1. One popular concept on stock market price is called "martingale pric- ing", which is relevant to "rational expectations" or "efficient market hypothesis" in economic theories. A rough statement on this is that, a (log of) stock market price pt (e.g., S&P 500 index) is supposed to reflect all the relevant information available up to time t, i.e., therefore pe log is MG. (a) Under the martingale pricing, show that continuously compounded (cc) return for stock market price is mds (b) In what follows, we assume cc return r is a covariance stationary process. Prove the following statements: i. Ifrt ~ iid (0.2) (or indepedent white nose), then r, ~ mas (0, 2). ii. If rt ~ mds (0, 2), then r ~ ll'N (0, 2) (or weak white hoise). (c) Prove the following statements: i. If (r) is id, then it is strictly stationary i. If (r,) is strictly stationary, then it is covariance stationary 1. One popular concept on stock market price is called "martingale pric- ing", which is relevant to "rational expectations" or "efficient market hypothesis" in economic theories. A rough statement on this is that, a (log of) stock market price pt (e.g., S&P 500 index) is supposed to reflect all the relevant information available up to time t, i.e., therefore pe log is MG. (a) Under the martingale pricing, show that continuously compounded (cc) return for stock market price is mds (b) In what follows, we assume cc return r is a covariance stationary process. Prove the following statements: i. Ifrt ~ iid (0.2) (or indepedent white nose), then r, ~ mas (0, 2). ii. If rt ~ mds (0, 2), then r ~ ll'N (0, 2) (or weak white hoise). (c) Prove the following statements: i. If (r) is id, then it is strictly stationary i. If (r,) is strictly stationary, then it is covariance stationary