Bellman equation: Difference between revisions
imported>Luis (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...") |
imported>Luis No edit summary |
||
| (4 intermediate revisions by 2 users not shown) | |||
| Line 3: | Line 3: | ||
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 fully understood whether that such representation holds for integro-differential equations. | |||
[[Category:Fully nonlinear equations]] | |||
{{stub}} | |||
Latest revision as of 18:26, 7 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.
This article is a stub. You can help this nonlocal wiki by expanding it.