# Bellman equation

(Difference between revisions)
 Revision as of 00:26, 8 February 2012 (view source)Luis (Talk | contribs)← Older edit Latest revision as of 00:26, 8 February 2012 (view source)Luis (Talk | contribs) Line 7: Line 7: The equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_a$. Under some conditions on the operators $L_a$, the solution is always smooth due to the [[nonlocal Evans-Krylov theorem|nonlocal version of Evans-Krylov theorem]] The equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_a$. Under some conditions on the operators $L_a$, the solution is always smooth due to the [[nonlocal Evans-Krylov theorem|nonlocal version of Evans-Krylov theorem]] - Note that any '''convex''' fully nonlinear elliptic PDE of second order $F(D^2u, Du, u, x)$ can be written as a Bellman equation by taking the supremum of all supporting planes of $F$. It is not known whether that such representation holds for integro-differential equations. + Note that any '''convex''' fully nonlinear elliptic PDE of second order $F(D^2u, Du, u, x)$ can be written as a Bellman equation by taking the supremum of all supporting planes of $F$. It is not fully understood whether that such representation holds for integro-differential equations. [[Category:Fully nonlinear equations]] [[Category:Fully nonlinear equations]]

## Latest revision as of 00:26, 8 February 2012

The Bellman equation is the equality $\sup_{a \in \mathcal{A}} \, L_a u(x) = f(x),$ where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.

The equation appears naturally in problems of stochastic control with Levy processes.

The equation is uniformly elliptic with respect to any class $\mathcal{L}$ that contains all the operators $L_a$. Under some conditions on the operators $L_a$, the solution is always smooth due to the nonlocal version of Evans-Krylov theorem

Note that any convex fully nonlinear elliptic PDE of second order $F(D^2u, Du, u, x)$ can be written as a Bellman equation by taking the supremum of all supporting planes of $F$. It is not fully understood whether that such representation holds for integro-differential equations.