Formulate the following barrier option pricing problems as partial differential equations with suitable boundary and final conditions:

(a) The option has barriers at levels S_u and S_d, above and below the initial asset price, respectively. If the asset touches both barriers before expiry, then the option has payoff max(S − E, 0). Otherwise the option does not pay out.

(b) The option has barriers at levels S_u and Sd, above and below the initial asset price, respectively. If the asset price first rises to S_u and then falls to S_d before expiry, then the option pays out $1 at expiry.