Hello I have to calculate: $\mathbb{E}(\hat{S}(t)|F_u^W)$ where $F_u^W$ - natural filtration for Wiener Process, and where $\hat{S}(t)=e^{-rt}S(0)e^{\mu t -\frac{1}{2}\sigma^2t+\sigma W_t}$. Here is solution (exercise 2.1) but i don't undertand why final result is $\hat{S}(u)e^{(\mu-r)t}$ instead of $\hat{S}(u)e^{(\mu-r)(t-u)}$ Is there a mistake or do I not understand something?
https://www.cambridge.org/sa/files/5213/6680/0138/BSM-solutions.pdf