Stochastic Control Interpretation of PDEs
Many of the PDEs studied here have an interpretation as the value function of a Stochastic Control problem.
This is the case for the
- Pucci Maximal and Minimal Operators
- Convex Envelope
Stochastic Control Interpretation of the Convex Envelope
The convex envelope has an interpretation an optimal stopping problem for a controlled diffusion, where the control is the choice of direction for a one dimensional diffusion (in the ambient higher dimensional space) and the function whose convex envelope is sought plays the role of the obstacle. See Convex Envelope.
Two Operator Stochastic Control Problems
A toy stochastic control problem was also solved in
Also refer to A Free Boundary Problem from Stochastic Control for a model problem. See also [Homogenization of Linear Elliptic]] for the interpretation of the toy problem as a bound on a PDE with random (or unknown) coefficients.
An open research problem is finding Invariant Measures for Controlled Diffusions
This can be interpreted as the nonlinear analogue of the (Kolmogorov) backward equation for the evolution of measures. See Stochastic Differential Equations for the case without controls.