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| The Bellman equation is the equality
| | #REDIRECT [[Fully nonlinear integro-differential equations]] |
| \[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
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| where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.
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| The equation appears naturally in problems of stochastic control with [[Levy processes]].
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| The equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_a$.
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| 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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| [[Category:Fully nonlinear equations]]
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| {{stub}}
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Latest revision as of 11:38, 29 May 2011