2,716 research outputs found
Stochastic control problems for systems driven by normal martingales
In this paper we study a class of stochastic control problems in which the
control of the jump size is essential. Such a model is a generalized version
for various applied problems ranging from optimal reinsurance selections for
general insurance models to queueing theory. The main novel point of such a
control problem is that by changing the jump size of the system, one
essentially changes the type of the driving martingale. Such a feature does not
seem to have been investigated in any existing stochastic control literature.
We shall first provide a rigorous theoretical foundation for the control
problem by establishing an existence result for the multidimensional structure
equation on a Wiener--Poisson space, given an arbitrary bounded jump size
control process; and by providing an auxiliary counterexample showing the
nonuniqueness for such solutions. Based on these theoretical results, we then
formulate the control problem and prove the Bellman principle, and derive the
corresponding Hamilton--Jacobi--Bellman (HJB) equation, which in this case is a
mixed second-order partial differential/difference equation. Finally, we prove
a uniqueness result for the viscosity solution of such an HJB equation.Comment: Published in at http://dx.doi.org/10.1214/07-AAP467 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On a continuous time game with incomplete information
For zero-sum two-player continuous-time games with integral payoff and
incomplete information on one side, one shows that the optimal strategy of the
informed player can be computed through an auxiliary optimization problem over
some martingale measures. One also characterizes the optimal martingale
measures and compute it explicitely in several examples
H\"older regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations with super-quadratic growth in the gradient
Viscosity solutions of fully nonlinear, local or non local, Hamilton-Jacobi
equations with a super-quadratic growth in the gradient variable are proved to
be H\"older continuous, with a modulus depending only on the growth of the
Hamiltonian. The proof involves some representation formula for nonlocal
Hamilton-Jacobi equations in terms of controlled jump processes and a weak
reverse inequality
A probabilistic representation for the value of zero-sum differential games with incomplete information on both sides
We prove that for a class of zero-sum differential games with incomplete
information on both sides, the value admits a probabilistic representation as
the value of a zero-sum stochastic differential game with complete information,
where both players control a continuous martingale. A similar representation as
a control problem over discontinuous martingales was known for games with
incomplete information on one side (see Cardaliaguet-Rainer [8]), and our
result is a continuous-time analog of the so called splitting-game introduced
in Laraki [20] and Sorin [27] in order to analyze discrete-time models. It was
proved by Cardaliaguet [4, 5] that the value of the games we consider is the
unique solution of some Hamilton-Jacobi equation with convexity constraints.
Our result provides therefore a new probabilistic representation for solutions
of Hamilton-Jacobi equations with convexity constraints as values of stochastic
differential games with unbounded control spaces and unbounded volatility
Games with incomplete information in continuous time and for continuous types
We consider a two-player zero-sum game with integral payoff and with
incomplete information on one side, where the payoff is chosen among a
continuous set of possible payoffs. We prove that the value function of this
game is solution of an auxiliary optimization problem over a set of
measure-valued processes. Then we use this equivalent formulation to
characterize the value function as the viscosity solution of a special type of
a Hamilton-Jacobi equation. This paper generalizes the results of a previous
work of the authors, where only a finite number of possible payoffs is
considered
Differential games with asymmetric information and without Isaacs condition
We investigate a two-player zero-sum differential game with asymmetric
information on the payoff and without Isaacs condition. The dynamics is an
ordinary differential equation parametrised by two controls chosen by the
players. Each player has a private information on the payoff of the game, while
his opponent knows only the probability distribution on the information of the
other player. We show that a suitable definition of random strategies allows to
prove the existence of a value in mixed strategies. Moreover, the value
function can be characterised in term of the unique viscosity solution in some
dual sense of a Hamilton-Jacobi-Isaacs equation. Here we do not suppose the
Isaacs condition which is usually assumed in differential games
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