142 research outputs found

### Existence of optimal controls for singular control problems with state constraints

We establish the existence of an optimal control for a general class of
singular control problems with state constraints. The proof uses weak
convergence arguments and a time rescaling technique. The existence of optimal
controls for Brownian control problems \citehar, associated with a broad family
of stochastic networks, follows as a consequence.Comment: Published at http://dx.doi.org/10.1214/105051606000000556 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org

### Singular control with state constraints on unbounded domain

We study a class of stochastic control problems where a cost of the form
\begin{equation}\mathbb{E}\int_{[0,\infty)}e^{-\beta s}[\ell(X_s)
ds+h(Y^{\circ}_s) d|Y|_s]\end{equation} is to be minimized over control
processes $Y$ whose increments take values in a cone $\mathbb{Y}$ of
$\mathbb{R}^p$, keeping the state process $X=x+B+GY$ in a cone $\mathbb{X}$ of
$\mathbb{R}^k$, $k\le p$. Here, $x\in\mathbb{X}$, $B$ is a Brownian motion with
drift $b$ and covariance $\Sigma$, $G$ is a fixed matrix, and $Y^{\circ}$ is
the Radon--Nikodym derivative $dY/d|Y|$. Let $\mathcal{L}=-(1/2)trace(\Sigma
D^2)-b\cdot D$ where $D$ denotes the gradient. Solutions to the corresponding
dynamic programming PDE,
\begin{equation}[(\mathcal{L}+\beta)f-\ell]\vee\sup_{y\in\mathbb{Y}:|Gy|=1
}[-Gy\cdot Df-h(y)]=0,\end{equation} on $\mathbb{X}^o$ are considered with a
polynomial growth condition and are required to be supersolution up to the
boundary (corresponding to a ``state constraint'' boundary condition on
$\partial\mathbb{X}$). Under suitable conditions on the problem data, including
continuity and nonnegativity of $\ell$ and $h$, and polynomial growth of
$\ell$, our main result is the unique viscosity-sense solvability of the PDE by
the control problem's value function in appropriate classes of functions. In
some cases where uniqueness generally fails to hold in the class of functions
that grow at most polynomially (e.g., when $h=0$), our methods provide
uniqueness within the class of functions that, in addition, have compact level
sets. The results are new even in the following special cases: (1) The
one-dimensional case $k=p=1$, $\mathbb{X}=\mathbb{Y}=\mathbb{R}_+$; (2) The
first-order case $\Sigma=0$; (3) The case where $\ell$ and $h$ are linear. The
proofs combine probabilistic arguments and viscosity solution methods. Our
framework covers a wide range of diffusion control problems that arise from
queueing networks in heavy traffic.Comment: Published at http://dx.doi.org/10.1214/009117906000000359 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Some Fluctuation Results for Weakly Interacting Multi-type Particle System

A collection of $N$-diffusing interacting particles where each particle
belongs to one of $K$ different populations is considered. Evolution equation
for a particle from population $k$ depends on the $K$ empirical measures of
particle states corresponding to the various populations and the form of this
dependence may change from one population to another. In addition, the drift
coefficients in the particle evolution equations may depend on a factor that is
common to all particles and which is described through the solution of a
stochastic differential equation coupled, through the empirical measures, with
the $N$-particle dynamics. We are interested in the asymptotic behavior as
$N\to \infty$. Although the full system is not exchangeable, particles in the
same population have an exchangeable distribution. Using this structure, one
can prove using standard techniques a law of large numbers result and a
propagation of chaos property. In the current work we study fluctuations about
the law of large number limit. For the case where the common factor is absent
the limit is given in terms of a Gaussian field whereas in the presence of a
common factor it is characterized through a mixture of Gaussian distributions.
We also obtain, as a corollary, new fluctuation results for disjoint
sub-families of single type particle systems, i.e. when $K=1$. Finally, we
establish limit theorems for multi-type statistics of such weakly interacting
particles, given in terms of multiple Wiener integrals.Comment: 47 page

### Large deviations for multidimensional state-dependent shot noise processes

Shot noise processes are used in applied probability to model a variety of
physical systems in, for example, teletraffic theory, insurance and risk theory
and in the engineering sciences. In this work we prove a large deviation
principle for the sample-paths of a general class of multidimensional
state-dependent Poisson shot noise processes. The result covers previously
known large deviation results for one dimensional state-independent shot noise
processes with light tails. We use the weak convergence approach to large
deviations, which reduces the proof to establishing the appropriate convergence
of certain controlled versions of the original processes together with relevant
results on existence and uniqueness

### Near critical catalyst reactant branching processes with controlled immigration

Near critical catalyst-reactant branching processes with controlled
immigration are studied. The reactant population evolves according to a
branching process whose branching rate is proportional to the total mass of the
catalyst. The bulk catalyst evolution is that of a classical continuous time
branching process; in addition there is a specific form of immigration.
Immigration takes place exactly when the catalyst population falls below a
certain threshold, in which case the population is instantaneously replenished
to the threshold. Such models are motivated by problems in chemical kinetics
where one wants to keep the level of a catalyst above a certain threshold in
order to maintain a desired level of reaction activity. A diffusion limit
theorem for the scaled processes is presented, in which the catalyst limit is
described through a reflected diffusion, while the reactant limit is a
diffusion with coefficients that are functions of both the reactant and the
catalyst. Stochastic averaging principles under fast catalyst dynamics are
established. In the case where the catalyst evolves "much faster" than the
reactant, a scaling limit, in which the reactant is described through a one
dimensional SDE with coefficients depending on the invariant distribution of
the reflected diffusion, is obtained. Proofs rely on constrained martingale
problem characterizations, Lyapunov function constructions, moment estimates
that are uniform in time and the scaling parameter and occupation measure
techniques.Comment: Published in at http://dx.doi.org/10.1214/12-AAP894 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### A large deviations approach to asymptotically optimal control of crisscross network in heavy traffic

In this work we study the problem of asymptotically optimal control of a
well-known multi-class queuing network, referred to as the ``crisscross
network,'' in heavy traffic. We consider exponential inter-arrival and service
times, linear holding cost and an infinite horizon discounted cost criterion.
In a suitable parameter regime, this problem has been studied in detail by
Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996) 2133-2171] using
viscosity solution methods. In this work, using the pathwise solution of the
Brownian control problem, we present an elementary and transparent treatment of
the problem (with the identical parameter regime as in [SIAM J. Control Optim.
34 (1996) 2133-2171]) using large deviation ideas introduced in [Ann. Appl.
Probab. 10 (2000) 75-103, Ann. Appl. Probab. 11 (2001) 608-649]. We obtain an
asymptotically optimal scheduling policy which is of threshold type. The proof
is of independent interest since it is one of the few results which gives the
asymptotic optimality of a control policy for a network with a more than
one-dimensional workload process.Comment: Published at http://dx.doi.org/10.1214/105051605000000250 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org

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