^{2}dW + \beta ^{2} x^{2} dt$
My understanding is that under the chain rule the differential equation $\frac{d}{dt}(x^{2})$ is equal to $2x\frac{dx}{dt}$, so $d(x^{2})= 2x\,dx$. But if I apply this to the equation for $dx$, I get
$d(x^{2})= -2 \gamma x^{2}dt + 2\beta^{2}x^{2}dW$
which is missing the term $\beta^{2}x^{2}dt$.
Where did I go wrong? Is there a rule of Ito's calculus that I'm not following? I know that from Ito's rule $(\beta x dW)^{2} = \beta^{2}x^{2}dt$, but I don't see where it comes into play
Yes, you need to take a second order Taylor expansion in order to appropriately apply the Ito rule.
So you have $$ d(x^2) = 2x dx + dx^2$$ and you get an additional term $$ dx^2 = (-\gamma xdt + \beta x dW)^2 = \beta^2 x^2 dW^2 = \beta^2x^2dt$$