Bellman equation - Revision history

imported>Luis at 00:26, 8 February 2012

2012-02-08T00:26:46Z

← Older revision		Revision as of 19:26, 7 February 2012
Line 7:		Line 7:
	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]]		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 ~~known~~ whether that such representation holds for integro-differential equations.		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]]

imported>Luis at 00:26, 8 February 2012

2012-02-08T00:26:34Z

← Older revision		Revision as of 19:26, 7 February 2012
Line 3:		Line 3:
	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 known whether that such representation holds for integro-differential equations.		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.

imported>Luis at 02:35, 6 February 2012

2012-02-06T02:35:46Z

← Older revision		Revision as of 21:35, 5 February 2012
Line 10:		Line 10:

	[[Category:Fully nonlinear equations]]		[[Category:Fully nonlinear equations]]


			{{stub}}

imported>Nestor at 06:23, 2 June 2011

2011-06-02T06:23:28Z

← Older revision		Revision as of 01:23, 2 June 2011
Line 8:		Line 8:

	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.		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.

			[[Category:Fully nonlinear equations]]

imported>Luis at 19:49, 27 May 2011

2011-05-27T19:49:04Z

← Older revision		Revision as of 14:49, 27 May 2011
Line 6:		Line 6:

	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 $L_a$.

			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.

imported>Luis: Created page with "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 arbitrar..."

2011-05-27T19:38:35Z

Created page with "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 arbitrar..."

New page

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 $L_a$.