Efficient Markets and Contingent Claims Valuation: An Information Theoretic Approach.

This research article shows how the pricing of derivative securities can be seen from the context of stochastic optimal control theory and information theory. The financial market is seen as an information processing system, which optimizes an information functional. An optimization problem is constructed, for which the linearized Hamilton-Jacobi-Bellman equation is the Black-Scholes pricing equation for financial derivatives. The model suggests that one can define a reasonable Hamiltonian for the financial market, which results in an optimal transport equation for the market drift. It is shown that in such a framework, which supports Black-Scholes pricing, the market drift obeys a backwards Burgers equation and that the market reaches a thermodynamical equilibrium, which minimizes the free energy and maximizes entropy.

Quantum Mechanics can be understood through stochastic optimization on spacetimes.

The main contribution of this paper is to explain where the imaginary structure comes from in quantum mechanics. It is shown how the demand of relativistic invariance is key and how the geometric structure of the spacetime together with the demand of linearity are fundamental in understanding the foundations of quantum mechanics. We derive the Stueckelberg covariant wave equation from first principles via a stochastic control scheme. From the Stueckelberg wave equation a Telegrapher's equation is deduced, from which the classical relativistic and nonrelativistic equations of quantum mechanics can be derived in a straightforward manner. We therefore provide meaningful insight into quantum mechanics by deriving the concepts from a coordinate invariant stochastic optimization problem, instead of just stating postulates.

Spatial dependencies between local luminance and contrast in natural images.

Previous research has suggested only weak statistical dependencies between local luminance and contrast in natural images. Here we study luminance and contrast in natural images using established measures and show that when multiple measurements of these two local quantities are taken in different spatial locations across the visual field, strong dependencies are revealed that were not apparent in previous pointwise (single-site) analyses. We present a few simple experiments demonstrating this spatial dependency of luminance and contrast and show that the luminance measurements can be used to approximate the contrast measurements. We also show that relying on higher-order statistics, independent component analysis learns paired spatial features for luminance and contrast. These features are shown to share orientation and localization, with the filters corresponding to the features dependent in their outputs. Finally, we demonstrate that the found dependencies also exist in artificial images generated from a dead leaves model, implying that simple image phenomena may suffice to account for the dependencies. Our results indicate that local luminance and contrast computations do not recover independent information sources from the visual signal. Subsequently, our results predict spatial processing of local luminance and contrast to be non-separable in visual systems.

As above, so below? Towards understanding inverse models in BCI.

OBJECTIVE: In brain-computer interfaces (BCI), measurements of the user's brain activity are classified into commands for the computer. With EEG-based BCIs, the origins of the classified phenomena are often considered to be spatially localized in the cortical volume and mixed in the EEG. We investigate if more accurate BCIs can be obtained by reconstructing the source activities in the volume. APPROACH: We contrast the physiology-driven source reconstruction with data-driven representations obtained by statistical machine learning. We explain these approaches in a common linear dictionary framework and review the different ways to obtain the dictionary parameters. We consider the effect of source reconstruction on some major difficulties in BCI classification, namely information loss, feature selection and nonstationarity of the EEG. MAIN RESULTS: Our analysis suggests that the approaches differ mainly in their parameter estimation. Physiological source reconstruction may thus be expected to improve BCI accuracy if machine learning is not used or where it produces less optimal parameters. We argue that the considered difficulties of surface EEG classification can remain in the reconstructed volume and that data-driven techniques are still necessary. Finally, we provide some suggestions for comparing approaches. SIGNIFICANCE: The present work illustrates the relationships between source reconstruction and machine learning-based approaches for EEG data representation. The provided analysis and discussion should help in understanding, applying, comparing and improving such techniques in the future.

Attending to Visual Stimuli versus Performing Visual Imagery as a Control Strategy for EEG-based Brain-Computer Interfaces.

Currently the most common imagery task used in Brain-Computer Interfaces (BCIs) is motor imagery, asking a user to imagine moving a part of the body. This study investigates the possibility to build BCIs based on another kind of mental imagery, namely "visual imagery". We study to what extent can we distinguish alternative mental processes of observing visual stimuli and imagining it to obtain EEG-based BCIs. Per trial, we instructed each of 26 users who participated in the study to observe a visual cue of one of two predefined images (a flower or a hammer) and then imagine the same cue, followed by rest. We investigated if we can differentiate between the different subtrial types from the EEG alone, as well as detect which image was shown in the trial. We obtained the following classifier performances: (i) visual imagery vs. visual observation task (71% of classification accuracy), (ii) visual observation task towards different visual stimuli (classifying one observation cue versus another observation cue with an accuracy of 61%) and (iii) resting vs. observation/imagery (77% of accuracy between imagery task versus resting state, and the accuracy of 75% between observation task versus resting state). Our results show that the presence of visual imagery and specifically related alpha power changes are useful to broaden the range of BCI control strategies.

simBCI - A framework for studying BCI methods by simulated EEG.

Brain-Computer Interface (BCI) methods are commonly studied using Electroencephalogram (EEG) data recorded from human experiments. For understanding and developing BCI signal processing techniques real data is costly to obtain and its composition is apriori unknown. The brain mechanisms generating the EEG are not directly observable and their states cannot be uniquely identified from the EEG. Subsequently, we do not have generative ground truth for real data. In this paper we propose a novel convenience framework called simBCI to alleviate testing and studying BCI signal processing methods in simulated, controlled conditions. The framework can be used to generate artificial BCI data and to test classification pipelines with such data. Models and parameters on both data generation and the signal processing side can be iterated over to examine the interplay of different combinations. The framework provides the first time open source implementations of several models and methods. We invite researchers to insert more advanced models. The proposed system does not intend to replace human experiments. Instead, it can be used to discover hypotheses, study algorithms, to educate about BCI, and to debug signal processing pipelines of other BCI systems. The proposed framework is modular, extensible and freely available as open source1. It currently requires Matlab.