Browsing by Author "Ernst, Philip A."

Now showing 1 - 6 of 6
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    MEXIT: Maximal un-coupling times for stochastic processes
    (Elsevier, 2019) Ernst, Philip A.; Kendall, Wilfrid S.; Roberts, Gareth O.; Rosenthal, Jeffrey S.
    Classical coupling constructions arrange for copies of the same Markov process started at two different initial states to become equal as soon as possible. In this paper, we consider an alternative coupling framework in which one seeks to arrange for two different Markov (or other stochastic) processes to remain equal for as long as possible, when started in the same state. We refer to this “un-coupling” or “maximal agreement” construction as MEXIT, standing for “maximal exit”. After highlighting the importance of un-coupling arguments in a few key statistical and probabilistic settings, we develop an explicit MEXIT construction for stochastic processes in discrete time with countable state-space. This construction is generalized to random processes on general state-space running in continuous time, and then exemplified by discussion of MEXIT for Brownian motions with two different constant drifts.
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    On longest consecutive patterns in Markov chains
    (2019-11-11) Xia, Yizhou; Ernst, Philip A.
    The length of longest consecutive head in Bernoulli trials L(n) has been studied extensively and has been found applications in biology, finance and non-parametric statistics. The study of longest consecutive successes in random trials dates the work of de Moivre. Limiting theorems and large deviation results are provided for L(n) with the assumption of existence of stationary distribution. Given a discrete-time homogeneous Markov chain with initial state i, one extension from previous Bernoulli case is to study the distribution of L(j,n), the length of the longest consecutive visits of this chain to state j until time n. Our work focuses on studying L(j,n) for both homogeneous and time-nonhomogeneous Markov chains. In the existing literature, no limiting theorems of L(j,n) are derived under the case of time nonhomogeneous Markov chains. We are able to solve this by first deriving a new exact formula of the distribution of L(j,n) and then derive an upper and lower bound of P(L(j,n)
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    The value of foresight
    (Elsevier, 2017) Ernst, Philip A.; Rogers, L.C.G.; Zhou, Quan
    Suppose you have one unit of stock, currently worth 1, which you must sell before time . 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 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.
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    Tukey's transformational ladder for portfolio management
    (Springer, 2017) Ernst, Philip A.; Thompson, James R.; Miao, Yinsen
    Over the past half-century, the empirical finance community has produced vast literature on the advantages of the equally weighted Standard and Poor (S&P 500) portfolio as well as the often overlooked disadvantages of the market capitalization weighted S&P 500’s portfolio (see Bloomfield et al. in J Financ Econ 5:201–218, 1977; DeMiguel et al. in Rev Financ Stud 22(5):1915–1953, 2009; Jacobs et al. in J Financ Mark 19:62–85, 2014; Treynor in Financ Anal J 61(5):65–69, 2005). However, portfolio allocation based on Tukey’s transformational ladder has, 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 (Tukey in Exploratory data analysis, Addison-Wesley, Boston, 1977): 1/x2,1/x,1/x−−√,log(x),x−−√,x,andx2, where x is defined as the market capitalization weighted S&P 500 portfolio. Accounting for dividends and transaction fees, we find that the 1/x2 weighting strategy produces cumulative returns that significantly dominate all other portfolio returns, 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/x2 weighting strategy are superior to those of the 1 / x weighting strategy, which are in turn superior to those of the 1/x−−√ weighted portfolio, and so forth, ending with the x2 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.
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    Two Random Walk Problems
    (2022-04-22) Huang, Dongzhou; Ernst, Philip A.
    Among numerous probabilistic objects, random walk is one of the most fundamental but most favourable. This dissertation concerns two problems related to random walk theory. The first problem regards $d$ independent Bernoulli random walks. We investigate the first “rencontre-time” (i.e. the first time all of the $d$ Bernoulli random walks arrive in the same state) and derive its distribution. Further, relying on the probability generating function, we discuss the probability of the first “rencontre-time” equaling infinity, whose positivity depends on the dimension $d$ and the success-parameters of these $d$ Bernoulli random walks. We then investigate the conditional expectations of the first “rencontre-time” by studying their bounds. In the second problem, we investigate Yule's “nonsense correlation” for two independent Gaussian random walks. The original problem, calculating the second moment of Yule's “nonsense correlation” for two independent Bernoulli random walks, has proved elusive. Relevant work in this topic includes two papers by Ernst et al., with the former first calculating explicitly the second moment of its asymptotic distribution and the latter providing the first approximation to the density of the asymptotic distribution by exploiting its moments up to order 16. We replace the Bernoulli random walks with Gaussian random walks. Relying on the property that the distribution of Gaussian random vector is invariant under orthonormal transformation, we successfully derive the distribution of Yule's “nonsense correlation” of Gaussian random walks. We also provide rates of convergence of the empirical correlation of two independent Gaussian random walks to the empirical correlation of two independent Wiener processes. At the level of distributions, in Wasserstein distance, the convergence rate is the inverse $n^{-1}$ of the number of the data points $n$.
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    Yule’s “nonsense correlation” for Gaussian random walks
    (Elsevier, 2023) Ernst, Philip A.; Huang, Dongzhou; Viens, Frederi G.
    This paper provides an exact formula for the second moment of the empirical correlation (also known as Yule’s “nonsense correlation”) for two independent standard Gaussian random walks, as well as implicit formulas for higher moments. We also establish rates of convergence of the empirical correlation of two independent standard Gaussian random walks to the empirical correlation of two independent Wiener processes.