350 research outputs found
A finite-dimensional approximation for partial differential equations on Wasserstein space
This paper presents a finite-dimensional approximation for a class of partial
differential equations on the space of probability measures. These equations
are satisfied in the sense of viscosity solutions. The main result states the
convergence of the viscosity solutions of the finite-dimensional PDE to the
viscosity solutions of the PDE on Wasserstein space, provided that uniqueness
holds for the latter, and heavily relies on an adaptation of the Barles &
Souganidis monotone scheme to our context, as well as on a key precompactness
result for semimartingale measures. We illustrate this result with the example
of the Hamilton-Jacobi-Bellman and Bellman-Isaacs equations arising in
stochastic control and differential games, and propose an extension to the case
of path-dependent PDEs
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 differential games for fully coupled FBSDEs with jumps
This paper is concerned with stochastic differential games (SDGs) defined
through fully coupled forward-backward stochastic differential equations
(FBSDEs) which are governed by Brownian motion and Poisson random measure. For
SDGs, the upper and the lower value functions are defined by the controlled
fully coupled FBSDEs with jumps. Using a new transformation introduced in [6],
we prove that the upper and the lower value functions are deterministic. Then,
after establishing the dynamic programming principle for the upper and the
lower value functions of this SDGs, we prove that the upper and the lower value
functions are the viscosity solutions to the associated upper and the lower
Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. Furthermore, for
a special case (when do not depend on ), under the
Isaacs' condition, we get the existence of the value of the game.Comment: 33 page
Lyapunov stabilizability of controlled diffusions via a superoptimality principle for viscosity solutions
We prove optimality principles for semicontinuous bounded viscosity solutions
of Hamilton-Jacobi-Bellman equations. In particular we provide a representation
formula for viscosity supersolutions as value functions of suitable obstacle
control problems. This result is applied to extend the Lyapunov direct method
for stability to controlled Ito stochastic differential equations. We define
the appropriate concept of Lyapunov function to study the stochastic open loop
stabilizability in probability and the local and global asymptotic
stabilizability (or asymptotic controllability). Finally we illustrate the
theory with some examples.Comment: 22 page
Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications
In this paper, we prove a comparison result between semicontinuous viscosity
sub and supersolutions growing at most quadratically of second-order degenerate
parabolic Hamilton-Jacobi-Bellman and Isaacs equations. As an application, we
characterize the value function of a finite horizon stochastic control problem
with unbounded controls as the unique viscosity solution of the corresponding
dynamic programming equation
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