An investor investing in a stock $S_t$, continually rebalances his portfolio, such that at any time $t$, the number of shares he holds is proportional to $ S_t^r$. Here $r$ is a constant real number. The change in his portfolio value between time $0$ and $T$ is therfore $$P(T) - P(0) = \int_0^TS^r dS $$
Assume the stock follows a geometric Brownian motion, $$ dS = \mu S dt + \sigma S dW $$ with constants $\mu$ and $\sigma$.
How do you calculate the portfolio value $P(T)$? By this I mean, can you find an expression for the integral $P(T)$ that only involves $r$, $\mu$, $\sigma$ and a standard normal random variable?
The case $r=0$ is found in many textbooks and has solution $P(T) = S(T) = S_0e^{(\mu -\frac{1}{2}\sigma^2)T + \sigma W_T} $
In particular, how do you do solve this for the cases $r=-1, -\frac{1}{2},\frac{1}{2},1,2$ or can it be done for $r$ in general?