(a) Define a complete market.

The price process of a traded security satisfies the following stochastic differential equation (SDE)

dS_t = S_tdt + S_tdZ_t

where Z_t is a Brownian motion under the real-world probability measure P. Let r > 0 be the continuously compounded risk-free rate of interest, with r not equal to .

(b)Show that the discounted stock price e^rtS_t is not a martingale under the real-world probability measure P.

(c)Demonstrate how the discounted asset price can be a martingale under an equivalent martingale measure Q.

Let V_t be the value at t of a self-financing portfolio, consisting of _t shares of stocks and _t units of cash bonds.

(d) Show that d(e^rtV_t) = _td(e^rtS_t)