Publications
34 results found
Bednarz E, Ernst PA, Osękowski A, 2024, On the diameter of the stopped spider process, Mathematics of Operations Research, Vol: 49, Pages: 346-365, ISSN: 0364-765X
We consider the Brownian “spider process,” also known as Walsh Brownian motion, first introduced by J. B. Walsh [Walsh JB (1978) A diffusion with a discontinuous local time. Asterisque 52:37–45]. The paper provides the best constant Cn for the inequalityEDτ≤CnEτ−−−√,where τ is the class of all adapted and integrable stopping times and D denotes the diameter of the spider process measured in terms of the British rail metric. This solves a variant of the long-standing open “spider problem” due to L. E. Dubins. The proof relies on the explicit identification of the value function for the associated optimal stopping problem.Funding: P. A. Ernst thanks the Royal Society Wolfson Fellowship (RSWF\R2\222005) and the U.S. Office of Naval Research (ONR N00014-21-1-2672) for their support of this research.
Ernst PA, Huang D, Viens FG, 2023, Yule’s “nonsense correlation” for Gaussian random walks, Stochastic Processes and their Applications, Vol: 162, Pages: 423-455, ISSN: 0304-4149
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.
Ernst P, Mei H, 2023, Exact optimal stopping for multidimensional linear switching diffusions, Mathematics of Operations Research, Vol: 48, Pages: 1589-1606, ISSN: 0364-765X
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. 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.
Ernst PA, Peskir G, 2022, Quickest real-time detection of a Brownian coordinate drift, The Annals of Applied Probability, Vol: 32, ISSN: 1050-5164
Ernst PA, Imerman MB, Shepp L, et al., 2022, Fiscal stimulus as an optimal control problem, Stochastic Processes and their Applications, Vol: 150, Pages: 1091-1108, ISSN: 0304-4149
During the Great Recession, Democrats in the United States argued that government spending could be utilized to “grease the wheels” of the economy in order to create wealth and to increase employment; Republicans, on the other hand, contended that government spending is wasteful and discourages investment, thereby increasing unemployment. This past year we have found ourselves in the midst of another crisis where government spending and fiscal stimulus is again being considered as a solution. In the present paper, we address this question by formulating an optimal control problem generalizing the model of Radner and Shepp (1996). The model allows for the company to borrow continuously from the government. We prove that there exists an optimal strategy; rigorous verification proofs for its optimality are provided. We proceed to prove that government loans increase the expected value of a company. We also examine the consequences of different profit-taking behaviors among firms who receive fiscal stimulus.
Thomas Bruss F, Ernst PA, Huang D, 2022, The rencontre problem, Stochastic Processes and their Applications, Vol: 150, Pages: 938-971, ISSN: 0304-4149
Let{X 1k}∞k=1 ,{X 2k}∞k=1 , . . . ,{X dk}∞k=1 be d independent sequences of Bernoulli random variableswith success-parameters p1, p2, . . . , pd respectively, where d ≥ 2 is a positive integer, and 0 < p j < 1for all j = 1, 2, . . . , d. LetS j (n) =n∑i=1X ji = X j1 + X j2 + · · · + X jn , n = 1, 2, . . . .We declare a “rencontre” at time n, or, equivalently, say that n is a “rencontre time,” ifS1(n) = S2(n) = · · · = Sd (n).We motivate and study the distribution of the first (provided it is finite) rencontre time
Ernst PA, Kagan AM, Rogers LCG, 2022, The least favorable noise, Electronic Communications in Probability, Vol: 27, Pages: 1-11, ISSN: 1083-589X
Suppose that a random variable X of interest is observed perturbed by independent additive noise Y. This paper concerns the “the least favorable perturbation” ˆYε, which maximizes the prediction error E(X−E(X|X+Y))2 in the class of Y with var(Y)≤ε. We find a characterization of the answer to this question, and show by example that it can be surprisingly complicated. However, in the special case where X is infinitely divisible, the solution is complete and simple. We also explore the conjecture that noisier Y makes prediction worse.
Gerber S, Markowitz HM, Ernst PA, et al., 2022, The Gerber Statistic: A Robust Co-Movement Measure for Portfolio Optimization, The Journal of Portfolio Management, Vol: 48, Pages: 87-102, ISSN: 0095-4918
Ernst PA, Franceschi S, 2021, Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary, The Annals of Applied Probability, Vol: 31, ISSN: 1050-5164
Ernst PA, Franceschi S, Huang D, 2021, Escape and absorption probabilities for obliquely reflected Brownian motion in a quadrant, Stochastic Processes and their Applications, Vol: 142, Pages: 634-670, ISSN: 0304-4149
Ernst PA, Rogers LCG, 2020, The Value of Insight, Mathematics of Operations Research, Vol: 45, Pages: 1193-1209, ISSN: 0364-765X
<jats:p> An investor may invest in a riskless bank account and in a stock that is a standard Black–Scholes asset with occasional Gaussian jumps of the log price, as proposed by Merton [Merton RC ( 1976 ) Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 3(1):125–144.]. It is well known how to solve the standard running consumption problem for this investor, which we take as a benchmark for comparing the performance of two different insiders, one who knows in advance of each jump exactly when the jump will happen, and the other who has information in advance of each jump about the size of the jump but no information about the time. These considerations give rise to two novel and concrete stochastic control problems. For each problem, rigorous verification proofs for optimality are presented. </jats:p>
Ernst PA, Peskir G, Zhou Q, 2020, Optimal real-time detection of a drifting Brownian coordinate, The Annals of Applied Probability, Vol: 30, ISSN: 1050-5164
Ernst PA, Rogers LCG, Zhou Q, 2020, When is it best to follow the leader?, Stochastic Processes and their Applications, Vol: 130, Pages: 3394-3407, ISSN: 0304-4149
Ernst P, Viens F, 2019, In memory of Larry Shepp: An editorial, High Frequency, Vol: 2, Pages: 74-75, ISSN: 2470-6981
Ernst PA, Soleymani F, 2019, A Legendre-based computational method for solving a class of Itô stochastic delay differential equations, Numerical Algorithms, Vol: 80, Pages: 1267-1282, ISSN: 1017-1398
Ernst PA, Kendall WS, Roberts GO, et al., 2019, MEXIT: Maximal un-coupling times for stochastic processes, Stochastic Processes and their Applications, Vol: 129, Pages: 355-380, ISSN: 0304-4149
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.
Ernst PA, Shaman P, 2019, The bias mapping of the Yule-Walker estimator is a contraction, Statistica Sinica, ISSN: 1017-0405
Ernst PA, Asmussen S, Hasenbein JJ, 2018, Stability and busy periods in a multiclass queue with state-dependent arrival rates, Queueing Systems, Vol: 90, Pages: 207-224, ISSN: 0257-0130
Ernst PA, Kimmel M, Kurpas M, et al., 2018, Heavy-tailed distributions in branching process models of secondary cancerous tumors, Advances in Applied Probability, Vol: 50, Pages: 99-114, ISSN: 0001-8678
Recent progress in microdissection and in DNA sequencing has facilitated the subsampling of multi-focal cancers in organs such as the liver in several hundred spots, helping to determine the pattern of mutations in each of these spots. This has led to the construction of genealogies of the primary, secondary, tertiary, and so forth, foci of the tumor. These studies have led to diverse conclusions concerning the Darwinian (selective) or neutral evolution in cancer. Mathematical models of the development of multi-focal tumors have been devised to support these claims. We offer a model for the development of a multi-focal tumor: it is a mathematically rigorous refinement of a model of Ling et al. (2015). Guided by numerical studies and simulations, we show that the rigorous model, in the form of an infinite-type branching process, displays distributions of tumor size which have heavy tails and moments that become infinite in finite time. To demonstrate these points, we obtain bounds on the tails of the distributions of the process and an infinite series expansion for the first moments. In addition to its inherent mathematical interest, the model is corroborated by recent literature on apparent super-exponential growth in cancer metastases.
Zhou Q, Ernst PA, Morgan KL, et al., 2018, Sequential rerandomization, Biometrika, Vol: 105, Pages: 745-752, ISSN: 0006-3444
Ernst PA, Rogers LCG, Zhou Q, 2017, The value of foresight, Stochastic Processes and their Applications, Vol: 127, Pages: 3913-3927, ISSN: 0304-4149
Suppose you have one unit of stock, currently worth 1, which you must sell before time T . The OptionalSampling Theorem tells us that whatever stopping time we choose to sell, the expected discounted value weget when we sell will be 1. Suppose however that we are able to see a units of time into the future, and baseour stopping rule on that; we should be able to do better than expected value 1. But how much better canwe do? And how would we exploit the additional information? The optimal solution to this problem willnever be found, but in this paper we establish remarkably close bounds on the value of the problem, and wederive a fairly simple exercise rule that manages to extract most of the value of foresight
Ernst PA, 2017, Minimizing Fisher information with absolute moment constraints, Statistics & Probability Letters, Vol: 129, Pages: 167-170, ISSN: 0167-7152
Ernst PA, Shepp LA, Wyner AJ, 2017, Yule’s “nonsense correlation” solved!, The Annals of Statistics, Vol: 45, Pages: 1789-1809, ISSN: 0090-5364
In this paper, we resolve a longstanding open statistical problem. The problem is to mathematically prove Yule’s 1926 empirical finding of “nonsense correlation” [J. Roy. Statist. Soc. 89 (1926) 1–63], which we do by analytically determining the second moment of the empirical correlation coefficientθ:=∫10W1(t)W2(t)dt−∫10W1(t)dt∫10W2(t)dt√∫10W21(t)dt−(∫10W1(t)dt)2√∫10W22(t)dt−(∫10W2(t)dt)2,of two independent Wiener processes, W1,W2. Using tools from Fredholm integral equation theory, we successfully calculate the second moment of θ to obtain a value for the standard deviation of θ of nearly 0.5. 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.
Ernst PA, Thompson JR, Miao Y, 2017, Tukey’s transformational ladder for portfolio management, Financial Markets and Portfolio Management, Vol: 31, Pages: 317-355, ISSN: 1934-4554
Ernst PA, Grigorescu I, 2017, Asymptotics for the time of ruin in the war of attrition, Advances in Applied Probability, Vol: 49, Pages: 388-410, ISSN: 0001-8678
<jats:title>Abstract</jats:title><jats:p>We consider two players, starting with<jats:italic>m</jats:italic>and<jats:italic>n</jats:italic>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<jats:italic>p</jats:italic>(<jats:italic>m</jats:italic>,<jats:italic>n</jats:italic>) that the first player wins. When<jats:italic>m</jats:italic>~<jats:italic>Nx</jats:italic><jats:sub>0</jats:sub>,<jats:italic>n</jats:italic>~<jats:italic>Ny</jats:italic><jats:sub>0</jats:sub>, we prove the fluid limit as<jats:italic>N</jats:italic>→ ∞. When<jats:italic>x</jats:italic><jats:sub>0</jats:sub>=<jats:italic>y</jats:italic><jats:sub>0</jats:sub>,<jats:italic>z</jats:italic>→<jats:italic>p</jats:italic>(<jats:italic>N</jats:italic>,<jats:italic>N</jats:italic>+<jats:italic>z</jats:italic>√<jats:italic>N</jats:italic>) converges to the standard normal cumulative distribution function and the difference in fortunes scales diffusively. The exact limit of the time of ruin τ<jats:sub><jats:italic>N</jats:italic></jats:sub>is established as (<jats:italic>T</jats:italic>- τ<jats:sub><jats:italic>N</jats:italic></jats:sub>) ~<jats:italic>N</jats:italic><jats:sup>-β</jats:sup><jats:italic>W</jats:italic><jats:sup>1/β</jats:sup>, β = ¼,<jats:italic>T</jats:italic>=<jats:italic>x</jats:italic><jats:sub>0</jats:sub>+<jats:italic>y</jats:italic><jats:sub>0</jats:su
Ernst PA, Shepp L, 2017, On occupation times of the first and third quadrants for planar Brownian motion, Journal of Applied Probability, Vol: 54, Pages: 337-342, ISSN: 0021-9002
<jats:title>Abstract</jats:title><jats:p>In Bingham and Doney (1988) the authors presented the applied probability community with a question which is very simply stated, yet is extremely difficult to solve: what is the distribution of the quadrant occupation time of planar Brownian motion? In this paper we study an alternate formulation of this long-standing open problem: let <jats:italic>X</jats:italic>(<jats:italic>t</jats:italic>), <jats:italic>Y</jats:italic>(<jats:italic>t</jats:italic>) <jats:italic>t</jats:italic>≥0, be standard Brownian motions starting at <jats:italic>x</jats:italic>, <jats:italic>y</jats:italic>, respectively. Find the distribution of the total time <jats:italic>T</jats:italic>=Leb{<jats:italic>t</jats:italic>∈[0,1]: <jats:italic>X</jats:italic>(<jats:italic>t</jats:italic>)×<jats:italic>Y</jats:italic>(<jats:italic>t</jats:italic>)>0}, when <jats:italic>x</jats:italic>=<jats:italic>y</jats:italic>=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 <jats:italic>T</jats:italic> follows the arcsine law.</jats:p>
Ernst PA, Brown LD, Shepp L, et al., 2017, Stationary Gaussian Markov processes as limits of stationary autoregressive time series, Journal of Multivariate Analysis, Vol: 155, Pages: 180-186, ISSN: 0047-259X
Ernst P, 2017, On the arbitrage price of European call options, Stochastic Models, Vol: 33, Pages: 48-58, ISSN: 1532-6349
Ernst P, Pemantle R, Satopää V, et al., 2016, Bayesian aggregation of two forecasts in the partial information framework, Statistics & Probability Letters, Vol: 119, Pages: 170-180, ISSN: 0167-7152
Ernst P, Shepp L, 2016, Eliminating a loophole in the National Flood Insurance Program, Law, Probability and Risk, Vol: 15, Pages: 251-258, ISSN: 1470-8396
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