4,289 research outputs found
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
OPTIMIZING COMPLEX BIOECONOMIC SIMULATIONS USING AN EFFICIENT SEARCH HEURISTIC
For simulation to be truly useful for investigating many problems in agricultural economics, non-simplifying optimization techniques need to be employed. General methods for simulation optimization that do not inhibit system characterization or analysis are available, and they would appear to provide much of the mathematical and optimizing rigor demanded by economists. This paper describes the theory and algorithm of a robust and efficient simulation optimization approach, the Complex Method. An example of implementing the algorithm is illustrated using a pest management problem.simulation, optimization, Complex Method, hill-climbing, Research Methods/ Statistical Methods,
On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects
We consider a class of dynamic advertising problems under uncertainty in the
presence of carryover and distributed forgetting effects, generalizing a
classical model of Nerlove and Arrow. In particular, we allow the dynamics of
the product goodwill to depend on its past values, as well as previous
advertising levels. Building on previous work of two of the authors, the
optimal advertising model is formulated as an infinite dimensional stochastic
control problem. We obtain (partial) regularity as well as approximation
results for the corresponding value function. Under specific structural
assumptions we study the effects of delays on the value function and optimal
strategy. In the absence of carryover effects, since the value function and the
optimal advertising policy can be characterized in terms of the solution of the
associated HJB equation, we obtain sharper characterizations of the optimal
policy.Comment: numerical example added; minor revision
Optimal stopping under adverse nonlinear expectation and related games
We study the existence of optimal actions in a zero-sum game
between a stopper and a controller choosing a
probability measure. This includes the optimal stopping problem
for a class of sublinear expectations
such as the -expectation. We show that the game has a
value. Moreover, exploiting the theory of sublinear expectations, we define a
nonlinear Snell envelope and prove that the first hitting time
is an optimal stopping time. The existence of a saddle
point is shown under a compactness condition. Finally, the results are applied
to the subhedging of American options under volatility uncertainty.Comment: Published at http://dx.doi.org/10.1214/14-AAP1054 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I
The main objective of this paper and the accompanying one \cite{ETZ2} is to
provide a notion of viscosity solutions for fully nonlinear parabolic
path-dependent PDEs. Our definition extends our previous work \cite{EKTZ},
focused on the semilinear case, and is crucially based on the nonlinear optimal
stopping problem analyzed in \cite{ETZ0}. We prove that our notion of viscosity
solutions is consistent with the corresponding notion of classical solutions,
and satisfies a stability property and a partial comparison result. The latter
is a key step for the wellposedness results established in \cite{ETZ2}. We also
show that the value processes of path-dependent stochastic control problems are
viscosity solutions of the corresponding path dependent dynamic programming
equation.Comment: 42 page
Tightness and duality of martingale transport on the Skorokhod space
The martingale optimal transport aims to optimally transfer a probability
measure to another along the class of martingales. This problem is mainly
motivated by the robust superhedging of exotic derivatives in financial
mathematics, which turns out to be the corresponding Kantorovich dual. In this
paper we consider the continuous-time martingale transport on the Skorokhod
space of cadlag paths. Similar to the classical setting of optimal transport,
we introduce different dual problems and establish the corresponding dualities
by a crucial use of the S-topology and the dynamic programming principle
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