Probability Commons^™

On The Total Duration Of Negative Surplus Of A Risk Process With Two-Step Premium Function, Pavlina Jordanova

Applications and Applied Mathematics: An International Journal (AAM)

We consider a risk reserve process whose premium rate reduces from c_d to c_u when the reserve comes above some critical value v. In the model of Cramer-Lundberg with initial capital u ≥ 0, we obtain the probability that ruin does not occur before the first up-crossing of level v. When u < v, following H. Gerber and E. Shiu (1997), we derive the probability that starting with initial capital u ruin occurs and the severity of ruin is not bigger than v. Further we express the probability of ruin in the two step premium function model - ψ (u,v), by the last two probabilities. Our assumptions imply that the surplus process will go to infinity almost surely. This entails that the process will stay below zero only temporarily. We derive the distribution of the total duration of negative surplus and obtain its Laplace transform and mean value. As a consequence of these results, under certain conditions in the Model of Cramer-Lundberg we obtain the expected value of the severity of ruin. In the end of the paper we give examples with exponential claim sizes.

Radical Impact Of Change In Actions And Confidence Index On Reverse Decision Making An Application Based Study, Swatee Trimbak Paithankar

Engineering Management & Systems Engineering Theses & Dissertations

While making decisions under uncertainty, people are often unaware of the logical approach to form the decision process. It is assumed that collecting details, analyzing and evaluating data is enough to make 'proper' decisions. However, past research in the decision making arena has significantly validated that there exists a class of decision problems which is complex, ill-structured and not defined to the level where decision makers can draw logical conclusions based on existing traditional decision approaches. RDM (reverse decision making), one of the novel approaches of decision making under conditions of uncertainty, has shown potential towards addressing some of these …

The Shift From Defined Benefit Pensions To 401(K) Plans And The Pension Assets Of The Baby Boom Cohort, James Poterba, Steven Venti, David A. Wise

Dartmouth Scholarship

The rise of 401(k) plans and the decline of defined benefit plans will have an important effect on the wealth of future retirees. Changing demographic structure also will affect the aggregate stock of retirement wealth. We project the stock of assets held in retirement plans and the average retirement saving of retirees through 2040. Our projections show large increases in wealth at retirement, especially if the returns on corporate equities are comparable with historical returns. Retirement wealth will grow, however, even if equity returns fall substantially below their historical level.

Green And Poisson Functions With Wentzell Boundary Conditions, José-Luis Menaldi, Luciano Tubaro

Mathematics Faculty Research Publications

We discuss the construction and estimates of the Green and Poisson functions associated with a parabolic second order integro-di erential operator with Wentzell boundary conditions.

A Distributed Parabolic Control With Mixed Boundary Conditions, Jose-Luis Menaldi, Domingo Alberto Tarzia

Mathematics Faculty Research Publications

We study the asymptotic behavior of an optimal distributed control problem where the state is given by the heat equation with mixed boundary conditions. The parameter α intervenes in the Robin boundary condition and it represents the heat transfer coefficient on a portion Γ₁ of the boundary of a given regular n-dimensional domain. For each α, the distributed parabolic control problem optimizes the internal energy g. It is proven that the optimal control ĝ_α with optimal state u_ĝαα and optimal adjoint state p_ĝαα are convergent as α → 1 …

Why De Minimis?, Matthew D. Adler

Faculty Scholarship

De minimis cutoffs are a familiar feature of risk regulation. This includes the quantitative individual risk thresholds for fatality risks employed in many contexts by EPA, FDA, and other agencies, such as the 1-in-1 million lifetime cancer risk cutoff; extreme event cutoffs for addressing natural hazards, such as the 100 - year - flood or 475 - year - earthquake; de minimis failure probabilities for built structures; the exclusion of low - probability causal models; and other policymaking criteria. All these tests have a common structure, as I show in the Article. A de minimis test, broadly defined, tells the …

First Passage Time Problem For Multivariate Jump-Diffusion Processes: Models, Computation, And Applications In Finance, Di Zhang

Theses and Dissertations (Comprehensive)

The first passage time (FPT) problems are ubiquitous in many applications, from physics to finance. Mathematically, such problems are often reduced to the evaluation of the probability density of the time for a process to cross a certain level, a boundary, or to enter a certain region. While in other areas of applications the FPT problems can often be solved analytically, in finance we usually have to resort to the application of numerical procedures, in particular when we deal with jump-diffusion stochastic processes (JDP). The application of the conventional Monte-Carlo procedure is possible for the solution of the resulting model, …

Modeling And Efficient Estimation Of Intra-Family Correlations, Roy Sabo

Mathematics & Statistics Theses & Dissertations

Familial data occur when observations are taken on multiple members of the same family. Due to relationships between these members, both genetic and by cohabitation, their response variables will likely exhibit some form of dependence. Most of the existing literature models this dependence with an equicorrelated structure. This structure is appropriate when the dependencies between family members are similar, such as in genetic studies, but not in cases where we expect the dependencies to differ, such as behavioral comparisons across different age groups. In this dissertation we first discuss an alternative structure based upon first-order autoregressive correlation. Specifically we create …

The Stable Manifold Theorem For Semilinear Stochastic Evolution Equations And Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang, Huaizhong Zhao

Articles and Preprints

The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2.

In Part 1, we prove general existence and compactness theorems for C^k-cocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with Lipschitz and non-Lipschitz terms such as stochastic reaction diffusion equations and the …

Remarks On Risk-Sensitive Control Problems, José Luis Menaldi, Maurice Robin

Mathematics Faculty Research Publications

The main purpose of this paper is to investigate the asymptotic behavior of the discounted risk-sensitive control problem for periodic diffusion processes when the discount factor α goes to zero. If u_α(θ, x) denotes the optimal cost function, being the risk factor, then it is shown that lim_α→0αu_α(θ, x) = ξ(θ) where ξ(θ) is the average on ]0, θ[ of the optimal cost of the (usual) in nite horizon risk-sensitive control problem.

Penalty Approximation And Analytical Characterization Of The Problem Of Super-Replication Under Portfolio Constraints, Alain Bensoussan, Nizar Touzi, José Luis Menaldi

Mathematics Faculty Research Publications

In this paper, we consider the problem of super-replication under portfolio constraints in a Markov framework. More specifically, we assume that the portfolio is restricted to lie in a convex subset, and we show that the super-replication value is the smallest function which lies above the Black-Scholes price function and which is stable for the so-called face lifting operator. A natural approach to this problem is the penalty approximation, which not only provides a constructive smooth approximation, but also a way to proceed analytically.

Random Walks On The Torus With Several Generators, Timothy Prescott '02, Francis E. Su

All HMC Faculty Publications and Research

Given n vectors {_i} ∈ [0, 1)^d, consider a random walk on the d-dimensional torus ^d = ℝ^d/ℤ^d generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*^k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*^k) ≥ C₁k^−n/2, where C₁ = C(n, d) is …

Mathematical And Empirical Modeling Of Chemical Reactions In A Microreactor, Jing Hu

Doctoral Dissertations

This dissertation is concerned with mathematical and empirical modeling to simulate three important chemical reactions (cyclohexene hydrogenation and dehydrogenation, preferential oxidation of carbon monoxide, and the Fischer-Tropsch (F-T) synthesis in a microreaction system.

Empirical modeling and optimization techniques based on experimental design (Central Composite Design (CCD)) and response surface methodology were applied to these three chemical reactions. Regression models were built, and the operating conditions (such as temperature, the ratio of the reactants, and total flow rate) which maximize reactant conversion and product selectivity were determined for each reaction.

A probability model for predicting the probability that a certain species …

Discrete-Time Approximations Of Stochastic Delay Equations: The Milstein Scheme, Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan

Articles and Preprints

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDE's). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the non-anticipating nature of the SDDE, the use of anticipating calculus methods appears to be novel.

Reconstructability Analysis With Fourier Transforms, Martin Zwick

Complex Systems Faculty Publications and Presentations

Fourier methods used in two‐ and three‐dimensional image reconstruction can be used also in reconstructability analysis (RA). These methods maximize a variance‐type measure instead of information‐theoretic uncertainty, but the two measures are roughly collinear and the Fourier approach yields results close to that of standard RA. The Fourier method, however, does not require iterative calculations for models with loops. Moreover, the error in Fourier RA models can be assessed without actually generating the full probability distributions of the models; calculations scale with the size of the data rather than the state space. State‐based modeling using the Fourier approach is also …

A Monte Carlo Analysis Of Hedonic Models Using Traditional And Spatial Approaches, Helen R. Neill, David M. Hassenzahl, Djeto D. Assane

Public Policy and Leadership Faculty Research

Hedonic regression analysis of single family homes typically includes structural variables, locational variables and neighborhood quality characteristics. When nearby properties are related, Dubin (1988) reports that error terms are spatially autocorrelated. Estimation methods for these spatially autocorrelated error terms or hereafter, spatial approaches, include maximum likelihood estimation (MLE) and kriging techniques such as kriged maximum likelihood estimation (KMLE). Unfortunately these spatial methods require massive computer resources and are limited to significantly fewer observations than traditional ordinary least squares (OLS). This paper investigates the combination of spatial approaches and Monte Carlo analysis, a method that approximates large data sets. A question …

Transient Analysis And Applications Of Markov Reward Processes, Jeffrey A. Sipe

Theses and Dissertations

In this thesis, the problem of computing the cumulative distribution function (cdf) of the random time required for a system to first reach a specified reward threshold when the rate at which the reward accrues is controlled by a continuous time stochastic process is considered. This random time is a type of first passage time for the cumulative reward process. The major contribution of this work is a simplified, analytical expression for the Laplace-Stieltjes Transform of the cdf in one dimension rather than two. The result is obtained using two techniques: i) by converting an existing partial differential equation to …

Gaussian Mixture Reduction Of Tracking Multiple Maneuvering Targets In Clutter, Jason L. Williams

Theses and Dissertations

The problem of tracking multiple maneuvering targets in clutter naturally leads to a Gaussian mixture representation of the Provability Density Function (PDF) of the target state vector. State-of-the-art Multiple Hypothesis Tracking (MHT) techniques maintain the mean, covariance and probability weight corresponding to each hypothesis, yet they rely on ad hoc merging and pruning rules to control the growth of hypotheses.

Impulse Control Of Stochastic Navier-Stokes Equations, J. L. Menaldi, S. S. Sritharan

Mathematics Faculty Research Publications

In this paper we study stopping time and impulse control problems for stochastic Navier-Stokes equation. Exploiting a local monotonicity property of the nonlinearity, we establish existence and uniqueness of strong solutions in two dimensions which gives a Markov-Feller process. The variational inequality associated with the stopping time problem and the quasi-variational inequality associated with the impulse control problem are resolved in a weak sense, using semigroup approach with a convergence uniform over path.

Estimation Of Cumulative Incidence Functions In Competing Risks Studies Under An Order Restriction, Hammou El Barmi, Subhash C. Kochar, Hari Mukerjee, Francisco J. Samaniego

Mathematics and Statistics Faculty Publications and Presentations

In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t for a particular type of failure in the presence of other risks. Its estimation and asymptotic distribution theory have been studied by many. In some cases there are reasons to believe that the CIFs due to two types of failure are order restricted. Several procedures have appeared in the literature for testing for such orders. In this paper we initiate the study of estimation of two CIFs subject to a type …

On Choosing And Bounding Probability Metrics, Alison L. Gibbs, Francis E. Su

All HMC Faculty Publications and Research

When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a means of deriving bounds for another one in an applied problem. Considering other metrics can also provide alternate insights. We also give examples that show that rates of convergence can strongly depend on the metric chosen. Careful consideration is necessary when choosing a metric.

Multi-Level Decomposition Of Probabilistic Relations, Stanislaw Grygiel, Martin Zwick, Marek Perkowski

Complex Systems Faculty Publications and Presentations

Two methods of decomposition of probabilistic relations are presented in this paper. They consist of splitting relations (blocks) into pairs of smaller blocks related to each other by new variables generated in such a way so as to minimize a cost function which depends on the size and structure of the result. The decomposition is repeated iteratively until a stopping criterion is met. Topology and contents of the resulting structure develop dynamically in the decomposition process and reflect relationships hidden in the data.

Stochastic 2-D Navier-Stokes Equation, J. L. Menaldi, S. S. Sritharan

Mathematics Faculty Research Publications

In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier-Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions-Prodi solutions to the deterministic Navier-Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this signi cantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier-Stokes martingale problem where the probability space is also obtained as a part of the solution.

Autoassociative-Heteroassociative Neural Network, Claudia V. Kropas-Hughes, Steven K. Rogers, Mark E. Oxley, Matthew Kabrisky

AFIT Patents

An efficient neural network computing technique capable of synthesizing two sets of output signal data from a single input signal data set. The method and device of the invention involves a unique integration of autoassociative and heteroassociative neural network mappings, the autoassociative neural network mapping enabling a quality metric for assessing the generalization or prediction accuracy of the heteroassociative neural network mapping.

Fuzzy Product -Limit Estimators: Soft Computing In The Presence Of Very Small And Highly Censored Data Sets, Kian Lawrence Pokorny

Doctoral Dissertations

When very few data are available and a high proportion of the data is censored, accurate estimates of reliability are problematic. Standard statistical methods require a more complete data set, and with any fewer data, expert knowledge or heuristic methods are required. In the current research a computational system is developed that obtains a survival curve, point estimate, and confidence interval about the point estimate.

The system uses numerical methods to define fuzzy membership functions about each data point that quantify uncertainty due to censoring. The “fuzzy” data are then used to estimate a survival curve, and the mean survival …

Using Simulated Annealing In Geostatistics, Wesley Wells

Theses : Honours

Simulation methods are now used extensively for estimation and prediction in mining and petroleum industries and also in environmental management. In this thesis we describe the method of simulated annealing and examine in detail the GSLIB implementation algorithm SASJM. In the context of two case studies involving both sample and exhaustive data sets, we demonstrate this algorithm and then investigate the effect on the outcome of varying the different algorithm parameters. We also consider the effect of varying the weighting given in the simulated annealing objective function to the reproduction of each of the sample histogram and semivariogram. For the …

Stochastic Functional Differential Equations On Manifolds, Rémi Léandre, Salah-Eldin A. Mohammed

Articles and Preprints

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters.

Empirical Spectral Analysis Of Random Number Generators, David Zeitler

Dissertations

Computer simulation procedures have become a staple of research and development in many fields, including statistics. The generation of pseudo random number sequences is the core of computer simulation procedures. Validity of research results often depend on the underlying validity of the generator being used.

In this work we develop the machinery for a class of tests of spatial uniformity based on a multi-dimensional Fourier transform of the empirical probability density function. The test can be adapted to specific requirements and has the added advantage that it has computational complexity that is relatively independent of the number of data points …

Discrepancy Convergence For The Drunkard's Walk On The Sphere, Francis E. Su

All HMC Faculty Publications and Research

We analyze the drunkard's walk on the unit sphere with step size θ and show that the walk converges in order C/sin²(θ) steps in the discrepancy metric (C a constant). This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.

A Coin Flipping Game With Non Transitive Odds, Stacy Jurgens

Honors Capstones

Capstone submitted as a graduation requirement for the BSU Honors Program.