50 research outputs found
Stochastic Differential Games and Viscosity Solutions of Hamilton-Jacobi-Bellman-Isaacs Equations
In this paper we study zero-sum two-player stochastic differential games with
the help of theory of Backward Stochastic Differential Equations (BSDEs). At
the one hand we generalize the results of the pioneer work of Fleming and
Souganidis by considering cost functionals defined by controlled BSDEs and by
allowing the admissible control processes to depend on events occurring before
the beginning of the game (which implies that the cost functionals become
random variables), on the other hand the application of BSDE methods, in
particular that of the notion of stochastic "backward semigroups" introduced by
Peng allows to prove a dynamic programming principle for the upper and the
lower value functions of the game in a straight-forward way, without passing by
additional approximations. The upper and the lower value functions are proved
to be the unique viscosity solutions of the upper and the lower
Hamilton-Jacobi-Bellman-Isaacs equations, respectively. For this Peng's BSDE
method is translated from the framework of stochastic control theory into that
of stochastic differential games.Comment: The results were presented by Rainer Buckdahn at the "12th
International Symposium on Dynamic Games and Applications" in
Sophia-Antipolis (France) in June 2006; They were also reported by Juan Li at
2nd Workshop on "Stochastic Equations and Related Topics" in Jena (Germany)
in July 2006 and at one seminar in the ETH of Zurich in November 200
Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition
In the present work, we consider 2-person zero-sum stochastic differential
games with a nonlinear pay-off functional which is defined through a backward
stochastic differential equation. Our main objective is to study for such a
game the problem of the existence of a value without Isaacs condition. Not
surprising, this requires a suitable concept of mixed strategies which, to the
authors' best knowledge, was not known in the context of stochastic
differential games. For this, we consider nonanticipative strategies with a
delay defined through a partition of the time interval . The
underlying stochastic controls for the both players are randomized along
by a hazard which is independent of the governing Brownian motion, and knowing
the information available at the left time point of the subintervals
generated by , the controls of Players 1 and 2 are conditionally
independent over . It is shown that the associated lower and
upper value functions and converge uniformly on compacts to
a function , the so-called value in mixed strategies, as the mesh of
tends to zero. This function is characterized as the unique viscosity
solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation.Comment: Published in at http://dx.doi.org/10.1214/13-AOP849 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Interior H\"older and Calder\'on-Zygmund estimates for fully nonlinear equations with natural gradient growth
We establish local H\"older estimates for viscosity solutions of fully
nonlinear second order equations with quadratic growth in the gradient and
unbounded right-hand side in spaces, for an integrability threshold
guaranteeing the validity of the maximum principle. This is done through a
nonlinear Harnack inequality for nonhomogeneous equations driven by a uniformly
elliptic Isaacs operator and perturbed by a Hamiltonian term with natural
growth in the gradient. As a byproduct, we derive a new Liouville property for
entire viscosity solutions of fully nonlinear equations as well as a
nonlinear Calder\'on-Zygmund estimate for strong solutions of such equations
Differential games through viability theory : old and recent results.
This article is devoted to a survey of results for differential games obtained through Viability Theory. We recall the basic theory for differential games (obtained in the 1990s), but we also give an overview of recent advances in the following areas : games with hard constraints, stochastic differential games, and hybrid differential games. We also discuss several applications.Game theory; Differential game; viability algorithm;
Carathéodory's method for a class of dynamic games
AbstractThe method of equivalent variational methods, originally due to Carathéodory for free problems in the calculus of variations is extended to investigate feedback Nash equilibria for a class of n-person differential games. Both the finite-horizon and infinite-horizon cases are considered. Examples are given to illustrate the presented results