# Bellman equation

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where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$. | 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 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 $L_a$. | + | 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 known whether that such representation holds for integro-differential equations. |

## 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 known whether that such representation holds for integro-differential equations.

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