Stony Brook AMS - Downloadable Preprints, 2000

In this paper we study theoretically an inter-relation between 3D and 2D periodic structures in the Rayleigh-Taylor (RTI) and Richtmeyer-Meshkov instability (RMI). We consider the local dynamics of a 3D flow with rectangular symmetry in the plane within a potential approximation. Under the Layzer-type approach, in the intermediate 3D-2D region there are no regular late-time solutions and there is no continuous 3D-2D dimensional crossover in either the RTI or the RMI cases. We show that for low symmetric bubbles in RTI and RMI, there is a two-parameter family of regular late-time solutions. The solutions of the familiy are smooth functions of the parameters and a continuous 3D-2D transition may occur. We discuss the separation criteria for the family of asymptotes, structural stability of highly symmetric 3D and 2D flows in RTI and RMI and the influence of the initial value of the Froude number on the RT or RM type of late-time behavior.

The paper analyzes stochastic optimization problems involving random fields on infinite directed graphs. The primary focus is on a problem of maximizing a concave functional of the field subject to a system of convex and linear constraints. The latter are specified in terms of linear operators acting in the space $L_\infty $. We examine conditions under which these constraints can be relaxed by using dual variables in $L_1$ -- stochastic Lagrange multipliers. We develop a method for constructing the Lagrange multipliers. In contrast to the conventional methods employed for such purposes (relying on the Yosida-Hewitt theorem), our technique is based on an elementary measure-theoretic fact, the ''biting lemma''.

Automated dendritic spine detection and analysis is one of the applications of the 3DMA code, which has been designed for stochastic, geometric analysis of two phase, two and three dimensional images. This guide provides instructions for users interested in the spine detection/analysis facility of the 3DMA software.
This is a companion document to the 3DMA General Users Manual (SUNYSB-AMS-99-20), and provides documentation only for those algorithms utilized in dendritic spine analysis. For general installation and execution instructions of the 3DMA code, see the General Users Manual.

We report solutions of the problem of the nonlinear motion of ideal fluid with a free surface and with no external forces. The motion of the free surface is associated with generation of bubbles and spikes by the Richtmeyer-Meshkov instability. At late time parameters of the regular bubble are not uniquely determined by the value of spatial period of the flow and there exists a family of regular asymptotic solutions. We made the local stability analysis for the solution and show that bubbles with a flattened surface are faster and more stable than narrow bubbles with the radius of curvature of order of half of spatial period both in 3D and 2D.

The paper analyzes a stochastic model of an economy with locally interacting agents. The mathematical basis of the study is a control theory for random fields on a directed graph. The graph involved in the model describes directions of commodity flows in the economy. We consider equilibria of the economic system, i.e., those states of it in which material and financial balance constraints are satisfied and all the agents choose their most preferred programs. Conditions are examined under which such states exist and are unique. In the present paper, results obtained previously for finite graphs are extended to infinite graphs.

This paper represents a model for the financial valuation of a firm which has control on the dividend payment stream and its risk as well as potential profit by choosing different business activities among those available to it. Furthermore the company invests its free reserve in an asset, which may or may not contain an element of risk. The company chooses a dividend payment policy and we associate the value of the company with the expected present value of the net dividend distributions (under the optimal policy).
One of the examples could be a large corporation such as an insurance company, whose liquid assets in the absence of control and investments fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, while drift is understood as potential profit. At each moment of time there is an option to reduce risk exposure, simultaneously reducing the potential profit, like using proportional reinsurance with another carrier for an insurance company. The company invests its reserve in a financial asset, whose price evolve as a geometric Brownian motion, with mean rate r > 0 and diffusion constant sigma_P >= 0. Thus sigma_P = 0 corresponds to investments in a riskless bank account. The objective is to find a policy, consisting of risk control and dividend payment scheme, which maximizes the expected total discounted dividends paid out until the time of bankruptcy. We apply the theory of controlled diffusions to solve the problem. We show that if the discount rate c is less than r then the optimal return function is infinite. If r=c the return function is finite for all x finite, but no optimal policy exists. If r is less than c, then there is a finite level u_1 > 0, such that the optimal action is to distribute all reserve exceeding u_1 as dividends. Furthermore there exists a constant x_0, with x_0 < u_1 such that the risk exposure monotonically increases on (0,x_0) from 0 to maximum possible.

Preprint #SUNYSB-AMS-00-06.
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The paper examines set-valued random dynamical systems defined by convex homogeneous stochastic operators. The main results are existence and uniqueness theorems for infinite paths growing in a certain sense faster than others (rapid paths). The study is motivated by problems related to stochastic analogues of the von Neumann--Gale model of economic growth.

Preprint #SUNYSB-AMS-00-07.
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The paper studies a stochastic model of an economy with locally interacting agents, generalizing a deterministic economic model proposed by Polterovich. Equilibrium states of the system under consideration are solutions to certain variational inequalities in spaces of random vectors. By analyzing these inequalities, we establish an existence theorem for equilibrium, which extends and refines a number of previous results.

Preprint #SUNYSB-AMS-00-08.
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The main objective of the study is to extend the classical arbitrage pricing and hedging theorems to securities market models taking into account trading constraints and transaction costs. A general framework suitable for the analysis of these questions is developed. The framework is suggested by a parallelism between dynamic models of financial markets and stochastic analogues of the von Neumann--Gale model of economic growth.

Preprint #SUNYSB-AMS-00-09.
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We study the optimal investment policy for an investor who has available one bank account and n risky assets modelled by log-normal diffusions. The objective is to maximize the long run average growth of wealth for a logarithmic utility function in presence of proportional transaction costs. This problem is formulated as an ergodic singular stochastic control problem and interpreted as the limit of a discounted control problem for vanishing discount factor. The variational inequalities for the discounted control problem and the limiting ergodic problem are established in the viscosity sense. The ergodic variational inequality is solved by using a numerical algorithm based on policies iterations and multigrid methods. A numerical example is displayed for 2 risky assets.

Preprint #SUNYSB-AMS-00-10.
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The current paper presents a short survey of stochastic models of risk control and dividend optimization techniques for a financial corporation. While being close to consumption/investment models of Mathematical Finance, dividend optimization models possess special features which do not allow them to be treated as a particular case of consumption/investment models.

In a typical model of this sort, in the absence of control, the reserve (surplus) process, which represents the liquid assets of the company, is governed by a Brownian motion with constant drift and diffusion coefficient. This is a limiting case of the classical Cramer-Lundberg model in which the reserve is a compound Poisson process, amended by a linear term, representing a constant influx of the insurance premiums. Risk control action corresponds to reinsuring part of the claims the cedent is required to pay simultaneously diverting part of the premiums to a reinsurance company. This translates into controlling the drift and the diffusion coefficient of the approximating process. The dividend distribution policy consists of choosing the times and the amounts of dividends to be paid put to shareholders. Mathematically, the cumulative dividend process is described by an increasing functional which may or may not be continuous with respect to time.

The objective in the models presented here is maximization of the dividend pay-outs. We will discuss models with different types of conditions imposed upon a company and different types of reinsurances available, such as proportional, noncheap, proportional in a presence of a constant debt liability, excess-of-loss. We will show that in most cases the optimal dividend distribution scheme is of a barrier type, while the risk control policy depends substantially on the nature of reinsurance available.

Preprint #SUNYSB-AMS-00-11.
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We consider numerical solutions of the Darcy and Buckley-Leverett equations for flow in porous media. These solutions depend on a realization of a random field that describes the reservoir permeability. The main content of this paper is to formulate and analyze a probability model for the numerical coarse grid solution error. We explore the extent to which the coarse grid oil production rate is sufficient to predict future oil production rates. We find that very early oil production data is sufficient to reduce the prediction error in oil production by about 30\%, relative to the prior probability prediction.

Preprint #SUNYSB-AMS-00-12.
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We propose a pair of ordinary differential equations to describe the motion of the two edges of a Rayleigh-Taylor (RT) or Richtmyer-Meshkov (RM) mixing zone. These model equations give a simple physics-based description of the RT and RM mixing rates. The equations are in agreement with all available experiments, including the recent LEM RT and RM experiments for spikes as well as bubbles. In particular, the model equations predict that as the Atwood number A tends to 1, the scaling constant alpha_s tends to 0.5, while theta_s tends to 1.

Preprint #SUNYSB-AMS-00-13.
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We built a supercomputer called Galaxy by connecting Intel Pentium-based computer nodes with Fast and Gigabit Ethernet switches. Each node has two processors at clock speeds varying from 300MHz to 600MHz, up to 512MB of memory, and small 2Gb local disk. All nodes run the standard RedHat Linux and inter-node communication is handled by a message passing interface called MPI. Local tools are written to visualize the system performance and to balance loads. We have benchmarked a sub-Galaxy with 72 processors by NAS and Parallel LINPACK benchmark suites. We achieved 16.9 Gflops in a standard single precision LU decomposition for 46848 by 46848 matrix parallel LINPACK benchmark. A Galaxy with 128 processors costs approximately $250,000 and it delivers 40 Gflops of performance. This leads to a cost-performance ratio of 160 Kflops-per-dollar, which is to improve further due to increase in processor speeds and network bandwidth at similar cost. Our final system with 512 processors is expected to reach several Tflops. This article first describes the Galaxy architectural details, and then present and analyze its performance in terms of floating point number crunching, network bandwidth, and IO throughput.

Preprint #SUNYSB-AMS-00-14.
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A thorough understanding and analysis of geometry and topology of three-dimensional fiber networks from high resolution images is an important and challenging task due to the enormous complexity and randomness of the structure. In this paper we propose a technique that is aimed at structural analysis of fiber mats, both for quality evaluation and improvement of fiber products.
A sequence of image processing techniques is applied to the images, to obtain the medial axis of the fiber network. A description of the network is then determined from the medial axis. We demonstrate computational algorithms that can efficiently identify individual fibers from a network of randomly oriented and curled fibers that are bonded irregularly with each other. We can accurately measure the orientation, location, curl, length, bonds, and crossing angles of the identified fibers as well as the density of the fibers contained in a given volume.
The performance of the proposed technique is presented for simulated fiber data and for a synthetic (polymer) fiber mat.

This paper represents a model for the financial valuation of a firm which has control on its risk as well as potential profit by choosing different business activities among those available to it. Furthermore the firm has the option of investing its reserve in a financial market consisting of a riskless asset (bond) and a risky asset (stock). The company chooses a dividend payment policy and the value of the company is associated with the expected present value of the net dividend distributions (under the optimal policy).

The example we consider is that of a large corporation such as an insurance company, whose liquid assets in the absence of control and investments fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, while drift is understood as potential profit. At each moment of time there is an option to reduce risk exposure, simultaneously reducing the potential profit, like using proportional reinsurance with another carrier for an insurance company. The company invests its reserve in a financial market, which is modelled by a classical Black-Scholes model. The management of the company also controls the dividend pay-outs to shareholders. The objective is to find a policy consisting of investment strategy, risk control and dividend pay-out scheme, which maximizes the expected total discounted dividends paid out until the time of bankruptcy.

We apply the theory of controlled diffusions to solve the problem and show that there is a level u₁ >0, such that the optimal action is to distribute all reserve exceeding u₁ as dividends. Furthermore there exists a constant x₀, with x₀ < u₁ such that the risk exposure monotonically increases on (0,x₀) from 0 to maximum possible.The optimal choice of investments depends on the market price of risk m_p=(r₁-r₀)/sigma_P^2, where r₀,r₁ denotes the mean rate of return of the bond and stock respectively and sigma_P denotes the diffusion coefficient of the stock price. We get the following result:

1. m_p<= 0: Invest everything in the bond.
2. m_plarge: Invest everything in the stock.
3. m_psmall: There exists x₁ with x₀ < x₁, such that the optimal strategy is to invest a fixed fraction of the reserve in the stock when the reserve level does not exceed x₁ and all of the reserve when it does exceed x₁.

Methods of isotonic regression are applied to increase the power of common trend tests in situations where a monotonicity constraint is imposed upon the dose-response function. Isotonic versions of Cochran-Armitage type trend tests for binary response data are developed, and the bootstrap method is used in finding the empirical distributions of the test statistics and their critical values. The isotonic likelihood ratio test with a survival adjustment is also proposed. This survival adjustment can be applied to the likelihood ratio test for either the order-restricted or unrestricted parameter cases. To achieve the isotonic modifications, an amalgamation algorithm is applied when the observed dose-response is non-monotonic. A Monte Carlo simulation study comparing these trend tests shows the advantages of the isotonic modifications and survival adjustment. By applying the proposed methods to data from a toxicology and carcinogenesis study conducted as part of the National Toxicology Program, the effect of C.I. Pigment Red 23 is investigated

We use diffusion approximation for this optimal control problem, where two situations are considered:
(a) The rate of dividend pay-out are unrestricted and in this case mathematically the problem becomes a mixed singular-regular control problem for diffusion processes. Its analytical part is related to a free boundary (Stephan) problem for a linear second order differential equation. The optimal policy prescribes to reinsure using a certain retention level (depending on the reserve) and pay no dividends when the reserve is below some critical level x₁ and to pay out everything that exceeds x₁. Reinsurance will stop at a level x₀<= x₁ depending on the claim size distribution.
(b) The rate of dividend pay-out is bounded by some positive constant M, in which case the problem becomes a regular control problem. Here the optimal policy is to reinsure at a certain rate and pay no dividends when the reserve is below x₁ and pay out at maximum rate when the reserve exceeds x₁. In this case reinsurance may or may not stop depending on the claim size distribution and the size of M, but in all cases the retention level will remain constant when the reserve exceeds x₁.

In this work, we describe two models of permeability hysteresis. Common to both models is that the scanning flow regime is modeled with a family of curves along which the flow is reversible. In the Scanning Hysteresis Model (SHM), the scanning curves are bounded by two curves, the drainage and imbibition curves, where the flow can only occur in a specific direction. The SHM is a heuristic model consistent with experiments, but it does not have a nice mathematical specification. For instance, the algorithm for constructing solutions of Riemann problems involves several ad hoc assumptions.

The Scanning Hysteresis Model with Relaxation (SHMR) augments the SHM by (a) allowing the scanning flow to extend beyond the drainage and imbibition curves and (b) treating these two curves merely as attractors of states outside the scanning region. The attraction, or relaxation, occurs on a time scale that corresponds to the redistribution of phases within the pores of the medium driven by capillary forces. By means of a formal Chapman-Enskog expansion, we show that the SHM with additional viscosity arises from the SHMR in the limit of vanishing relaxation time, provided that the diffusion associated with capillarity exceeds that induced by relaxation. Moreover, through a rigorous study of traveling waves in the SHMR, we show that the shock waves used to solve Riemann problems in the SHM are precisely those that have diffusive profiles. Thus the analysis of the SHMR justifies the SHM model. Simulations based on a simple numerical method for the simulation of flow with hysteresis confirm our analysis.

Preprint #SUNYSB-AMS-00-25.
Appeared in June, 2000 issue of Journal of Business
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Preprint #SUNYSB-AMS-00-26.
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Winner, student paper competition, International Nonwovens Technical Conference, Sept. 25-28, 2000, Dallas TX; sponsored by the Technical Association of the Pulp and Paper Industry (TAPPI) and the International Nonwovens and Disposables Association (INDA).

Preprint #SUNYSB-AMS-00-27.
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