There are two parts to this question. Firstly, I would like to calculate the integral of
$\text{d} \theta_{t} = Kr \sin \theta_{t}\text{d}t + \sigma \text{d} W_{t}$
from $t$ to $t+T$, i.e.,
$\theta_{t+T} - \theta_{t} = K\int_{t}^{t+T} \sin \theta_{s}\text{d}s + \sigma \int_{t}^{t+T}\text{d} W_{s}$.
Here $K$ and $r$ are constants, $\sigma$ is the noise strength and $W_{t}$ is a standard Brownian motion. Furthermore, all $\theta_{t}$'s should be read modulo $2\pi$.
Secondly, I would like to calculate
$\text{Var}[\theta_{t+T} - \theta_{t}]$.
For some context: I have data of different realizations of $\theta_{t+T} - \theta_{t}$ and I was hoping to find a way to estimate $\sigma$ using the variance of these data.