Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

Fractional Differential Equations

Full-waveform inversion FWI is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.

/pdf/an-overview-of-full-waveform-inversion-in-exploration-59sei7fzvh.pdf

An overview of full-waveform inversion in exploration geophysics

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Here we develop an option pricing method for European options based on the Fourier-cosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fourier-cosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers different underlying dynamics, including L\'evy processes and Heston stochastic volatility model, and various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal with European options in particular. In a follow-up paper we will present its application to options with early-exercise features.

A novel pricing method for european options based on fourier-cosine series expansions

Here we develop an option pricing method for European options based on the Fourier-cosine series and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fourier-cosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers underlying asset processes for which the characteristic function is known and various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal with European options in particular. In a follow-up paper we will present its application to options with early-exercise features.

/pdf/a-novel-pricing-method-for-european-options-based-on-fourier-2w9rx9dt1j.pdf

A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions

Forecasting financial time series using past observations has been a significant topic of interest. While temporal relationships in the data exist, they are difficult to analyze and predict accurately due to the non-linear trends and noise present in the series. We propose to learn these dependencies by a convolutional neural network. In particular the focus is on multivariate time series forecasting. Effectively, we use multiple financial time series as input in the neural network, thus conditioning the forecast of a time series x(t) on both its own history as well as that of a second (or third) time series y(t). Training a model on multiple stock series allows the network to exploit the correlation structure between these series so that the network can learn the market dynamics in shorter sequences of data. We show that long-term temporal dependencies in and between financial time series can be learned by means of a deep convolutional neural network based on the WaveNet model [2]. The network makes use of dilated convolutions applied to multiple time series so that the receptive field of the network is wide enough to learn both short and long-term dependencies. The architecture includes batch normalization and uses a 1 × k convolution with parametrized skip connections from the input time series as well as the time series we condition on, in this way learning long-term interdependencies in an efficient manner [1]. This improves the forecast, while at the same time limiting the requirement for a long historical price series and reducing the noise. Knowing the strong performance of CNNs on classification problems we show that they can be applied successfully to forecasting financial time series, without the need of large samples of data. We compare the performance of the WaveNet model to a state-of-the-art fully convolutional network (FCN), and an autoregressive model popular in econometrics and show that our model is much better able to learn important dependencies in between financial time series resulting in a more robust and accurate forecast.

/pdf/conditional-time-series-forecasting-with-convolutional-1lddhuznwx.pdf

Conditional time series forecasting with convolutional neural networks

An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtz-type differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with Fourier analysis. Multigrid analysis results are verified by numerical experiments. High wavenumber Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner.

/pdf/a-novel-multigrid-based-preconditioner-for-heterogeneous-243hkyqpjx.pdf

A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems

We present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Levy asset price models. The error convergence is exponential for processes characterized by very smooth ($${{\rm{C}}^{\infty}[a,b]\in\mathbb {R}}$$) transitional probability density functions. The computational complexity is O((M − 1)N log N) with N a (small) number of terms from the series expansion, and M, the number of early-exercise/monitoring dates. This paper is the follow-up of (Fang and Oosterlee in SIAM J Sci Comput 31(2):826–848, 2008) in which we presented the impressive performance of the Fourier-cosine series method for European options.

https://mpra.ub.uni-muenchen.de/9248/1/MPRA_paper_9248.pdf

Cornelis W. Oosterlee

Papers

A novel pricing method for european options based on fourier-cosine series expansions

A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions

Conditional time series forecasting with convolutional neural networks

A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems

Pricing early-exercise and discrete barrier options by fourier-cosine series expansions