Isaacs equation and Bellman equation: Difference between pages

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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 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 $L_{ab}$.
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.
 
 
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Revision as of 14:49, 27 May 2011

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.