# Bellman equation

(Difference between revisions)
 Revision as of 19:38, 27 May 2011 (view source)Luis (Talk | contribs) (Created page with "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 arbitrar...")← Older edit Revision as of 19:49, 27 May 2011 (view source)Luis (Talk | contribs) Newer edit → Line 6: Line 6: The equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_a$. The equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_a$. + + 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.

## Revision as of 19:49, 27 May 2011

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 $L_a$.

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.