Do you know any good papers on methods of Stochastic Optimal Control and Hamilton-Jacobi-Bellman(HJB) for optimization of different ratios(Sharpe, M2, Sortino, Sterling, etc.)? Meaning that using known stochastic dynamics of asset, we want to find optimal trading strategy for maximization of these ratios. I've found only this thread on Wilmott.
You should probably start with Merton's portfolio problem; it is exactly what you have in mind. Few features of this model: