Merton Portfolio Optimization as a Starting Point for Ratio Objectives

Summary

The question asks for research on using stochastic optimal control and Hamilton–Jacobi–Bellman methods to maximize risk-adjusted performance measures, including Sharpe, M-squared, Sortino, and Sterling ratios. The response points to Merton's continuous-time portfolio problem as a natural starting framework for choosing investment allocations under specified asset dynamics.

It notes that in an independent and identically distributed setting, the problem connects to the classical Markowitz portfolio model. It also directs readers toward extensions that incorporate drawdown constraints or transaction costs, both of which add realistic complications to portfolio control. The reply is a concise pointer rather than a worked optimization: it does not derive an HJB equation, explain how to encode each ratio as an objective, compare ratio definitions, or cite specific papers. Thus, Merton's problem offers a foundation for further study, but the document does not show that it directly solves every requested ratio-maximization problem.

Key ideas

• Merton's portfolio problem is suggested as a foundation for continuous-time portfolio optimization.
• Under independent and identically distributed dynamics, the framework connects to Markowitz portfolio selection.
• Drawdown constraints and transaction costs are cited as extensions of the basic portfolio problem.
• The response does not derive an HJB formulation or show how to maximize the named performance ratios directly.

Tags

# Stochastic Optimal Control for ratios


# Stochastic Optimal Control for ratios












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.

## Answer by lehalle (score 2, accepted)

https://quant.stackexchange.com/a/40460

You should probably start with Merton's portfolio problem; it is exactly what you have in mind. Few features of this model:

- When dynamics are i.i.d. (i.e. no dynamics...) it recovers Markowitz portfolio.

- You will easily find version including draw down constraints;

- or with transaction costs.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.