Consider an American up-and-out put option with barrier level B() = B0 e and strike price

Consider an American up-and-out put option with barrier level B(τ) = B0^e−ατ and strike price X. Assuming that the underlying asset pays a continuous dividend yield q, find the integral representation of the early exercise premium. What would be the effect on the optimal exercise price S^∗(τ ;B(τ)) when B₀ decreases?