Under the condition that there exist traded securities with returns that are instantaneously perfectly correlated with the changes in all state variables in the economy, the continuous time model is equivalent to Arrow-Debreu complete markets model and the dynamic trading in securities can be a substitute for a full set of market for pure securities. Discuss the necessary and sufficient conditions under which these claims are true under

(a) Three-fund separation theorem (see Merton (1990), Sec. 15.7).

(b) \(m+2\)-fund separation theorem (see Merton (1990), Sec. 15.10, Eqs. (15.48) and (15.50) in particular).