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This soft/reference question is related to this MO post of a similar nature.

What are some examples of elliptic PDEs appearing in control and BSDEs?

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I’ll see if I have time to write a proper answer later, but if you consider stochastic games a form of control problem, then tug of war games are intimately connected to the $p$-Laplacian, which is an elliptic partial differential operator.

More or less, the $p$-Laplace equation $\nabla_p u = 0$ is the dynamic programming equation for tug of war games. The link above goes into more detail on this connection.

Here is a relatively recently discovered example of the $\infty$-Laplacian operator appearing naturally in option pricing theory, which is an area of stochastic control heavily involving BSDE and FBSDE theory.

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