Higher-order semi-implicit Taylor schemes for Itô stochastic differential equations

AbstractThe paper considers the derivation of families of semi-implicit schemes of weak order N=3.0 (general case) and N=4.0 (additive noise case) for the numerical solution of Itô stochastic differential equations. The degree of implicitness of the schemes depends on the selection of N parameters which vary between 0 and 1 and the families contain as particular cases the 3.0 and 4.0 weak order explicit Taylor schemes. Since the implementation of the multiple integrals that appear in these theoretical schemes is difficult, for the applications they are replaced by simpler random variables, obtaining simplified schemes. In this way, for the multidimensional case with one-dimensional noise, we present an infinite family of semi-implicit simplified schemes of weak order 3.0 and for the multidimensional case with additive one-dimensional noise, we give an infinite family of semi-implicit simplified schemes of weak order 4.0. The mean-square stability of the 3.0 family is analyzed, concluding that, as in the deterministic case, the stability behavior improves when the degree of implicitness grows. Numerical experiments confirming the theoretical results are shown

Cleaning large correlation matrices: tools from random matrix theory

This review covers recent results concerning the estimation of large covariance matrices using tools from Random Matrix Theory (RMT). We introduce several RMT methods and analytical techniques, such as the Replica formalism and Free Probability, with an emphasis on the Marchenko-Pastur equation that provides information on the resolvent of multiplicatively corrupted noisy matrices. Special care is devoted to the statistics of the eigenvectors of the empirical correlation matrix, which turn out to be crucial for many applications. We show in particular how these results can be used to build consistent "Rotationally Invariant" estimators (RIE) for large correlation matrices when there is no prior on the structure of the underlying process. The last part of this review is dedicated to some real-world applications within financial markets as a case in point. We establish empirically the efficacy of the RIE framework, which is found to be superior in this case to all previously proposed methods. The case of additively (rather than multiplicatively) corrupted noisy matrices is also dealt with in a special Appendix. Several open problems and interesting technical developments are discussed throughout the paper.Comment: 165 pages, article submitted to Physics Report

BER and outage probability of DPSK subcarrier intensity modulated free space optics in fully developed speckle.

In this paper a differential phase shift keying (DPSK) subcarrier intensity modulated (SIM) free space optical (FSO) link is considered in negative exponential atmospheric turbulence environment. To mitigate the scintillation effect, the selection combining spatial diversity scheme (SelC) is employed at the receiver. Bit error rate (BER) and outage probability (Pout) analysis are presented with and without the SelC spatial diversity. It is shown that at a BER of 10-6, a maximum diversity gain 25 dB is predicted. And about 60 dBm signal power is required to achieve an outage probability of 10-6, based on a threshold BER of 10-4

Shuttle Ku-band signal design study

Carrier synchronization and data demodulation of Unbalanced Quadriphase Shift Keyed (UQPSK) Shuttle communications' signals by optimum and suboptimum methods are discussed. The problem of analyzing carrier reconstruction techniques for unbalanced QPSK signal formats is addressed. An evaluation of the demodulation approach of the Ku-Band Shuttle return link for UQPSK when the I-Q channel power ratio is large is carried out. The effects that Shuttle rocket motor plumes have on the RF communications are determined also. The effect of data asymmetry on bit error probability is discussed

Differential Signalling in Free-Space Optical Communication Systems

In this paper, we review the differential signalling technique and investigate its implementation of in free-space optical (FSO) communication systems. The paper is an extended version of our previous works, where the effects of background noise, weak turbulence and pointing errors (PEs) were investigated separately. Here, for the first time, we present a thorough description of the differential signalling scheme including for combined effects. At first, we present an extension of the analysis of differential signalling to the case of moderate to strong atmospheric turbulence. Next, we investigate a more general case where both channel turbulence and PEs are taken into consideration. We provide closed-form expressions for the optimal detection threshold and the average bit-error-rate, and present a set of numerical results to illustrate the performance improvement offered by the proposed differential signalling under various turbulence and PEs conditions