I want to derive the original state price density from its dynamics by using Ito's lemma.
The dynamics of state price density $H$ is:
$$dH = -rH\,dt - \theta H\,dW(t)$$
The lecturer's solution to this is:
$$H(T) = H(0)\exp\left[(-r-0.5 \,\theta^2)T - \theta\,dW(T)\right]$$
But every time I compute it, my solution is always:
$$H(T) = H(0)\exp\left[(-r+0.5 \, \theta^2)T - θdW(T)\right]$$
The way I used to compute it is set its log function and apply Ito's lemma to the log function, which is the same as in the website Use Ito's Lemma to compute $d(\log S(t)$ and use this to find the closed form solution of S(t)
Is the lecturer's solution wrong? Or the method I used is problematic?
Thank you for your help.