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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.

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Bjoerk - Arbitrage Theory in Continuous Time describes this extensively in Chapter 19. – phi Jan 27 '14 at 13:01
up vote 1 down vote accepted

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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