Consider a market with a risky asset S and a bond B with dynamics dS_t = Mu S_t dt + Sigma S_t dt + Sigma S_t dW_t, dB_t = rB_tdt, Where W is a Brownian motion under the physical measure P. Consider an option with payoff (S_T). The price of the option u(t,S_t) satisfies u(t, S_t)/B_t = E Median [ (S_T)/B_T|f_t], Where E Median denoted expectation under the risk-neutral measure P Median, and f_t is a filtration for W. Derive the PDE satisfied by the function u(t, s). Let X_t denote the value of an investor's portfolio at time t. Let Delta_t denote the number of shares of S_t the investor owns and let denote the number of bounds the investor owns. The total value of the investors portfolio is X_t = Delta_tS_t + B_t. We say that the portfolio X_t is self-financing if we have dX_t = Delta_tdS_t + DB_t. Using your result from part (a), show that if Delta_t = u(t, S-t)/s, X_t - Delta_tS_t/B_t then the portfolio is self-financing