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The problem of when to exercise an option with Bermudan features is an optimal stopping problem. I have a done a lot of work on how to do these things when the state space is high dimensional. There are various more complicated problems where the contract is more difficult eg swing options.
Of course, optimal control is at the core of math finance. Take few applications: Option Pricing: you have an exposure to a time dependent combination of market factors; you have some knowledge of their dynamics. They are partly deterministic, partly stochastic (i.e. random). At each "time step" you can adjust your portfolio at a given cost. Your goal is ...
Actually, a lot of finance and economics are centered around optimal control problems. Traditionally, most economies are modeled as dynamic systems. In finance, portfolio optimizations, advanced option pricing etc are all optimal control problems. You could look at the book Non Linear Option Pricing, it has a lot of optimal control problems.
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