Isaacs equation - Revision history

imported>Luis at 14:14, 26 June 2015

2015-06-26T14:14:41Z

← Older revision		Revision as of 09:14, 26 June 2015
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	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.		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.

			Under very general assumptions, [[fully nonlinear integro-differential equations]] can be written in the form of an Isaacs equation.


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imported>Luis at 00:24, 8 February 2012

2012-02-08T00:24:51Z

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	where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.		where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.

	The equation appears naturally in zero sum stochastic games with [[Levy processes]].		The equation appears naturally in [[stochastic control\|zero sum stochastic games]] 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 $L_{ab}$. Under some conditions on that class, there are interior [[differentiability estimates\|$C^{1,\alpha}$ estimates]] for the solution.

imported>Luis at 00:22, 8 February 2012

2012-02-08T00:22:56Z

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	The equation appears naturally in zero sum stochastic games with [[Levy processes]].		The equation appears naturally in zero sum stochastic games 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 $L_{ab}$. Under some conditions on that class, there are interior [[differentiability estimates\|$C^{1,\alpha}$ estimates]] for the solution.

	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 second order fully nonlinear uniformly elliptic PDE $F(D^2 u)=0$ can be written as an Isaacs equation by the following two steps:

imported>Luis at 02:37, 6 February 2012

2012-02-06T02:37:18Z

← Older revision		Revision as of 21:37, 5 February 2012
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	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.		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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imported>Luis at 05:52, 23 January 2012

2012-01-23T05:52:05Z

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	# 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 $\|\|A\|\|\leq \Lambda$.		# 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 ~~linear~~ with respect to all parameters.		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]]~~

imported>Nestor at 06:23, 2 June 2011

2011-06-02T06:23:10Z

← Older revision		Revision as of 01:23, 2 June 2011
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	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 linear with respect to all parameters.		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 linear with respect to all parameters.

			[[Category:Fully nonlinear equations]]

imported>Luis at 19:46, 27 May 2011

2011-05-27T19:46:34Z

← Older revision		Revision as of 14:46, 27 May 2011
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	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 $L_{ab}$.

			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:
			# $F(X)$ 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 linear with respect to all parameters.

imported>Luis: Created page with "The Isaacs equation is the equality \[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \] where $L_{ab}$ is some family of linear integro-differential o..."

2011-05-27T19:38:04Z

Created page with "The Isaacs equation is the equality \[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \] where $L_{ab}$ is some family of linear integro-differential o..."

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The Isaacs equation is the equality
\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.

The equation appears naturally in zero sum stochastic games with [[Levy processes]].

The equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$.