# Proving an Expectation

Assume the risk-free bond $$B_t$$ and the stock $$S_t$$ follow the dynamics of the Black & Scholes model without dividends. Consider the perpetual American put option with payoff $$(K-S_\tau)^+$$ when exercised at time $$\tau >0$$. Given that $$0, consider the stopping time $$\tau_L=inf(u\geq0:S_u\leq L)$$.

For any $$\lambda \in \Re$$, $$Y_{\lambda,t} = (S_t/S_0)^\lambda e^{-(r\lambda-\lambda(1-\lambda)\sigma^2/2)t}$$.

For $$\lambda=\lambda_-$$, $$Y_{\lambda,t} = (S_t/S_0)^\lambda e^{-rt}$$

If $$S_0\geq L$$, then prove that: $$E^Q[e^{-r\tau L}] = (S_0/L)^\lambda$$

Was thinking to make use of the process $$Y_{\lambda,t}$$ in the formulation of the expectation. However, I can't seem to link it to the part regarding $$\tau_L$$.

I would really appreciate all the help I can get! Thank you!