SAAM Algorithm

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After having another attention to the (*), we know that when \(\mu\) and \(\Sigma\) are unknown and treated as state variables whose uncertainties are specified by their posterior distributions given the observations \(r_1,r_2,\cdots,r_n\) in a Bayesian framework. Note that if the prior distribution puts all its mass at\((\mu_0,\Sigma_0) \), then the minimization problem (*) reduces to Markowitz’s portfolio optimization problem that assumes \(\mu_0\) and \(\Sigma_0\) are given. The Lagrange multiplier \(\lambda\) in (*) can be regarded as the investor’s risk-aversion index when the variance is used to measure risk.

The problem (*) is not a standard stochastic optimization problem because of \([E(\omega^T r_{n+1})]^2\) ,which is included in \(Var(\omega^T r_{n+1})=E[(\omega^T r_{n+1})^2]-[E(\omega^T r_{n+1})]^2\). The method hereof solving (*) is to convert it to a standard stochastic control problem by using an additional parameter. Note that \(E(W)-\lambda Var(W) = h(EW,EW^2)\), where \(h(x,y) = x+\lambda x^2-\lambda y\). Let \(W_B =\omega_B^T r_{n+1}\) and \(\eta =1+2\lambda E(W_B)\), where \(\omega_B\) is the Bayes weight vector. Then

\begin{equation}
0  \geq   h(EW,EW^2)-h(EW_B,EW_B^2) \\
=\eta\{E(W)-E(W_B)\} +\lambda \{E(W_B^2)-E(W^2)\} +\lambda \{E(W)-E(W_B)\}^2 \\
\geq \{\lambda E(W_B^2)-\eta E(W_B)\}-  \{\lambda E(W^2)-\eta E(W)\}
\end{equation}

Since \(\eta =1+2\lambda E(W_B)\) and the target is \(max\{E(\omega^T r_{n+1}）-\lambda Var(\omega^T r_{n+1})\}\), so we have an equivalence problem as follows:

\(\underset{\eta}{max} \{E[\omega^T(\eta) r_{n+1}]-\lambda Var[\omega^T(\eta) r_{n+1}]\}\)

, \(\omega(\eta)\) is the solution of the stochastic optimization problem

\(\omega(\eta) =\underset{\omega}{argmin} \{\lambda E[(\omega^T r_{n+1})^2]-\eta E(\omega^T r_{n+1})\}\)