Consider the Black-Scholes model (for the risky stock S) with volatility sigma>0 and interest rate r>-1 under the EMM Q,so that

dS_t=S_t{rdt+(sigma)dB_t^Q}, S_0>0.

fir T>0, derive the joint density if S_T and M_T:= max_{0<=t<=T}S_t.

Hint: first derive the joint density of the Brownian Motion with a drift mu in R, i.e. (mu)t+B_t^Q, and its maximum?