Journal ArticleDOI
Existence of stochastic control under state constraints
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In this paper, the authors give a necessary and sufficient condition for the existence of control that keeps the corresponding trajectory of the related stochastic control system within a prescribed closed subset of the state space.Abstract:
In this Note, we give a necessary and sufficient condition for the existence of control that keeps the corresponding trajectory of the related stochastic control system within a prescribed closed subset of the state space. The problem of existence of stochastic control under a state-constraint is also called the viability property of the underlying control system. Our result is: the square of the distance function of this constraint is a viscosity supersolution of a Hamilton-Jacobi-Bellman equation if and only if the system enjoys the viability property.read more
Citations
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Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations
Shige Peng,Xuehong Zhu +1 more
TL;DR: In this article, the authors presented a new approach to obtain the comparison theorem of two 1-dimensional SDEs with diffusion and jumps, where the comparison requirement was regarded as to keep the solution ( X t 1, X t 2 ) within the constraint K = { ( x 1, x 2 ) ; x 1 ⩽ x 2 }.
Book
Global Carleman Estimates for Degenerate Parabolic Operators With Applications
TL;DR: In this paper, Carleman estimates for weakly degenerate operators with Dirichlet boundary conditions are presented, as well as a proof of observability and controllability results.
Journal ArticleDOI
Invariance of stochastic control systems with deterministic arguments
TL;DR: In this paper, it was shown that a closed set K of a finite-dimensional space is invariant under the stochastic control system with two control agents, if and only if the diffusion σ is C 1, 1 regularized.
Journal ArticleDOI
A Representation Formula for the Mean Curvature Motion
TL;DR: This result is generalized to several geometric evolution equations and given a representation formula for the mean curvature motion in terms of the value function of some stochastic optimal control problem.
Journal ArticleDOI
History Path Dependent Optimal Control and Portfolio Valuation and Management
J.-P. Aubin,G. Haddad +1 more
TL;DR: In this article, the authors provide an explicit formula of the valuation function and show that it is the solution of a ''Clio Hamilton-Jacobi-Bellman'' equation.
References
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Journal ArticleDOI
User’s guide to viscosity solutions of second order partial differential equations
TL;DR: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorem, and continuous dependence may now be proved by very efficient and striking arguments as discussed by the authors.
Journal ArticleDOI
Compactification methods in the control of degenerate diffusions: existence of an optimal control
TL;DR: In this article, the authors prove the existence of an optimal Markovian relaxed control for a degenerate diffusion, i.e. a control which takes values in the space of probability measures on a compact space.
Journal ArticleDOI
The viability theorem for stochastic differential inclusions 2
TL;DR: In this paper, the authors combine two ways for representing uncertainty through stochastic differential inclusions: a "stochastic uncertainty" driven by a Wiener process, and a "contingent uncertainty"driven by a set-valued map, as well as to consider stochastically control problems with continuous dynamir and state dependent controls.
Journal Article
Stochastic viability and invariance
TL;DR: In this article, the authors implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php).
Journal Article
Viability for constrained stochastic differential equations
Serge Gautier,Lionel Thibault +1 more