Steps to understand that $ \sigma \int_t^T e^{\kappa(s-T)} dW_s $ is distributed $ \sigma \sqrt{\frac{1-e^{-2\kappa (T-t)}}{2 \kappa}} N_{0,1}$

Question

What are the steps to see that $ \sigma \int_t^T e^{\kappa(s-T)} dW_s $ is distributed $ \sigma \sqrt{\frac{1-e^{-2\kappa (T-t)}}{2 \kappa}} N_{0,1}$?

My question stems from the euler scheme for $ S_T = S_t e^{-\kappa(T-t)} + \mu \left( 1- e^{-\kappa(T-t)} \right) + \sigma \int_t^T e^{\kappa(s-T)} dW_s $.

Which is the solution to: $dS_t = \kappa (\mu - S_t) dt + \sigma dW_t $.

Answer 1

$\newcommand{\exp}{ \mathbb{ E } }$

Use Itô isometry on the expectation of the square, you already know that the expectation of the integral is zero and that it is a Gaussian.

$\exp[ (\sigma \int_t^T e^{-\kappa (T-s) } dW_s)^2 ] = \exp[ \sigma^2 \int_t^T e^{-2 \kappa (T-s) } ds) ] = \frac{\sigma^2(1-e^{-2\kappa(T-t)})}{2\kappa}$