modeling terminal asset value

Attached is the equation which shows the initial model of the terminal asset value with the assumption that r is a risk-free rate. Now, I would like to change the assumption and make it with a risky variable. I would need following:
1. Develop an optimal investment strategy of initial capital X into risky assets.
2. Study a method for predicting the payoffs of the strategy.
3. Numerical prediction of methods
4. Compare risk-free vs risky