19,433 research outputs found
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
New insights on stochastic reachability
In this paper, we give new characterizations of the stochastic reachability problem for stochastic hybrid systems in the language of different theories that can be employed in studying stochastic processes (Markov processes, potential theory, optimal control). These characterizations are further used to obtain the probabilities involved in the context of stochastic reachability as viscosity solutions of some variational inequalities
Coupling, local times, immersions
This paper answers a question of \'{E}mery [In S\'{e}minaire de
Probabilit\'{e}s XLII (2009) 383-396 Springer] by constructing an explicit
coupling of two copies of the Bene\v{s} et al. [In Applied Stochastic Analysis
(1991) 121-156 Gordon & Breach] diffusion (BKR diffusion), neither of which
starts at the origin, and whose natural filtrations agree. The paper commences
by surveying probabilistic coupling, introducing the formal definition of an
immersed coupling (the natural filtration of each component is immersed in a
common underlying filtration; such couplings have been described as co-adapted
or Markovian in older terminologies) and of an equi-filtration coupling (the
natural filtration of each component is immersed in the filtration of the
other; consequently the underlying filtration is simultaneously the natural
filtration for each of the two coupled processes). This survey is followed by a
detailed case-study of the simpler but potentially thematic problem of coupling
Brownian motion together with its local time at . This problem possesses its
own intrinsic interest as well as being closely related to the BKR coupling
construction. Attention focusses on a simple immersed (co-adapted) coupling,
namely the reflection/synchronized coupling. It is shown that this coupling is
optimal amongst all immersed couplings of Brownian motion together with its
local time at , in the sense of maximizing the coupling probability at all
possible times, at least when not started at pairs of initial points lying in a
certain singular set. However numerical evidence indicates that the coupling is
not a maximal coupling, and is a simple but non-trivial instance for which this
distinction occurs. It is shown how the reflection/synchronized coupling can be
converted into a successful equi-filtration coupling, by modifying the coupling
using a deterministic time-delay and then by concatenating an infinite sequence
of such modified couplings. The construction of an explicit equi-filtration
coupling of two copies of the BKR diffusion follows by a direct generalization,
although the proof of success for the BKR coupling requires somewhat more
analysis than in the local time case.Comment: Published at http://dx.doi.org/10.3150/14-BEJ596 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Comparing hitting time behaviour of Markov jump processes and their diffusion approximations
Markov jump processes can provide accurate models in many applications, notably chemical and biochemical kinetics, and population dynamics. Stochastic differential equations offer a computationally efficient way to approximate these processes. It is therefore of interest to establish results that shed light on the extent to which the jump and diffusion models agree. In this work we focus on mean hitting time behavior in a thermodynamic limit. We study three simple types of reactions where analytical results can be derived, and we find that the match between mean hitting time behavior of the two models is vastly different in each case. In particular, for a degradation reaction we find that the relative discrepancy decays extremely slowly, namely, as the inverse of the logarithm of the system size. After giving some further computational results, we conclude by pointing out that studying hitting times allows the Markov jump and stochastic differential equation regimes to be compared in a manner that avoids pitfalls that may invalidate other approaches
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