optimal strategy problem (using Jensen's inequality)

Question

I have a strategy in Samuelson model with zero safe rate defined as $$Z_t^{\Pi}=\frac{X_t^{\Pi}}{X_t^{\rho}} \quad \quad (1)$$ where $$\frac{dX_t^{\Pi}}{X_t^{\Pi}} = \mu \pi dt + \sigma \pi \ dW_t \quad \quad (2)$$ $$ \frac{dX_t^{\rho}}{X_t^{\rho}} = \mu \rho dt + \sigma \rho \ dW_t \quad \quad (3)$$

what gives the following dynamic $$\frac{dZ_t^{\Pi}}{Z_t^{\Pi}} = (\mu -\sigma^2 \rho )(\pi - \rho) dt + \sigma (\pi - \rho)\ dW_t \quad \quad (4)$$

To prove that $\rho=\frac{\mu}{\sigma^2} $ is the optimal strategy for the $$\max_{\Pi} E [ logX_T^{\Pi}] \quad \quad (5)$$

Using (1), logarithmic property, Jensen's inequality and supermartingale property I can derive the below inequality $$ E \big{[} log(X_t^{\Pi}) \big{]} \leq E \big{[} log(X_t^{\rho}) \big{]} \quad \quad (6)$$

The question I have is how the inequality (5) implies that the optimal strategy is $\rho=\frac{\mu}{\sigma^2} $?

Answer 1

I assume that the problem is $$\max_{\pi} E\left(\ln Z_T^{\Pi} \right).$$ Note that $\ln Z_t^{\Pi} = \ln X_t^{\Pi} -\ln X_t^{\rho}$. Moreover, \begin{align*} d\ln Z_t^{\Pi} &= d\ln X_t^{\Pi} -d\ln X_t^{\rho}\\ &=\Big[\big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big) - \big(\mu \rho- \frac{1}{2}\sigma^2 \rho^2\big) \Big]dt + \sigma(\pi-\rho)dW_t. \end{align*} Then \begin{align*} E\left(\ln Z_T^{\Pi} \right) = \Big[\big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big) - \big(\mu \rho- \frac{1}{2}\sigma^2 \rho^2\big) \Big]T. \end{align*} Consequently, \begin{align*} \max_{\pi}E\left(\ln Z_T^{\Pi} \right) &= \max_{\pi}\Big[\big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big) - \big(\mu \rho- \frac{1}{2}\sigma^2 \rho^2\big) \Big]T\\ &=\max_{\pi}\Big[-\frac{1}{2}\sigma^2\big( \pi - \frac{\mu}{\sigma^2} \big)^2 + \frac{1}{2}\mu^2- \big(\mu \rho- \frac{1}{2}\sigma^2 \rho^2\big) \Big]T, \end{align*} which is a maximization problem for a quadratic function of $\pi$. It is then clear that the maximum is achieved at $\pi = \frac{\mu}{\sigma^2}$.

EDIT

Consider the problem $\max_{\pi}E\left(\ln X_T^{\Pi} \right)$. Note that \begin{align*} d\ln X_t^{\Pi} = \big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big)dt + \sigma \pi dW_t. \end{align*} Then, \begin{align*} \max_{\pi}E\left(\ln X_T^{\Pi} \right) &= \max_{\pi}\big(\mu \pi - \frac{1}{2}\sigma^2 \pi^2\big)T\\ &=\max_{\pi}\Big[-\frac{1}{2}\sigma^2\big( \pi - \frac{\mu}{\sigma^2} \big)^2 + \frac{1}{2}\mu^2 \Big]T, \end{align*} which is again a maximization problem for a quadratic function of $\pi$, and the maximum is achieved at $\pi = \frac{\mu}{\sigma^2}$.