# Maximizing the expected log utility

Let's assume that we have a self-financing portfolio made by $$\delta_t$$ shares and $$M_t$$ cash, so that its infinitesimal variation is:

$$dW_t = rM_t \, dt + \delta_t \, dS_t$$

We define $$\alpha_t$$ as the fraction of wealth $$W_t$$ that is invested in stocks, namely $$\alpha_t = \frac{\delta_t S_t}{W_t}$$ . In particular, $$\alpha_t$$ is not constant for any time $$t$$, but it is known at time $$t$$.

In this way,we get another representation of the law of motion of the wealth $$W_t$$, given by:

$$dW_t = W_t[r+\alpha_t(\mu-r)] \, dt + W_t\alpha_t\sigma dB$$ where $$dB$$ refers to the standard Brownian motion.

The goal is to maximize the following function, i.e. to maximize the expected log-utility with respect to $$\alpha_t$$: $$$$\begin{array}{l} \displaystyle \ {V(t,W_t)}=\max_{\alpha} {E(log(W_T))}\\ \ \end{array}$$$$

At the very end the result that is obtained is that $$V(0, W_0)\geq E(log(W_t))$$ with the equality that is reached only if $$\alpha_t = \frac{\mu-r}{\sigma^2}$$

What I can't understand is not the computational or analytical side, but the interpretation that I should give to this result.
The only explaination that I have supposed arises from the fact that the log utiliy is a special case of the power utility when $$\eta=1$$ , where $$\eta$$ is a measure of the risk aversion of the investor. For this reason, if the agent is risk averse, he prefers something "sure" today rather than something "stochastic" tomorrow, unless he applies a perfect optimization of his wealth.

You are right about the log function. Since the payoff is the log of the portfolio, you get "diminishing" returns. So if there is a $$p=q=0.5$$ of the stock going $$\pm 10$$, the upwards payoff is less than the downwards. So in this scenario, since the payoff is only dependent on the stock, the more volatility, the less you should be invested in the stock.
If you take the case where the utility function is $$(1-x)^2$$ for the SDE: $$dX_t = \alpha_t r dt + dW_t$$ Using HJB, you there is no $$\alpha_t \in \mathbb{R}$$ that maximises the portfolio because the payoff is "squared", so negative values of your stock still give you profitable payoffs (giving you 0 downside for investing everything in stocks).