1,832 research outputs found
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
Verification Theorems for Stochastic Optimal Control Problems via a Time Dependent Fukushima - Dirichlet Decomposition
This paper is devoted to present a method of proving verification theorems
for stochastic optimal control of finite dimensional diffusion processes
without control in the diffusion term. The value function is assumed to be
continuous in time and once differentiable in the space variable ()
instead of once differentiable in time and twice in space (), like in
the classical results. The results are obtained using a time dependent
Fukushima - Dirichlet decomposition proved in a companion paper by the same
authors using stochastic calculus via regularization. Applications, examples
and comparison with other similar results are also given.Comment: 34 pages. To appear: Stochastic Processes and Their Application
Controlled diffusion processes
This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions
We study a class of stochastic target games where one player tries to find a
strategy such that the state process almost-surely reaches a given target, no
matter which action is chosen by the opponent. Our main result is a geometric
dynamic programming principle which allows us to characterize the value
function as the viscosity solution of a non-linear partial differential
equation. Because abstract mea-surable selection arguments cannot be used in
this context, the main obstacle is the construction of measurable
almost-optimal strategies. We propose a novel approach where smooth
supersolutions are used to define almost-optimal strategies of Markovian type,
similarly as in ver-ification arguments for classical solutions of
Hamilton--Jacobi--Bellman equations. The smooth supersolutions are constructed
by an exten-sion of Krylov's method of shaken coefficients. We apply our
results to a problem of option pricing under model uncertainty with different
interest rates for borrowing and lending.Comment: To appear in MO
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