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

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

Using Travel Simulation to Investigate Driver Response to In-Vehicle Route Guidance Systems,

A major application for developed satellite navigation systems is the in-vehicle route guidance market. As systems become cheaper to purchase and easier to install and indeed car manufacturers begin to fit the equipment as standard in new vehicles, the potential market for such systems in the developed world is massive. But what are the consequences of giving navigational assistance to car drivers? How will drivers respond to this information? Such information is liable to have a big impact upon driver route choice behaviour and is also subject to their interpretation of the guidance and action upon receiving it. This response may change under different travel circumstances. The impact of collective response to driver guidance is also of importance to traffic engineers and city planners, since routing through environmentally sensitive areas or heavily congested corridors should be avoided. The overall network effects are therefore of key importance to ensure efficient routing and minimal disruption to the road network. It is quite difficult to observe real-life behaviour on a consistent basis, since there are so many confounding variables in the real-world, traffic is never the same two days running, let alone hour by hour and a rigorous experimental environment is required, since control of experimental conditions is paramount to being able to confidently predict driver behaviour in response to navigational aids. Also the take up of guidance systems is still in its infancy, so far available only to a niche market of specialist professionals and those with disposable income. A need to test the common publics’ response to route guidance systems is therefore required. The development of travel simulation techniques, using portable computers and specialist software, gives robust experimental advantages. Although not totally realistic of the driving task, these techniques are sufficient in their realism of the decision element of route selection, enough to conduct experimental studies into drivers’ route choice behaviour under conditions of receiving simulated guidance advice. In this manner driver response to in-vehicle route guidance systems can be tested under a range of hypothetical journey making travel scenarios. This paper will outline the development of travel simulation techniques as a tool for in-vehicle route guidance research, including different methods and key simulation design requirements. The second half of the paper will report in detail on the findings from a recently conducted experiment investigating drivers’ response to route guidance when in familiar and unfamiliar road networks. The results will indicate the importance of providing meaningful information to drivers under these two real-life circumstances and report on how demands for route guidance information may vary by type of journey. Findings indicate that the guidance acceptance need not only depend on the optimum route choice criteria, it is also affected by network familiarity, quality and credibility of guidance advice and personal attributes of the drivers

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