In this paper, the Dirac's quantum mechanical interaction picture is applied to option pricing to obtain a solution of the Black-Scholes equation in the presence of a time-dependent arbitrage bubble. In particular, for the case of a call perturbed by a square bubble, an approximate solution (valid up third order in a perturbation series) is given in terms of the three first Greeks: Delta, Gamma, and Speed. Then an exact solution is constructed in terms of all higher order $S$-derivatives of the Black-Scholes formula. It is also shown that the interacting Black-Scholes equation is invariant under a discrete transformation that interchanges the interest rate with the mean of the underlying asset and vice versa. This implies that the interacting Black-Scholes equation can be written in a 'low energy' and a 'high energy' form, in such a way that the high-interaction limit of the low energy form corresponds to the weak-interaction limit of the high energy form. One can apply a perturbative analysis to the high energy form to study the high-interaction limit of the low energy form.
In this work, we developed an efficient approach to compute ensemble averages in systems with pairwise-additive energetic interactions between the entities. Methods involving full enumeration of the configuration space result in exponential complexity. Sampling methods such as Markov Chain Monte Carlo (MCMC) algorithms have been proposed to tackle the exponential complexity of these problems; however, in certain scenarios where significant energetic coupling exists between the entities, the efficiency of the such algorithms can be diminished. We used a strategy to improve the efficiency of MCMC by taking advantage of the cluster structure in the interaction energy matrix to bias the sampling. We pursued two different schemes for the biased MCMC runs and show that they are valid MCMC schemes. We used both synthesized and real-world systems to show the improved performance of our biased MCMC methods when compared to the regular MCMC method. In particular, we applied these algorithms to the problem of estimating protonation ensemble averages and titration curves of residues in a protein.
Kitazawa (2013, 2016) showed that the common parameters in the panel logit AR(1) model with strictly exogenous covariates and fixed effects are estimable at the root-n rate using the Generalized Method of Moments. Honor\'e and Weidner (2020) extended his results in various directions: they found additional moment conditions for the logit AR(1) model and also considered estimation of logit AR(p) models with p>1. In this note we prove a conjecture in their paper and show that for given values of the initial condition, the covariates and the common parameters 2^{T}-2T of their moment functions for the logit AR(1) model are linearly independent and span the set of valid moment functions, which is a 2^{T}-2T-dimensional linear subspace of the 2^{T}-dimensional vector space of real valued functions over the outcomes y element of {0,1}^{T}. We also prove that when p=2 and T element of {3,4,5}, there are, respectively, 2^{T}-4(T-1) and 2^{T}-(3T-2) linearly independent moment functions for the panel logit AR(2) models with and without covariates.
The simulation of stochastic reaction-diffusion systems using fine-grained representations can become computationally prohibitive when particle numbers become large. If particle numbers are sufficiently high then it may be possible to ignore stochastic fluctuations and use a more efficient coarse-grained simulation approach. Nevertheless, for multiscale systems which exhibit significant spatial variation in concentration, a coarse-grained approach may not be appropriate throughout the simulation domain. Such scenarios suggest a hybrid paradigm in which a computationally cheap, coarse-grained model is coupled to a more expensive, but more detailed fine-grained model enabling the accurate simulation of the fine-scale dynamics at a reasonable computational cost. In this paper, in order to couple two representations of reaction-diffusion at distinct spatial scales, we allow them to overlap in a "blending region". Both modelling paradigms provide a valid representation of the particle density in this region. From one end of the blending region to the other, control of the implementation of diffusion is passed from one modelling paradigm to another through the use of complementary "blending functions" which scale up or down the contribution of each model to the overall diffusion. We establish the reliability of our novel hybrid paradigm by demonstrating its simulation on four exemplar reaction-diffusion scenarios.
This paper investigates the large sample properties of local regression distribution estimators, which include a class of boundary adaptive density estimators as a prime example. First, we establish a pointwise Gaussian large sample distributional approximation in a unified way, allowing for both boundary and interior evaluation points simultaneously. Using this result, we study the asymptotic efficiency of the estimators, and show that a carefully crafted minimum distance implementation based on "redundant" regressors can lead to efficiency gains. Second, we establish uniform linearizations and strong approximations for the estimators, and employ these results to construct valid confidence bands. Third, we develop extensions to weighted distributions with estimated weights and to local $L^{2}$ least squares estimation. Finally, we illustrate our methods with two applications in program evaluation: counterfactual density testing, and IV specification and heterogeneity density analysis. Companion software packages in Stata and R are available.
The use of above-100 GHz radio frequencies would be one of promising approaches to enhance the fifth-generation cellular further. Any air interface and cellular network designs require channel models, for which measured evidence of largescale parameters such as pathloss, delay and angular spreads, is crucial. This paper provides the evidence from quasi-static spatiotemporal channel sounding campaigns at two indoor hotspot (InH) scenarios at 140 GHz band, assuming short-range backhaul connectivity. The measured two InH sites are shopping mall and airport check-in hall. Our estimated omni-directional large-scale parameters from the measurements are found in good match with those of the Third Generation Partnership Project (3GPP) for new radios (NR) channel model in InH scenario, despite the difference of assumed link types and radio frequency range. The 3GPP NR channel model is meant for access links and said to be valid up to 100 GHz, while our measurements cover shortrange backhaul scenarios at 140 GHz. We found more deviation between our estimated large-scale parameters and those of the 3GPP NR channel model in the airport than in the shopping mall.
Coherent optical transmission systems naturally lead to a four dimensional (4D) signal space, i.e., two polarizations each with two quadratures. In this paper we derive an analytical model to quantify the impact of Kerr nonlinearity on such 4Dspaces, taking the interpolarization dependency into account. This is in contrast to previous models such as the GN and EGN models, which are valid for polarization multiplexed (PM)formats, where the two polarizations are seen as independent channels on which data is multiplexed. The proposed model agrees with the EGN model in the special case of independent two-dimensional modulation in each polarization. The model accounts for the predominant nonlinear terms in a WDM system, namely self-phase modulation and and cross-phase modulation. Numerical results show that the EGN model may inaccurately estimate the nonlinear interference of 4D formats. This nonlinear interference discrepancy between the results of the proposed model and the EGN model could be up to 2.8 dB for a system with 80 WDM channels. The derived model is validated by split-step Fourier simulations, and it is shown to follow simulations very closely.
We discovered a deficiency in Algorithm 1 and Theorem 3 of [1]. The algorithm called CEMA aims to solve an energy management problem distributively. However, by means of a counter example, we show that Theorem 2 and 3 of [1] contradict each other in the case of a valid scenario, proving that the suggested algorithm does not always find the optimum. Furthermore, we provide theoretic results, showing that Theorem 3 of [1] does not hold generally. At last, we provide a rectification by adjusting the algorithm and the corresponding proof of Theorem 3.
Motivated by applications to wireless communications, this paper addresses the propagation of waves transmitted by ambient noise sources and interacting with metamaterials. We discuss a generalized Helmholtz-Kirchhoff identity that is valid in dispersive media and we characterize the statistical properties of the empirical cross spectral density of the wave field. We can then introduce and analyze an original communication scheme between two passive arrays that uses only ambient noise illumination. The passive transmitter array does not transmit anything but it is a tunable metamaterial surface that can modulate its dispersive properties and encode a message in the modulation. The passive receiver array made of two receivers that are half-a-wavelength apart from each other can decode the message from the empirical cross spectral density of the wave field.
Multisite phosphorylation plays an important role in regulating switchlike protein activity and has been used widely in mathematical models. With the development of new experimental techniques and more molecular data, molecular phosphorylation processes emerge in many systems with increasing complexity and sizes. These developments call for simple yet valid stochastic models to describe various multisite phosphorylation processes, especially in large and complex biochemical networks. To reduce model complexity, this work aims to simplify the multisite phosphorylation mechanism by a stochastic Hill function model. Further, this work optimizes regions of parameter space to match simulation results from the stochastic Hill function with the distributive multisite phosphorylation process. While traditional parameter optimization methods have been focusing on finding the best parameter vector, in most circumstances modelers would like to find a set of parameter vectors that generate similar system dynamics and results. This paper proposes a general $\alpha$-$\beta$-$\gamma$ rule to return an acceptable parameter region of the stochastic Hill function based on a quasi-Newton stochastic optimization (QNSTOP) algorithm. Different objective functions are investigated characterizing different features of the simulation-based empirical data, among which the approximate maximum log-likelihood method is recommended for general applications. Numerical results demonstrate that with an appropriate parameter vector value, the stochastic Hill function model depicts the multisite phosphorylation process well except the initial (transient) period.
Large scale integration of distributed energy resources and electric vehicles in a transactive energy environment present new challenges in terms of voltage stability and fluctuations in a power distribution system. The impact of different level of DER/EV penetration on the voltages across the network is typically quantified through voltage sensitivity analyses. Existing methods of voltage sensitivity analysis are computationally expensive and prior efforts to develop analytical approximation lacks generality and have not been effectively validated. The objective of this work is to provide a new analytical method of voltage sensitivity analysis that has low computational cost and also allows for stochastic analysis of voltage change. This paper first derives an analytical approximation of change in voltage at a particular bus due to change in power consumption at other bus in a radial three phase unbalanced power distribution system. Then, the proposed method is shown to be valid for different load configurations, which demonstrates its generality. The results from our analytical approach is validated via classical load flow simulation of the test system based on IEEE 37 bus network. The proposed method is shown to have good accuracy, and computation complexity is of order O(1), compared to O(n3) in classical sensitivity analysis approaches.
Many species are unsustainable at small population densities (Allee Effect), i.e., below a threshold named Allee threshold, the population decreases instead of growing. In a closed local population, environmental fluctuations always lead to extinction. Here, we show how, in spatially extended habitats, dispersal can lead to a sustainable population in a region, provided the amplitude of environmental fluctuations is below an extinction threshold. We have identified two types of sustainable populations: high-density and low-density populations (through a mean-field approximation, valid in the limit of large dispersal length). Our results show that patches where population is high, low or extinct, coexist when the population is close to global extinction (even for homogeneous habitats). The extinction threshold is maximum for characteristic dispersal distances much larger than the spatial scale of synchrony of environmental fluctuations. The extinction threshold increases proportionally to the square root of the dispersal rate and decreases with the Allee threshold. The low-density population solution can allow understanding difficulties in recovery after harvesting. This theoretical framework provides a novel approach to address other factors, such as habitat fragmentation or harvesting, impacting population resilience to environmental fluctuations.
We introduce a new approach for comparing the predictive accuracy of two nested models that bypasses the difficulties caused by the degeneracy of the asymptotic variance of forecast error loss differentials used in the construction of commonly used predictive comparison statistics. Our approach continues to rely on the out of sample MSE loss differentials between the two competing models, leads to nuisance parameter free Gaussian asymptotics and is shown to remain valid under flexible assumptions that can accommodate heteroskedasticity and the presence of mixed predictors (e.g. stationary and local to unit root). A local power analysis also establishes its ability to detect departures from the null in both stationary and persistent settings. Simulations calibrated to common economic and financial applications indicate that our methods have strong power with good size control across commonly encountered sample sizes.
We propose an original method for vectorizing an image or zooming it at an arbitrary scale. The core of our method relies on the resolution of a geometric variational model and therefore offers theoretic guarantees. More precisely, it associates a total variation energy to every valid triangulation of the image pixels. Its minimization induces a trian-gulation that reflects image gradients. We then exploit this triangulation to precisely locate discontinuities, which can then simply be vectorized or zoomed. This new approach works on arbitrary images without any learning phase. It is particularly appealing for processing images with low quantization like pixel art and can be used for depixelizing such images. The method can be evaluated with an online demonstrator, where users can reproduce results presented here or upload their own images.
Whether aneuploid cells with diverse karyotypes have any properties in common has a been a subject of intense interest. A recent study by Terhorst et al. (1) reinvestigated the common aneuploidy gene expression (CAGE), disputing the conclusion of our recent work (2). In this short article, which has been submitted to PNAS as a Letter to the Editor, we explain our major concerns about Terhorst et al. and why we believe that our previous conclusion stands valid.
Applied macroeconomists often compute confidence intervals for impulse responses using local projections, i.e., direct linear regressions of future outcomes on current covariates. This paper proves that local projection inference robustly handles two issues that commonly arise in applications: highly persistent data and the estimation of impulse responses at long horizons. We consider local projections that control for lags of the variables in the regression. We show that lag-augmented local projections with normal critical values are asymptotically valid uniformly over (i) both stationary and non-stationary data, and also over (ii) a wide range of response horizons. Moreover, lag augmentation obviates the need to correct standard errors for serial correlation in the regression residuals. Hence, local projection inference is arguably both simpler than previously thought and more robust than standard autoregressive inference, whose validity is known to depend sensitively on the persistence of the data and on the length of the horizon.
In this paper we present an asset allocation strategy based on the maximization of the Sortino ratio. Unlike the Sharpe ratio, the Sortino ratio penalizes negative return variances only. The resulting allocation is valid for any time horizon unlike. The returns of a strategy based on such an allocation are empirically illustrated using historical Dow Jones data and display a significant upgrade on more traditional allocation strategies such as the Kelly criterion.
The power-spectrum subband energy ratio (PSER) has been applied in a variety of fields, but reports on its statistical properties have been limited. As such, this study investigates these characteristics in the presence of additive Gaussian white noise for both pure noise and mixed signals. By analyzing the probability and independence of power spectrum bins, and the relationship between the F and beta distributions, we develop a probability distribution for the PSER. Results showed that in the case of pure noise, the PSER follows a beta distribution. In addition, the probability density function and the quantile exhibited no relationship with the noise variance, only with the number of lines in the power spectrum, that is, PSER is not affected by noise. When Gaussian white noise was mixed with the known signal, the resulting PSER followed a doubly non-central beta distribution. In this case, it was difficult to identify the quantile, as the probability density and cumulative distribution functions were represented by infinite series. However, when a spectral bin did not contain the power spectrum of the known signal, an approximated quantile was found. This quantile is strictly proved to be in agreement with the quantile in the case of pure noise and offers a convenient methodology for identifying valid signals.
We consider a population evolving due to mutation, selection and recombination, where selection includes single-locus terms (additive fitness) and two-loci terms (pairwise epistatic fitness). We further consider the problem of inferring fitness in the evolutionary dynamics from one or several snap-shots of the distribution of genotypes in the population. In the recent literature this has been done by applying the Quasi-Linkage Equilibrium (QLE) regime first obtained by Kimura in the limit of high recombination. Here we show that the approach also works in the interesting regime where the effects of mutations are comparable to or larger than recombination. This leads to a modified main epistatic fitness inference formula where the rates of mutation and recombination occur together. We also derive this formula using by a previously developed Gaussian closure that formally remains valid when recombination is absent. The findings are validated through numerical simulations.
Instrumental variables (IV) regression is a popular method for the estimation of the endogenous treatment effects. Conventional IV methods require all the instruments are relevant and valid. However, this is impractical especially in high-dimensional models when we consider a large set of candidate IVs. In this paper, we propose an IV estimator robust to the existence of both the invalid and irrelevant instruments (called R2IVE) for the estimation of endogenous treatment effects. This paper extends the scope of Kang et al. (2016) by considering a true high-dimensional IV model and a nonparametric reduced form equation. It is shown that our procedure can select the relevant and valid instruments consistently and the proposed R2IVE is root-n consistent and asymptotically normal. Monte Carlo simulations demonstrate that the R2IVE performs favorably compared to the existing high-dimensional IV estimators (such as, NAIVE (Fan and Zhong, 2018) and sisVIVE (Kang et al., 2016)) when invalid instruments exist. In the empirical study, we revisit the classic question of trade and growth (Frankel and Romer, 1999).