1. My professor says that when the interest rate is a random variable, we are supposed to use Brownian Motion. Thus, use Brownian Motion for r(s).
2. please use clear handwriting since I am not a native speaker, and don't copy the existing answers from Course Hero and Chegg. I check them out, but hard to read them due to awful handwriting also follow the logic. Thanks!

Show that the no-arbitrage time-zero value of a zero coupon bond with maturity t > 0 is e 2 , when the risk-neutral instantaneous risk free rate follows r(s) = Z, where Z is a standard normal random variable, and 0 g s S t. (Assume continuous compounding.)