An optimal polynomial approximation of Brownian motion

In this paper, we will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. The coefficients that are obtained from the expansion of Brownian motion in this polynomial basis are independent Gaussian random variables. Therefore it is practical (requires $N$ independent Gaussian coefficients) to generate an approximate sample path of Brownian motion that respects integration of polynomials with degree less than $N$. Moreover, since these orthogonal polynomials appear naturally as eigenfunctions of an integral operator defined by the Brownian bridge covariance function, the proposed approximation is optimal in a certain weighted $L^{2}(\mathbb{P})$ sense. In addition, discretizing Brownian paths as piecewise parabolas gives a locally higher order numerical method for stochastic differential equations (SDEs) when compared to the standard piecewise linear approach. We shall demonstrate these ideas by simulating Inhomogeneous Geometric Brownian Motion (IGBM). This numerical example will also illustrate the deficiencies of the piecewise parabola approximation when compared to a new version of the asymptotically efficient log-ODE (or Castell-Gaines) method.Comment: 27 pages, 8 figure

Are Auditors\u27 Going-Concern Evaluations More Useful after SOX?

Bankruptcy risk is a crucial factor in auditors’ decisions whether or not to modify their audit opinion based on the going-concern assumption. SOX required more extensive audit procedures than those required before its passage. More extensive audit procedures should result in more meaningful audit reports. This study examines whether the auditors’ going-concern opinion provides more useful incremental information after SOX than before SOX in distinguishing between distressed companies that become bankrupt in the next year and those that do not. We find that an audit opinion variable adds more useful information to bankruptcy prediction models after SOX than before SOX. Our findings provide evidence that financial statement users have derived benefits from the costly procedures required under SOX

An asymptotic radius of convergence for the Loewner equation and simulation of SLEkSLE_k traces via splitting

In this paper, we shall study the convergence of Taylor approximations for the backward Loewner differential equation (driven by Brownian motion) near the origin. More concretely, whenever the initial condition of the backward Loewner equation (which lies in the upper half plane) is small and has the form $Z_{0} = \varepsilon i$, we show these approximations exhibit an $O(\varepsilon)$ error provided the time horizon is $\varepsilon^{2+\delta}$ for $\delta > 0$. Statements of this theorem will be given using both rough path and $L^{2}(\mathbb{P})$ estimates. Furthermore, over the time horizon of $\varepsilon^{2-\delta}$, we shall see that "higher degree" terms within the Taylor expansion become larger than "lower degree" terms for small $\varepsilon$. In this sense, the time horizon on which approximations are accurate scales like $\varepsilon^{2}$. This scaling comes naturally from the Loewner equation when growing vector field derivatives are balanced against decaying iterated integrals of the Brownian motion. As well as being of theoretical interest, this scaling may be used as a guiding principle for developing adaptive step size strategies which perform efficiently near the origin. In addition, this result highlights the limitations of using stochastic Taylor methods (such as the Euler-Maruyama and Milstein methods) for approximating $SLE_{\kappa}$ traces. Due to the analytically tractable vector fields of the Loewner equation, we will show Ninomiya-Victoir (or Strang) splitting is particularly well suited for SLE simulation. As the singularity at the origin can lead to large numerical errors, we shall employ the adaptive step size proposed by Tom Kennedy to discretize $SLE_{\kappa}$ traces using this splitting. We believe that the Ninomiya-Victoir scheme is the first high order numerical method that has been successfully applied to $SLE_{\kappa}$ traces.Comment: 24 pages, 2 figure

The Incremental Usefulness Of Income Tax Allocations In Predicting One-Year-Ahead Future Cash Flows

Interperiod income tax allocation has been a hotly debated financial accounting issue for a long time.  Critics of interperiod tax allocation frequently question the usefulness of the extra information, particularly considering the FASB’s decision usefulness approach stated in its Conceptual Framework.  This study extends the research of Cheung et al. (1997) and Krishnan and Largay (2000) by using the ability to predict future taxes paid and future cash flow as criteria to evaluate the usefulness of interperiod tax allocation. This study extends previous research by examining not only whether interperiod tax allocation included in financial statements is useful, but also by examining whether such information is incrementally useful beyond taxes paid. For predicting future taxes paid and operating cash flow, our analyses provides little evidence that interperiod tax allocation information included in financial statements adds incremental predictive value beyond taxes paid as reported on the cash flow statement

The shifted ODE method for underdamped Langevin MCMC

In this paper, we consider the underdamped Langevin diffusion (ULD) and propose a numerical approximation using its associated ordinary differential equation (ODE). When used as a Markov Chain Monte Carlo (MCMC) algorithm, we show that the ODE approximation achieves a $2$-Wasserstein error of $\varepsilon$ in $\mathcal{O}\big(d^{\frac{1}{3}}/\varepsilon^{\frac{2}{3}}\big)$ steps under the standard smoothness and strong convexity assumptions on the target distribution. This matches the complexity of the randomized midpoint method proposed by Shen and Lee [NeurIPS 2019] which was shown to be order optimal by Cao, Lu and Wang. However, the main feature of the proposed numerical method is that it can utilize additional smoothness of the target log-density $f$. More concretely, we show that the ODE approximation achieves a $2$-Wasserstein error of $\varepsilon$ in $\mathcal{O}\big(d^{\frac{2}{5}}/\varepsilon^{\frac{2}{5}}\big)$ and $\mathcal{O}\big(\sqrt{d}/\varepsilon^{\frac{1}{3}}\big)$ steps when Lipschitz continuity is assumed for the Hessian and third derivative of $f$. By discretizing this ODE using a third order Runge-Kutta method, we can obtain a practical MCMC method that uses just two additional gradient evaluations per step. In our experiment, where the target comes from a logistic regression, this method shows faster convergence compared to other unadjusted Langevin MCMC algorithms

A Research Note On The Issue Of Non-Articulation And The Method Used To Calculate Net Operating Cash Flow

Using a proxy for nonarticulation, prior researchers found evidence that many companies using the indirect method of reporting net cash flow from operations have a significant level of nonarticulation.  The purpose of this study is to determine if companies using the direct method of reporting net cash flow from operations experience significantly lower levels of nonarticulation than companies that use the indirect method of reporting net cash flow from operations.  Results show that companies using the direct method have significantly less nonarticulation than companies using the indirect method.  This finding suggests that the Financial Accounting Standards Board (FASB) should consider requiring companies to use the direct method of preparing the Statement of Cash Flows

A Research Note On The Issue Of Non-Articulation And The Method Used To Calculate Net Operating Cash Flow

Using a proxy for nonarticulation, prior researchers found evidence that many companies using the indirect method of reporting net cash flow from operations have a significant level of nonarticulation. The purpose of this study is to determine if companies using the direct method of reporting net cash flow from operations experience significantly lower levels of nonarticulation than companies that use the indirect method of reporting net cash flow from operations. Results show that companies using the direct method have significantly less nonarticulation than companies using the indirect method. This finding suggests that the Financial Accounting Standards Board (FASB) should consider requiring companies to use the direct method of preparing the Statement of Cash Flows

An exploration of some aspects of mystery

This thesis project consists of twenty-four paintings, drawings and lithographs dealing with three sub-themes of the larger subject of mystery: the mystery of existence; the mystery of religion; the mystery of the unknown. These themes are explored through manipulations of light, color, compositional arrangement and painting and drawing techniques

A Comparison of Specificity of Passive Transfer of Chemical Contact Hypersensitivity Following Different Methods of Sensitization

This thesis compares the specificity of passive transfer of chemical contact hypersensitivity following various methods of sensitization

Anomaly detection on streamed data

We introduce powerful but simple methodology for identifying anomalous observations against a corpus of `normal' observations. All data are observed through a vector-valued feature map. Our approach depends on the choice of corpus and that feature map but is invariant to affine transformations of the map and has no other external dependencies, such as choices of metric; we call it conformance. Applying this method to (signatures) of time series and other types of streamed data we provide an effective methodology of broad applicability for identifying anomalous complex multimodal sequential data. We demonstrate the applicability and effectiveness of our method by evaluating it against multiple data sets. Based on quantifying performance using the receiver operating characteristic (ROC) area under the curve (AUC), our method yields an AUC score of 98.9\% for the PenDigits data set; in a subsequent experiment involving marine vessel traffic data our approach yields an AUC score of 89.1\%. Based on comparison involving univariate time series from the UEA \& UCR time series repository with performance quantified using balanced accuracy and assuming an optimal operating point, our approach outperforms a state-of-the-art shapelet method for 19 out of 28 data sets