Special Stochastic Systems Seminar
State University of New York at Stony Brook | |
Distribution Policy for a Corporation with Controllable Risk | |
PLEASE NOTE SPECIAL DAY, TIME AND PLACE FOR THIS WEEK ONLY |
Abstract: One of specific features of insurance models is that the risk an insurance company faces is generated by incoming claims and it is not related to the available capital. As a result in the diffusion approximation of the surplus (or the risk) process the diffusion and the drift coefficients do not go to zero when the surplus is approaching zero. One can reduce the risk by applying reinsurance, which simultaneously reduces potential profits. The problems of dividends optimization in such models become akin to consumption/investment models of mathematical finance in a "Bachelier" world, where stocks are governed by arithmetic Brownian motion and all the utility functions are linear.
In the case when the insurance company can invest a part or all of its surplus in a stock market, the dividend optimization model becomes especially interesting. In a certain sense it is a consumption/investment model for an investor who has a linear utility function and who is required to deal with two types of markets: the classical Back-Scholes market and the Bachelier market.
More formally we consider a problem in which the liquid assets or a surplus of a company are modeled by a diffusion process. At each moment of time the management of the company makes a decision of the amount of dividend paid-out to the shareholders. There is also a possibility to reduce risk exposure by conducting a less aggressive business activity, which also results in a smaller potential profit. In the case of an insurance company different business activities correspond to different reinsurance levels, which the insurance company may choose. The company also chooses how much of its surplus to invest in a stock market whose returns are governed by an exponential Brownian motion. The objective is to chose the policy which maximizes the expected dividend pay-outs.
Mathematically this problem is a mixed singular/regular stochastic control problem. Its analytical part consists of a solution to a series of related free boundary problems for linear and nonlinear ODEs. The resulting optimal process is a diffusion process whose coefficients are determined via the solutions to those ODEs and which is reflected from one of the free boundaries.
Special LCDS Lecture Series and PDE Seminar
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Universitaet Bonn, Wegelerstr. 10, 53115 Bonn, Germany | |
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Abstract: The magnetization of ferromagnetic materials forms complex structures of different dimensionality and on a broad range of length scales: domains, walls of different internal structure, Bloch lines and vortices.
Predicting the formation of these structures, and understanding their interaction and overall effects is crucial for key technological applications.
The rich source of experimental data and the simple mathematical form of the well-accepted model makes this analysis of good model problem to develop new mathematical tools for multiscale problems in material science.
We report on some new analysis of cross-tie walls and on new dimensionally reduced theories for thin films, their numerical simulation and experimental validation.
Scientific Computing Seminar
Argonne National Laboratory | |
Change of time as of April 2, 2001 |
Abstract: We present algorithms and results for spectral element simulation of transitional (weakly turbulent) flows in carotid bifurcations and arterio-venous (AV) grafts. The spectral element method is based upon rapidly convergent, high-order weighted residual techniques employing tensor-project polynomial basas of degree N in each of K hexahedral elements. Because of its low numerical dissipation and dispersion, the method is well suited to transitional Reynolds number applications where the physical viscosity is small. For high Reynolds numbers, the Galerkin formulation is stabilized using a recently developed filter that is local and preserves both interelement continuity and spectral accuracy. Time advancement is based on a consistent third-order operator splitting that permits large time steps (typ. a convective CFL of 1--5) and yields independent convective, viscous, and pressure subproblems to be solved at each step. We discuss issues related to construction of hex-based meshes from MR and CT scans, including automated mesh generation, Fourier-based surface smoothing, and localized oct-tree refinement. We present comparisons with 'in vitro laser Doppler anemometry for transitional flow in a carotid bifurcation at Re=1032, and in an AV graft at Re=1060 and 1820.
Special LCDS Lecture Series and PDE Seminar
Universitaet Bonn, Wegelerstr. 10, 53115 Bonn, Germany | |
**PLEASE NOTE CHANGE IN TIME AND PLACE FOR TODAY ONLY |
Abstract: The capillarity-driven spreading of a thin droplet of a viscous liquid on a solid plane is modelled by the lubrication approximation, an evolution equation for the film height {\it h}. However, as a consequence of the no-slip boundary condition for the liquid at the solid plane, logarithmic divergences in the viscous dissipation rate occur if the support of {\it h} changes.
This well-known singularity is removed by relaxing the no-slip condition, thereby introducing a microscopic lengthscale {\it b}. Matched asymptotics suggests a relationship (Tanner's law) between the speed of the contact line (the boundary of the support of {\it h}) and the macroscopic contact angle (the slope of {\it h} near the boundary of its support), modulo a logarithm involving {\it b}. This dynamic contact angle condition, which balances viscous forces and surface tension, is quite different from the static contact angle condition (Young's law), which balances just the surface tensions.
Tanner's law predicts a specific scaling for the spreading of the droplet. In a joint work with L. Giacomelli, we rigorously derive the scaling of the spreading, which is consistent with the one predicted based on Tanner's law, including the logarithmic terms. Mathematically speaking, this amounts to estimates of appropriate integral quantities of the evolution equation, which comes in the form of a nonlinear parabolic equation of fourth order.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Co-Sponsored by the Brown University Computer Science Department
The Weizmann Institute of Science, Israel | |
Abstract: Video provides a continuous visual window into the space-time world. It captures the evolution of dynamic scenes in space and time. This makes video much more than just a collection of images of a scene taken from different view points. In this talk I will show that by treating video as a space-time data volume, one can perform tasks that are very difficult (and often impossible) to perform when only "slices" of this information, such as image frames, are used. In particular, I will demonstrate the power of this approach by two example problems; (i) I will show how by utilizing all available spatio-temporal information within the video sequences, one can align and integrate information across multiple video sequences both in time and in space. By combining the spatial and dynamic visual scene information within a single alignment framework, situations which are inherently ambiguous for traditional image-to-image alignment methods are uniquely resolved by sequence-to-sequence alignment. Moreover, coherent dynamic information can sometimes be used for aligning video sequences even in extreme cases when there is \underline ( no common spatial information ) across these sequences (e.g., when there is no spatial overlap between the camera fields of views). I will demonstrate applications of this approach to three real-world problems: (i) Alignment of non-overlapping sequences for generating wide-screen movies, (ii) Multi-sensor image alignment for multi-sensor fusion, and (iii) Alignment of images (sequences) obtained at significantly different zooms (e.g., 1:10), for surveillance applications. (ii) I will show how extended spatio-temporal scene representations can be very efficiently used to view, browse, index into, edit and enhance the video data. In raw video data the spatio-temporal scene information is implicitly and redundantly distributed across many video frames. This makes access and manipulation of video data very difficult. However, by analyzing the redundancy of visual information within the space-time data volume, the distributed scene information can be integrated into coherent and compact scene-based visual representations. These lead to very efficient methods for access and manipulation of visual information in video data.
Brown Analysis Seminar
Department of Mathematics Colloquium
PDE Seminar
New York University | |
Department of Mathematics Colloquium
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