233 research outputs found
Average Continuous Control of Piecewise Deterministic Markov Processes
This paper deals with the long run average continuous control problem of
piecewise deterministic Markov processes (PDMP's) taking values in a general
Borel space and with compact action space depending on the state variable. The
control variable acts on the jump rate and transition measure of the PDMP, and
the running and boundary costs are assumed to be positive but not necessarily
bounded. Our first main result is to obtain an optimality equation for the long
run average cost in terms of a discrete-time optimality equation related to the
embedded Markov chain given by the post-jump location of the PDMP. Our second
main result guarantees the existence of a feedback measurable selector for the
discrete-time optimality equation by establishing a connection between this
equation and an integro-differential equation. Our final main result is to
obtain some sufficient conditions for the existence of a solution for a
discrete-time optimality inequality and an ordinary optimal feedback control
for the long run average cost using the so-called vanishing discount approach.Comment: 34 page
Consistent Price Systems under Model Uncertainty
We develop a version of the fundamental theorem of asset pricing for
discrete-time markets with proportional transaction costs and model
uncertainty. A robust notion of no-arbitrage of the second kind is defined and
shown to be equivalent to the existence of a collection of strictly consistent
price systems.Comment: 19 page
Maximizing the probability of attaining a target prior to extinction
We present a dynamic programming-based solution to the problem of maximizing
the probability of attaining a target set before hitting a cemetery set for a
discrete-time Markov control process. Under mild hypotheses we establish that
there exists a deterministic stationary policy that achieves the maximum value
of this probability. We demonstrate how the maximization of this probability
can be computed through the maximization of an expected total reward until the
first hitting time to either the target or the cemetery set. Martingale
characterizations of thrifty, equalizing, and optimal policies in the context
of our problem are also established.Comment: 22 pages, 1 figure. Revise
Newton and Bouligand derivatives of the scalar play and stop operator
We prove that the play and the stop operator possess Newton and Bouligand
derivatives, and exhibit formulas for those derivatives. The remainder estimate
is given in a strenghtened form, and a corresponding chain rule is developed.
The construction of the Newton derivative ensures that the mappings involved
are measurable
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