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In this paper, we present an Uzawa-based heuristic that is adapted
to certain type of stochastic optimal control problems. More precisely,
we consider dynamical systems that can be divided into small-scale
subsystems linked through a static almost sure coupling constraint
at each time step. This type of problem is common in production/portfolio
management where subsystems are, for instance, power units, and one has
to supply a stochastic power demand at each time step. We outline the
framework...
We study the Dirichlet boundary value problem for eikonal type equations of ray
light propagation in an inhomogeneous medium with discontinuous
refraction index. We prove a comparison principle
that allows us to obtain existence and uniqueness of a continuous
viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander
type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value...
We study here the impulse control minimax problem. We allow the cost functionals and dynamics to be unbounded and hence the value functions can possibly be unbounded. We prove that the value function of the problem is continuous. Moreover, the value function is characterized as the unique viscosity solution of an Isaacs quasi-variational inequality. This problem is in relation with an application in mathematical finance.
In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position x. In the constrained...
In the present paper, we consider nonlinear optimal control problems
with constraints on the state of the system. We are interested in
the characterization of the value function without any
controllability assumption. In the unconstrained case, it is possible to derive a
characterization of the value function by means of a
Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the
behavior of the value function along the trajectories arriving or
starting from any position x. In...
Risk sensitive and risk neutral long run portfolio problems with consumption and proportional transaction costs are studied. Existence of solutions to suitable Bellman equations is shown. The asymptotics of the risk sensitive cost when the risk factor converges to 0 is then considered. It turns out that optimal strategies are stationary functions of the portfolio (portions of the wealth invested in assets) and of economic factors. Furthermore an optimal portfolio strategy for a risk neutral control...
This work analyzes a discrete-time Markov Control Model (MCM) on Borel spaces when the performance index is the expected total discounted cost. This criterion admits unbounded costs. It is assumed that the discount rate in any period is obtained by using recursive functions and a known initial discount rate. The classic dynamic programming method for finite-horizon case is verified. Under slight conditions, the existence of deterministic non-stationary optimal policies for infinite-horizon case...
We study a quasiconvex conjugation that transforms the level sum of functions into the pointwise sum of their conjugates and derive
new duality results for the minimization of the max of two quasiconvex functions. Following Barron and al., we show that the level
sum provides quasiconvex viscosity solutions for Hamilton-Jacobi equations in which the initial condition is a general continuous
quasiconvex function which is not necessarily Lipschitz or bounded.
We investigate the problem of power utility maximization considering risk management and strategy constraints. The aim of this paper is to obtain admissible dynamic portfolio strategies. In case the floor is guaranteed with probability one, we provide two admissible solutions, the option based portfolio insurance in the constrained model, and the alternative method and show that none of the solutions dominate the other. In case the floor is guaranteed partially, we provide one admissible solution,...
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