Assume stock price follows geometric Brownian motion: $$ \frac{dS_t}{S_t}=rdt+bdW_t$$ where $r,b$ are constants and $W_t$ is the standard Brownian motion. Define $$V_t=\sup_{\tau \in [t,T]}\{\, E[e^{-r(\tau-t)}(K-S_\tau)^{+}]|F_t\}\,$$ and $$\tau*=\inf\{\,u\in[t,T]| V_u=(K-S_u)^{+}\}\,$$ where $K, T$ are constants, $(F_t)_{t\ge 0}$ is a filtration and $(K-S_u)^{+}=\max(K-S_u,0)$.
I want to find following expectation.
$$E[e^{-r(u-t)}\mathbb{1}_{\{\,K>S_u,\tau^*>u\}\,}|F_t]$$ Here $u\in [t,T]$ and $\mathbb{1}_{\{\,K>S_u,\tau^*>u\}\,}$ is the indicator function.
Any comments, reference or suggestions appreciated. The only reference I found was Shreve(Stochastic Calculus for Finance II: Continuous-Time Models) section 3.7.3 which mention about joint density function. But I am not sure how to get a joint density function when you have $\tau*$ is defined as above.