3,165 research outputs found
Multigrid methods for two-player zero-sum stochastic games
We present a fast numerical algorithm for large scale zero-sum stochastic
games with perfect information, which combines policy iteration and algebraic
multigrid methods. This algorithm can be applied either to a true finite state
space zero-sum two player game or to the discretization of an Isaacs equation.
We present numerical tests on discretizations of Isaacs equations or
variational inequalities. We also present a full multi-level policy iteration,
similar to FMG, which allows to improve substantially the computation time for
solving some variational inequalities.Comment: 31 page
Uniqueness of Viscosity Solutions for Optimal Multi-Modes Switching Problem with Risk of default
In this paper we study the optimal m-states switching problem in finite
horizon as well as infinite horizon with risk of default. We allow the
switching cost functionals and cost of default to be of polynomial growth and
arbitrary. We show uniqueness of a solution for a system of m variational
partial differential inequalities with inter-connected obstacles. This system
is the deterministic version of the Verification Theorem of the Markovian
optimal m-states switching problem with risk of default. This problem is
connected with the valuation of a power plant in the energy market.Comment: 25 pages; Real options, Backward stochastic differential equations,
Snell envelope, Stopping times, Switching, Viscosity solution of PDEs,
Variational inequalities. arXiv admin note: text overlap with arXiv:0805.1306
and arXiv:0904.070
Viscosity Solutions for a System of PDEs and Optimal Switching
In this paper, we study the -states optimal switching problem in finite
horizon, when the switching cost functions are arbitrary and can be positive or
negative. This has an economic incentive in terms of central evaluation in
cases where such organizations or state grants or financial assistance to power
plants that promotes green energy in their production activity or what uses
less polluting modes in their production. We show existence for optimal
strategy via a verification theorem then we show existence and uniqueness of
the value processes by using an approximation scheme. In the markovian
framework we show that the value processes can be characterized in terms of
deterministic continuous functions of the state of the process. Those latter
functions are the unique viscosity solutions for a system of variational
partial differential inequalities with inter-connected obstacles.Comment: 26 pages. arXiv admin note: substantial text overlap with
arXiv:1102.1256, arXiv:0805.1306, arXiv:0904.0707, arXiv:1202.1108, and
arXiv:0707.2663 and arXiv:1104.2689 by other authors. IMA Journal of
Mathematical Control and Information (2016
Optimal Multi-Modes Switching Problem in Infinite Horizon
This paper studies the problem of the deterministic version of the
Verification Theorem for the optimal m-states switching in infinite horizon
under Markovian framework with arbitrary switching cost functions. The problem
is formulated as an extended impulse control problem and solved by means of
probabilistic tools such as the Snell envelop of processes and reflected
backward stochastic differential equations. A viscosity solutions approach is
employed to carry out a finne analysis on the associated system of m
variational inequalities with inter-connected obstacles. We show that the
vector of value functions of the optimal problem is the unique viscosity
solution to the system. This problem is in relation with the valuation of firms
in a financial market
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