Perhaps more than any other “-omics” endeavor, the accuracy and level of detail obtained from mapping the major connection pathways in the living human brain with diffusion MRI depends on the capabilities of the imaging technology used. The current tools are remarkable; allowing the formation of an “image” of the water diffusion probability distribution in regions of complex crossing fibers at each of half a million voxels in the brain. Nonetheless our ability to map the connection pathways is limited by the image sensitivity and resolution, and also the contrast and resolution in encoding of the diffusion probability distribution. The goal of our Human Connectome Project (HCP) is to address these limiting factors by re-engineering the scanner from the ground up to optimize the high b-value, high angular resolution diffusion imaging needed for sensitive and accurate mapping of the brain’s structural connections. Our efforts were directed based on the relative contributions of each scanner component. The gradient subsection was a major focus since gradient amplitude is central to determining the diffusion contrast, the amount of T2 signal loss, and the blurring of the water PDF over the course of the diffusion time. By implementing a novel 4-port drive geometry and optimizing size and linearity for the brain, we demonstrate a whole-body sized scanner with Gmax = 300mT/m on each axis capable of the sustained duty cycle needed for diffusion imaging. The system is capable of slewing the gradient at a rate of 200 T/m/s as needed for the EPI image encoding. In order to enhance the efficiency of the diffusion sequence we implemented a FOV shifting approach to Simultaneous MultiSlice (SMS) EPI capable of unaliasing 3 slices excited simultaneously with a modest g-factor penalty allowing us to diffusion encode whole brain volumes with low TR and TE. Finally we combine the multi-slice approach with a compressive sampling reconstruction to sufficiently undersample q-space to achieve a DSI scan in less than 5 minutes. To augment this accelerated imaging approach we developed a 64-channel, tight-fitting brain array coil and show its performance benefit compared to a commercial 32-channel coils at all locations in the brain for these accelerated acquisitions. The technical challenges of developing the over-all system are discussed as well as results from SNR comparisons, ODF metrics and fiber tracking comparisons. The ultra-high gradients yielded substantial and immediate gains in the sensitivity through reduction of TE and improved signal detection and increased efficiency of the DSI or HARDI acquisition, accuracy and resolution of diffusion tractography, as defined by identification of known structure and fiber crossing.

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Distributional assumptions for the returns on the underlying assets play a key role in valuation theories for derivative securities. Based on a data set consisting of daily prices of the 30 DAX shares over a three-year period, we investigate the distributional form of compound returns. After performing a number of statistical tests, it becomes clear that some of the standard assumptions cannot be justi ed. Instead, we introduce the class of hyperbolic distributions which can be tted to the empirical returns with high accuracy. Two models based on hyperbolic L evy motion are discussed. By studying the Esscher transform of the process with hyperbolic returns, we derive a valuation formula for derivative securities. The result suggests a correction of standard Black{Scholes pricing, especially for options close to expiration.

We investigate a new basic model for asset pricing, the hyperbolic model, which allows an almost perfect statistical t of stock return data. After a brief introduction into the theory supported by an appendix we use also secondary market data to compare the hyperbolic model to the classical Black-Scholes model. We study implicit volatilities, the smile e ect and the pricing performance. Exploiting the full power of the hyperbolic model, we construct an option value process from a statistical point of view by estimating the implicit risk-neutral density function from option data. Finally we present some new valueat-risk calculations leading to new perspectives to cope with model risk.

As a generalization of the Gaussian Heath-Jarrow-Morton term structure model, we present a new class of bond price models that can be driven by a wide range of Lévy processes. We deduce the forward and short rate processes implied by this model and prove that, under certain assumptions, the short rate is Markovian if and only if the volatility structure has either the Vasiček or the Ho-Lee form. Finally, we compare numerically forward rates and European call option prices in a model driven by a hyperbolic Lévy motion with those in the Gaussian model.