1,374 research outputs found
Asymptotic Control for a Class of Piecewise Deterministic Markov Processes Associated to Temperate Viruses
We aim at characterizing the asymptotic behavior of value functions in the
control of piece-wise deterministic Markov processes (PDMP) of switch type
under nonexpansive assumptions. For a particular class of processes inspired by
temperate viruses, we show that uniform limits of discounted problems as the
discount decreases to zero and time-averaged problems as the time horizon
increases to infinity exist and coincide. The arguments allow the limit value
to depend on initial configuration of the system and do not require dissipative
properties on the dynamics. The approach strongly relies on viscosity
techniques, linear programming arguments and coupling via random measures
associated to PDMP. As an intermediate step in our approach, we present the
approximation of discounted value functions when using piecewise constant (in
time) open-loop policies.Comment: In this revised version, statements of the main results are gathered
in Section 3. Proofs of the main results (Theorem 4 and Theorem 7) make the
object of separate sections (Section 5, resp. Section 6). The biological
example makes the object of Section 4. Notations are gathered in Subsection
2.1. This is the final version to be published in SICO
Large Deviations for Small Noise Diffusions in a Fast Markovian Environment
A large deviation principle is established for a two-scale stochastic system
in which the slow component is a continuous process given by a small noise
finite dimensional It\^{o} stochastic differential equation, and the fast
component is a finite state pure jump process. Previous works have considered
settings where the coupling between the components is weak in a certain sense.
In the current work we study a fully coupled system in which the drift and
diffusion coefficient of the slow component and the jump intensity function and
jump distribution of the fast process depend on the states of both components.
In addition, the diffusion can be degenerate. Our proofs use certain stochastic
control representations for expectations of exponential functionals of finite
dimensional Brownian motions and Poisson random measures together with weak
convergence arguments. A key challenge is in the proof of the large deviation
lower bound where, due to the interplay between the degeneracy of the diffusion
and the full dependence of the coefficients on the two components, the
associated local rate function has poor regularity properties.Comment: 42 page
On the stability of stochastic jump kinetics
Motivated by the lack of a suitable constructive framework for analyzing
popular stochastic models of Systems Biology, we devise conditions for
existence and uniqueness of solutions to certain jump stochastic differential
equations (SDEs). Working from simple examples we find reasonable and explicit
assumptions on the driving coefficients for the SDE representation to make
sense. By `reasonable' we mean that stronger assumptions generally do not hold
for systems of practical interest. In particular, we argue against the
traditional use of global Lipschitz conditions and certain common growth
restrictions. By `explicit', finally, we like to highlight the fact that the
various constants occurring among our assumptions all can be determined once
the model is fixed.
We show how basic long time estimates and some limit results for
perturbations can be derived in this setting such that these can be contrasted
with the corresponding estimates from deterministic dynamics. The main
complication is that the natural path-wise representation is generated by a
counting measure with an intensity that depends nonlinearly on the state
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