286 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
Stochastic Verification Theorem of Forward-Backward Controlled Systems for Viscosity Solutions
In this paper, we investigate the controlled system described by
forward-backward stochastic differential equations with the control contained
in drift, diffusion and generator of BSDE. A new verification theorem is
derived within the framework of viscosity solutions without involving any
derivatives of the value functions. It is worth to pointing out that this
theorem has wider applicability than the restrictive classical verification
theorems. As a relevant problem, the optimal stochastic feedback controls for
forward-backward system are discussed as well
Regularity properties for general HJB equations. A BSDE method
In this work we investigate regularity properties of a large class of
Hamilton-Jacobi-Bellman (HJB) equations with or without obstacles, which can be
stochastically interpreted in form of a stochastic control system which
nonlinear cost functional is defined with the help of a backward stochastic
differential equation (BSDE) or a reflected BSDE (RBSDE). More precisely, we
prove that, firstly, the unique viscosity solution of such a HJB
equation over the time interval with or without an obstacle, and with
terminal condition at time , is jointly Lipschitz in , for
running any compact subinterval of . Secondly, for the case that
solves a HJB equation without an obstacle or with an upper obstacle it is shown
under appropriate assumptions that is jointly semiconcave in .
These results extend earlier ones by Buckdahn, Cannarsa and Quincampoix [1].
Our approach embeds their idea of time change into a BSDE analysis. We also
provide an elementary counter-example which shows that, in general, for the
case that solves a HJB equation with a lower obstacle the semi-concavity
doesn't hold true.Comment: 30 page
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
Integral-Partial Differential Equations of Isaacs' Type Related to Stochastic Differential Games with Jumps
In this paper we study zero-sum two-player stochastic differential games with
jumps with the help of theory of Backward Stochastic Differential Equations
(BSDEs). We generalize the results of Fleming and Souganidis [10] and those by
Biswas [3] by considering a controlled stochastic system driven by a
d-dimensional Brownian motion and a Poisson random measure and by associating
nonlinear cost functionals defined by controlled BSDEs. Moreover, unlike the
both papers cited above we allow the admissible control processes of both
players to depend on all events occurring before the beginning of the game.
This quite natural extension allows the players to take into account such
earlier events, and it makes even easier to derive the dynamic programming
principle. The price to pay is that the cost functionals become random
variables and so also the upper and the lower value functions of the game are a
priori random fields. The use of a new method allows to prove that, in fact,
the upper and the lower value functions are deterministic. On the other hand,
the application of BSDE methods [18] allows to prove a dynamic programming
principle for the upper and the lower value functions in a very
straight-forward way, as well as the fact that they are the unique viscosity
solutions of the upper and the lower integral-partial differential equations of
Hamilton-Jacobi-Bellman-Isaacs' type, respectively. Finally, the existence of
the value of the game is got in this more general setting if Isaacs' condition
holds.Comment: 30 pages
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