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Philip Ernst

Publications and source records attributed to Philip Ernst.

At least 19 recordsLinked to original sources
arXiv2025

Asymptotics of Yule's nonsense correlation for Ornstein-Uhlenbeck paths: The correlated case

We study the continuous-time version of the empirical correlation coefficient between the paths of two possibly correlated Ornstein-Uhlenbeck processes, known as Yule's nonsense correlation for these paths. Using sharp tools from the analysis on Wiener chaos, we establish the asymptotic normality of the fluctuations of this correlation coefficient around its long-time limit, which is the mathematical correlation coefficient between the two processes. This asymptotic normality is quantified in Kolmogorov distance, which allows us to establish speeds of convergence in the Type-II error for two simple tests of independence of the paths, based on the empirical correlation, and based on its numerator. An application to independence of two observations of solutions to the stochastic heat equation is given, with excellent asymptotic power properties using merely a small number of the solutions' Fourier modes.

math.PR↗
arXiv2023

The Minimax Wiener Sequential Testing Problem

Consider the sample path of a one-dimensional diffusion for which the diffusion coefficient is given and where the drift may take on one of two values: $μ_0$ or $μ_1$. Suppose that the signal-to-noise ratio (defined as the difference between the two possible drifts divided by the diffusion coefficient) is non-constant. Given an initial state for the observed process, we consider a minimax formulation of the Wiener sequential testing problem for detecting the correct drift coefficient as soon as possible and with minimal probabilities of incorrect terminal decisions. We solve the problem in the Bayesian formulation, under any prior probabilities of the process having drift $μ_0$ or $μ_1$, when the passage of time is penalized linearly. In the case where the signal-to-noise ratio is assumed constant, we obtain an explicit formula for the least favorable distribution.

math.OC↗
arXiv2022

Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary

Let $π$ be the occupancy density of an obliquely reflected Brownian motion in the half plane and let ($ρ$, $α$) be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of $π$($ρ$, $α$) as $ρ$ $\rightarrow$ $\infty$ with $α$ $\in$ (0, $π$). We find explicit functions a, b, c such that $π$($ρ$, $α$) $\sim$ $ρ$$\rightarrow$$\infty$ a($α$)$ρ$ b($α$) e --c($α$)$ρ$. This closes an open problem first stated by Professor J. Michael Harrison in August 2013. We also compute the exact asymptotics for the tail distribution of the boundary occupancy measure and we obtain an explicit integral expression for $π$. We conclude by finding the Martin boundary of the process and giving all of the corresponding harmonic functions satisfying an oblique Neumann boundary problem.

math.PR↗
arXiv2021

Exact optimal stopping for multidimensional linear switching diffusions

The paper studies a class of multidimensional optimal stopping problems with infinite horizon for linear switching diffusions. There are two main novelties in the optimal problems considered: the underlying stochastic process has discontinuous paths and the cost function is not necessarily integrable on the entire time horizon, where the latter is often a key assumption in classical optimal stopping theory for diffusions, cf. [22, Corollary 2.9]. Under relatively mild conditions, we show, for the class of multidimensional optimal stopping problems under consideration, that the first entry time of the stopping region is an optimal stopping time. Further, we prove that the corresponding optimal stopping boundaries can be represented as the unique solution to a nonlinear integral equation. We conclude with an application of our results to the problem of quickest real-time detection of a Markovian drift.

math.PR↗
arXiv2021

Escape and absorption probabilities for obliquely reflected Brownian motion in a quadrant

We consider an obliquely reflected Brownian motion $Z$ with positive drift in a quadrant stopped at time $T$, where $T:=\inf \{ t>0 : Z(t)=(0,0) \}$ is the first hitting time of the origin. Such a process can be defined even in the non-standard case where the reflection matrix is not completely-$\mathcal{S}$. We show that in this case the process has two possible behaviors: either it tends to infinity or it hits the corner (origin) in a finite time. Given an arbitrary starting point $(u,v)$ in the quadrant, we consider the escape (resp. absorption) probabilities $\mathbb{P}_{(u,v)}[T=\infty]$ (resp. $\mathbb{P}_{(u,v)}[T<\infty]$). We establish the partial differential equations and the oblique Neumann boundary conditions which characterize the escape probability and provide a functional equation satisfied by the Laplace transform of the escape probability. We then give asymptotics for the absorption probability in the simpler case where the starting point in the quadrant is $(u,0)$. We exhibit a remarkable geometric condition on the parameters which characterizes the case where the absorption probability has a product form and is exponential. We call this new criterion the dual skew symmetry condition due to its natural connection with the skew symmetry condition for the stationary distribution. We then obtain an explicit integral expression for the Laplace transform of the escape probability. We conclude by presenting exact asymptotics for the escape probability at the origin.

math.PR↗
arXiv2018

Optimal real-time detection of a drifting Brownian coordinate

Consider the motion of a Brownian particle in three dimensions, whose two spatial coordinates are standard Brownian motions with zero drift, and the remaining (unknown) spatial coordinate is a standard Brownian motion with a non-zero drift. Given that the position of the Brownian particle is being observed in real time, the problem is to detect as soon as possible and with minimal probabilities of the wrong terminal decisions, which spatial coordinate has the non-zero drift. We solve this problem in the Bayesian formulation, under any prior probabilities of the non-zero drift being in any of the three spatial coordinates, when the passage of time is penalised linearly. Finding the exact solution to the problem in three dimensions, including a rigorous treatment of its non-monotone optimal stopping boundaries, is the main contribution of the present paper. To our knowledge this is the first time that such a problem has been solved in the literature.

math.PR↗
arXiv2018

Stability and busy periods in a multiclass queue with state-dependent arrival rates

We introduce a multiclass single-server queueing system in which the arrival rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain the necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with the multitype Galton-Watson processes, the Laplace-Stieltjes transform of busy periods in the system is given. We conclude with tail asymptotics for the busy period for heavy-tailed service time distributions for the regularly varying case.

math.PR↗
arXiv2018

Sequential rerandomization

The seminal work of Morgan and Rubin (2012) considers rerandomization for all the units at one time. In practice, however, experimenters may have to rerandomize units sequentially. For example, a clinician studying a rare disease may be unable to wait to perform an experiment until all the experimental units are recruited. Our work offers a mathematical framework for sequential rerandomization designs, where the experimental units are enrolled in groups. We formulate an adaptive rerandomization procedure for balancing treatment/control assignments over some continuous or binary covariates, using Mahalanobis distance as the imbalance measure. We prove in our key result, Theorem 3, that given the same number of rerandomizations (in expected value), under certain mild assumptions, sequential rerandomization achieves better covariate balance than rerandomization at one time.

stat.AP↗
arXiv2018

When is it best to follow the leader?

An object is hidden in one of $N$ boxes. Initially, the probability that it is in box $i$ is $π_i(0)$. You then search in continuous time, observing box $J_t$ at time $t$, and receiving a signal as you observe: if the box you are observing does not contain the object, your signal is a Brownian motion, but if it does contain the object your signal is a Brownian motion with positive drift $μ$. It is straightforward to derive the evolution of the posterior distribution $π(t)$ for the location of the object. If $T$ denotes the first time that one of the $π_j(t)$ reaches a desired threshold $1-\varepsilon$, then the goal is to find a search policy $(J_t)_{t \geq 0}$ which minimizes the mean of $T$. This problem was studied by Posner and Rumsey (1966) and by Zigangirov (1966), who derive an expression for the mean time of a conjectured optimal policy, which we call {\em follow the leader} (FTL); at all times, observe the box with the highest posterior probability. Posner and Rumsey assert without proof that this is optimal, and Zigangirov offers a proof that if the prior distribution is uniform then FTL is optimal. In this paper, we show that if the prior is not uniform, then FTL is {\em not} always optimal; for uniform prior, the question remains open.

math.OC↗
arXiv2017

Portfolio Selection: The Power of Equal Weight

We empirically show the superiority of the equally weighted S\&P 500 portfolio over Sharpe's market capitalization weighted S\&P 500 portfolio. We proceed to consider the MaxMedian rule, a non-proprietary rule designed for the investor who wishes to do his/her own investing on a laptop with the purchase of only 20 stocks. Rather surprisingly, over the 1958-2016 horizon, the cumulative returns of MaxMedian beat those of the equally weighted S\&P 500 portfolio by a factor of 1.15.

q-fin.PM↗
arXiv2017

Tukey's transformational ladder for portfolio management

Over the past half-century, the empirical finance community has produced vast literature on the advantages of the equally weighted S\&P 500 portfolio as well as the often overlooked disadvantages of the market capitalization weighted Standard and Poor's (S\&P 500) portfolio (see \cite{Bloom}, \cite{Uppal}, \cite{Jacobs}, \cite{Treynor}). However, portfolio allocation based on Tukey's transformational ladde have, rather surprisingly, remained absent from the literature. In this work, we consider the S\&P 500 portfolio over the 1958-2015 time horizon weighted by Tukey's transformational ladder (\cite{Tukey2}): $1/x^2,\,\, 1/x,\,\, 1/\sqrt{x},\,\, \text{log}(x),\,\, \sqrt{x},\,\, x,\,\, \text{and} \,\, x^2$, where $x$ is defined as the market capitalization weighted S\&P 500 portfolio. Accounting for dividends and transaction fees, we find that the 1/$x^2$ weighting strategy produces cumulative returns that significantly dominates all other portfolios, achieving a compound annual growth rate of 18\% over the 1958-2015 horizon. Our story is furthered by a startling phenomenon: both the cumulative and annual returns of the $1/x^2$ weighting strategy are superior to those of the $1/x$ weighting strategy, which are in turn superior to those of the 1/$\sqrt{x}$ weighted portfolio, and so forth, ending with the $x^2$ transformation, whose cumulative returns are the lowest of the seven transformations of Tukey's transformational ladder. The order of cumulative returns precisely follows that of Tukey's transformational ladder. To the best of our knowledge, we are the first to discover this phenomenon.

q-fin.PM↗
arXiv2017

Asymptotics for the Time of Ruin in the War of Attrition

We consider two players, starting with $m$ and $n$ units, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probability $p(m,n)$ that the first player wins. When $m\sim Nx_{0}$, $n\sim N y_{0}$, we prove the fluid limit as $N\to \infty$. When $x_{0}=y_{0}$, then $z\to p(N,N+z\sqrt{N})$ converges to the standard normal CDF and the difference in fortunes scales diffusively. The exact limit of the time of ruin $τ_{N}$ is established as $(T-τ_N) \sim N^{-β}W^{\frac{1}β}$, $β=\frac{1}{4}$, $T=x_{0}+y_{0}$. Modulo a constant, $W \sim χ^{2}_{1}(z_{0}^{2}/T^{2})$.

math.PR↗
arXiv2016

Yule's "Nonsense Correlation" Solved!

In this paper, we resolve a longstanding open statistical problem. The problem is to mathematically confirm Yule's 1926 empirical finding of "nonsense correlation" (\cite{Yule}). We do so by analytically determining the second moment of the empirical correlation coefficient \beqn θ:= \frac{\int_0^1W_1(t)W_2(t) dt - \int_0^1W_1(t) dt \int_0^1 W_2(t) dt}{\sqrt{\int_0^1 W^2_1(t) dt - \parens{\int_0^1W_1(t) dt}^2} \sqrt{\int_0^1 W^2_2(t) dt - \parens{\int_0^1W_2(t) dt}^2}}, \eeqn of two {\em independent} Wiener processes, $W_1,W_2$. Using tools from Fred- holm integral equation theory, we successfully calculate the second moment of $θ$ to obtain a value for the standard deviation of $θ$ of nearly .5. The "nonsense" correlation, which we call "volatile" correlation, is volatile in the sense that its distribution is heavily dispersed and is frequently large in absolute value. It is induced because each Wiener process is "self-correlated" in time. This is because a Wiener process is an integral of pure noise and thus its values at different time points are correlated. In addition to providing an explicit formula for the second moment of $θ$, we offer implicit formulas for higher moments of $θ$.

math.ST↗
arXiv2016

On occupation times of the first and third quadrants for planar Brownian motion

An open problem of interest, first infused into the applied probability community in the work of Bingham and Doney in 1988, (see \cite{Bingham}) is stated as follows: find the distribution of the quadrant occupation time of planar Brownian motion. In this short communication, we study an alternate formulation of this longstanding open problem: let $X(t), Y(t), t \geq 0$ be standard Brownian motions starting at $x,y$ respectively. Find the distribution of the total time $T=Leb\{t \in [0,1]: X(t) \times Y(t) >0\}$, when $x=y=0$, i.e., the occupation time of the union of the first and third quadrants. If two adjacent quadrants are used, the problem becomes much easier and the distribution of $T$ follows the arcsine law.

math.PR↗
arXiv2016

The value of foresight

Suppose you have one unit of stock, currently worth 1, which you must sell before time $T$. The Optional Sampling Theorem tells us that whatever stopping time we choose to sell, the expected discounted value we get when we sell will be 1. Suppose however that we are able to see $a$ units of time into the future, and base our stopping rule on that; we should be able to do better than expected value 1. But how much better can we do? And how would we exploit the additional information? The optimal solution to this problem will never be found, but in this paper we establish remarkably close bounds on the value of the problem, and we derive a fairly simple exercise rule that manages to extract most of the value of foresight.

math.PR↗
arXiv2016

Exercising Control When Confronted by a (Brownian) Spider

We consider the Brownian "spider," a construct introduced in \cite{Dubins} and in \cite{Pitman}. In this note, the author proves the "spider" bounds by using the dynamic programming strategy of guessing the optimal reward function and subsequently establishing its optimality by proving its excessiveness.

math.PR↗
arXiv2016

On the Time for Brownian Motion to Visit Every Point on a Circle

Consider a Wiener process $W$ on a circle of circumference $L$. We prove the rather surprising result that the Laplace transform of the distribution of the first time, $θ_L$, when the Wiener process has visited every point of the circle can be solved in closed form using a continuous recurrence approach.

math.PR↗