HJB Methods for Financial Control and Their Computational Limits

Summary

The document introduces Hamilton–Jacobi–Bellman equations as a framework for solving continuous-time financial control problems, including consumption choice, optimal stopping, and robust control. It points readers toward classic treatments of consumption and portfolio decisions, continuous-time arbitrage theory, and investment under uncertainty. These references indicate the breadth of applications, but the post does not derive the HJB equation or walk through a worked solution.

Its main methodological caution is the curse of dimensionality. As the number of state variables or assets grows, the associated partial differential equation can become impractical to solve numerically. The discussion contrasts conventional PDE approaches with Monte Carlo methods, citing research that applies simulation to intertemporal portfolio choice. It mentions a four-asset application and a Monte Carlo study, but gives no performance comparison or implementation details. Thus, it is a starting point for study rather than a complete explanation of HJB construction, boundary conditions, or numerical convergence.

Key ideas

• HJB equations provide a method for formulating continuous-time financial control problems.
• Applications mentioned include consumption, portfolio choice, optimal stopping, and robust control.
• PDE-based solutions can become impractical as the number of model dimensions increases.
• Monte Carlo methods are presented as an alternative for intertemporal portfolio problems, without detailed comparison or implementation guidance.

Tags

# How are the Hamilton–Jacobi–Bellman equations used to solve optimal control problems?


# How are the Hamilton–Jacobi–Bellman equations used to solve optimal control problems?












I would like to learn more on how optimal control problems are solved for financial applications.

The approach seems to have a lot of interesting applications such as

- optimal consumption

- choosing optimal stopping times

- robust control

What is the intuitive idea behind the HJB equations and how are they used?

A good introduction or book recommendation would be appreciated as well.

## Answer by user25064 (score 1, accepted)

https://quant.stackexchange.com/a/10074

See for reference

- Merton 1971 Optimum consumption and portfolio rules in a continuous-time model is an excellent application of the topic.

- As @phi mentioned Arbitrage theory in Continuous Time by Bjork is an excellent resource as well.

- Dixit and Pindyck Investment Under Uncertainty

The pitfall is essentially that in many problems we face the curse of dimensionality. This implies that PDE based approaches are less than practical (often practically impossible in many respects). There is one study which expands on Merton's 1971 work to FOUR ASSETS which essentially represented countries. Recent work by DeTemple Garcia and Rindisbacher 2003 solve the problem of intertemporal portfolio using Monte Carlo methods. See A Monte Carlo Method for Optimal Portfolios Parallel processing on existing infrastructure may make this a more practical approach to intertemporal asset allocation.

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