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Stochastic control is widely used in finance since it is about decision taking in a random environment. Option replication, portfolio construction, intraday trading are typical applications of stochastic control.

Optimal control is at the core of math finance. From a formal viewpoint, Markovian dynamics, Dynamic Programming Principle (DPP), and the Hamilton-Jacobi-Bellman master equation are the main tools of stochastic control. 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 to lower your risk. Of course it is a control program, see Continuous-time Stochastic Control and Optimization with Financial Applications, by Huyen Pham, for examples (and here for a summary by the same author).

The Black-Scholes PDE is a straightforward translation of the HJB to a portfolio containing a diffusive product.

Portfolio Construction

You have a given amount of money to invest, you will build a portfolio with it. You have some expectations in terms of the dynamics of returns of the available investment instruments (stocks, bonds, etc) and estimated for the associated risk. Changing your allocation has a cost at each time step. Again it is a control program, see for instance Dynamic Portfolio Choice with Frictions, Garleanu and Pedersen.

Intraday Trading

You have a large amount of stocks to buy or sell today. You have the whole day (around 8h) to do it, and you can take a decision every millisecond. The faster you go the more you move the market at your detriment, the slower you go the more your are exposed to a bad price change. It is nothing else than control again, see Market Microstructure in Practice, by L and Laruelle. Market Microstructure Knowledge Needed for Controlling an Intra-Day Trading Process for an overview (there is a preprint available here).

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