Bellman equation: Difference between revisions
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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 | 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 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.
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