An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests
- A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\sigma$, and the remaining proportion $1-p$ in the risk-free asset, with continuous rate r.
Q1.) How would one find the stochastic differential equation for $U(X_t)$, where $dX_t$ can be written by combining the proportions of the different assets ?
Q2.) By considering $U(X_{t+dt}) | F^X_t$, where $F^X_t$ is the filtration generated by the wealth process, how would one derive the optimal proportion p that the investor should use for the time period $[t; t + dt]$ based on Expected Utility Theory.