Fluctuations of the empirical quantiles of independent Brownian motions

We consider $n$ independent, identically distributed one-dimensional Brownian motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth density function $f$. The empirical quantiles, or pointwise order statistics, are denoted by $B_{j:n}(t)$, and we are interested in a sequence of quantiles $Q_n(t) = B_{j(n):n}(t)$, where $j(n)/n \to \alpha \in (0,1)$. This sequence converges in probability in $C[0,\infty)$ to $q(t)$, the $\alpha$-quantile of the law of $B_j(t)$. Our main result establishes the convergence in law in $C[0,\infty)$ of the fluctuation processes $F_n = n^{1/2}(Q_n - q)$. The limit process $F$ is a centered Gaussian process and we derive an explicit formula for its covariance function. We also show that $F$ has many of the same local properties as $B^{1/4}$, the fractional Brownian motion with Hurst parameter $H = 1/4$. For example, it is a quartic variation process, it has H\"older continuous paths with any exponent $\gamma < 1/4$, and (at least locally) it has increments whose correlation is negative and of the same order of magnitude as those of $B^{1/4}$.Comment: 40 page

Crowding of Brownian spheres

We study two models consisting of reflecting one-dimensional Brownian "particles" of positive radius. We show that the stationary empirical distributions for the particle systems do not converge to the harmonic function for the generator of the individual particle process, unlike in the case when the particles are infinitely small.Comment: 13 page

The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6

Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of c\`adl\`ag functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary It\^o integral with respect to a Brownian motion that is independent of $B$.Comment: 45 page

Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion II

This is the published version, also available here: http://dx.doi.org/10.1214/ECP.v18-2840.The purpose of this paper is to provide a complete description the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter H=1/6

A change of variable formula with It\^{o} correction term

We consider the solution $u(x,t)$ to a stochastic heat equation. For fixed $x$, the process $F(t)=u(x,t)$ has a nontrivial quartic variation. It follows that $F$ is not a semimartingale, so a stochastic integral with respect to $F$ cannot be defined in the classical It\^{o} sense. We show that for sufficiently differentiable functions $g(x,t)$, a stochastic integral $\int g(F(t),t)\,dF(t)$ exists as a limit of discrete, midpoint-style Riemann sums, where the limit is taken in distribution in the Skorokhod space of cadlag functions. Moreover, we show that this integral satisfies a change of variable formula with a correction term that is an ordinary It\^{o} integral with respect to a Brownian motion that is independent of $F$.Comment: Published in at http://dx.doi.org/10.1214/09-AOP523 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Overview of the Main Propulsion System for the NASA Ares I Upper Stage

A functional overview of the Main Propulsion System (MPS) of the NASA Ares I Upper Stage is provided. In addition to a simple overview of the key MPS functions and design philosophies, major lessons learned are discussed. The intent is to provide a technical overview with enough detail to allow engineers outside of the MPS Integrated Product Team (IPT) to develop a rough understanding of MPS operations, components, design philosophy, and lessons learned

Prescription Drug Use under Medicare Part D: A Linear Model of Nonlinear Budget Sets

Medicare Part D enrollees face a complicated decision problem: they must dynamically choose prescription drug consumption in each period given difficult- to-find prices and a non-linear budget set. We use Medicare Part D claims data from 2006-2009 to estimate a flexible model of consumption that accounts for non-linear budget sets, dynamic incentives due to myopia and uncertainty, and price salience. By using variation away from kink points, we are able to estimate structural models with a linear regression of consumption on coverage range prices. We then compare performance under several candidate models of expectations and coverage phase weighting. The estimates suggest small marginal price elasticities and substantial myopia; we also find evidence that salient plan characteristics impact consumption beyond their effect on out-of-pocket prices. A hyperbolic discounting model which allows for salient plan characteristics fits the data well, and outperforms both rational models and alternative behavioral models