Andrey Sarantsev

For a random variable N = 0, 1, 2, ... we study the following question: When does the sum of N many independent and identically distributed copies of a random variable X have the same law as a nontrivial rescaling of X? We show that such N-stable random variable exists if and only 1 < E[N] < ∞. Under an additional assumption E[N ln N] < ∞, we describe all N-stable X. We also study a converse problem: For a given X ≥ 0 with E[X] = 1, find the set of all N such that X is N-stable. Distributions of these N form a commuting semigroup with respect to composition of probability generating functions.

Running Median Subtraction Filter (RMSF) is a robust statistical tool for removing slowly varying baselines in data streams containing transients (short-duration signals) of interest. In this work, we explore the RMSF performance and properties using simulated time series and analytical methods. We study the RMSF fidelity in preserving the signal of interest in the data using (i) a Gaussian pulse and (ii) a transient oscillatory signal. Such signals may be generated by hypothetical exotic low-mass fields (ELFs) associated with intense astrophysical events like binary black hole or neutron star mergers. We consider and assess RMSF as a candidate method to extract transient ELF signals. RMSF operates by sliding a window across the data and subtracting the median value within each window from the data points. With a suitable choice of running window size, RMSF effectively filters out baseline variations without compromising the integrity of transients. The RMSF window width is a critical parameter: it must be wide enough to encompass a short transient but narrow enough to remove the slowly varying baseline. We show that the RMSF removes the mean of a normally distributed white noise while preserving its variance and higher-order moments in the limit of large windows. In addition, RMSF does not color the white noise stream, that is, it does not induce any significant correlation in the filtered data. Ideally, a filter would preserve both the signal of interest and the statistical characteristics of the stochastic component of the data, while removing the background clutter and outliers. We find the RMSF to satisfy these practical criteria for data preprocessing. While we rigorously prove several RMSF properties, the paper is organized as a tutorial with multiple illustrations of RMSF applications.

We generalize the classic Shiller cyclically adjusted price-earnings ratio (CAPE) used for prediction of future total returns of the stock market. We treat earnings growth as exogenous. The difference between log wealth and log earnings is modeled as an autoregression of order 1 with linear trend 4.6 and Gaussian innovations. Detrending gives us a new valuation measure. Our results disprove the Efficient Market Hypothesis. Therefore, long-run total returns equal long-run earnings growth plus 4.6 Ẇe apply results to retirement planning. A withdrawal process governs how a retired capital owner withdraws a certain fraction of wealth annually. We study the long-term behavior of such processes.

We model time series of VIX (monthly average) and monthly stock index returns. We use log-Heston model: logarithm of VIX is modeled as an autoregression of order 1. Our main insight is that normalizing monthly stock index returns (dividing them by VIX) makes them much closer to independent identically distributed Gaussian. The resulting model is mean-reverting, and the innovations are non-Gaussian. The combined stochastic volatility model fits well, and captures Pareto-like tails of real-world stock market returns. This works for small and large stock indices, for both price and total returns.

The Skorokhod reflection was used in 1961 to create a reflected diffusion on the half-line. Later, it was used for processes with jumps such as reflected Lévy processes. Like a Brownian motion, which is a weak limit of random walks, reflected processes on the half-line serve as weak limits of random walks with switching regimes at zero: one regime away from zero, the other around zero. In this article, we develop a general theory of this regime change and prove convergence to a function with generalized reflection. Our results are deterministic and can be applied to a wide class of stochastic processes. Applications include storage processes, heavy traffic limits, diffusion on a half-line with a combination of continuous reflection, jump exit, and a delay at 0.

In this note, we consider the performance of the classic method of moments for parameter estimation of symmetric variance-gamma (generalized Laplace) distributions. We do this through both theoretical analysis (multivariate delta method) and a comprehensive simulation study with comparison to maximum likelihood estimation, finding performance is often unsatisfactory. In addition, we modify the method of moments by taking absolute moments to improve efficiency; in particular, our simulation studies demonstrate that our modified estimators have significantly improved performance for parameter values typically encountered in financial modelling, and is also competitive with maximum likelihood estimation.

The Skorokhod reflection was used in 1961 to create a reflected diffusion on the half-line. Later, it was used for processes with jumps such as reflected Lévy processes. Like a Brownian motion, which is a weak limit of random walks, reflected processes on the half-line serve as weak limits of random walks with switching regimes at zero: one regime away from zero, the other around zero. In this article, we develop a general theory of this regime change and prove convergence to a function with generalized reflection. Our results are deterministic and can be applied to a wide class of stochastic processes. Applications include storage processes, heavy traffic limits, diffusion on a half-line with a combination of continuous reflection, jump exit, and a delay at 0.

We generalize the classic Shiller cyclically adjusted price-earnings ratio (CAPE) used for prediction of future total returns of the stock market. We treat earnings growth as exogenous. The difference between log wealth and log earnings is modeled as an autoregression of order 1 with linear trend 4.5\% and Gaussian innovations. Detrending gives us a new valuation measure. This autoregression is significantly different from the random walk. Therefore, our results disprove the Efficient Market Hypothesis. Therefore, long-run total returns equal long-run earnings growth plus 4.5\%. We apply results to retirement planning. A withdrawal process governs how a retired capital owner withdraws a certain fraction of wealth annually. The fraction can vary from year to year. We study the long-term behavior of such processes.