Isaacs equation and Bellman equation: Difference between pages

Revision as of 21:35, 5 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 $L_a$.

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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-The Isaacs equation is the equality
+The Bellman equation is the equality
-\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
+\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
-where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
+where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.
-The equation appears naturally in [[stochastic control|zero sum stochastic games]] 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 $L_{ab}$. Under some conditions on that class, there are interior [[differentiability estimates|$C^{1,\alpha}$ estimates]] for the solution.
+The equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_a$.
-Note that any second order fully nonlinear uniformly elliptic PDE $F(D^2 u)=0$ can be written as an Isaacs equation by the following two steps:
+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.
-# $F(X)$ is Lipschitz with constant $\Lambda$, so it is the infimum of all cones $C_{X_0}(x) = F(X_0) + \Lambda|X-X_0|$.
-# Each cone $C(X)$ is the supremum of all linear functions of the form $L(X) = F(X_0) + \mathrm{tr} \, A \cdot (X-X_0)$ for $||A||\leq \Lambda$.
-A more general second order fully nonlinear uniformly elliptic PDE $F(D^2 u, Du, u, x)=0$ can also be written as an Isaacs equation if it is Lipschitz with respect to all parameters.
+[[Category:Fully nonlinear equations]]
-Under very general assumptions, [[fully nonlinear integro-differential equations]] can be written in the form of an Isaacs equation.
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Isaacs equation and Bellman equation: Difference between pages

Revision as of 21:35, 5 February 2012

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