Proactive decisions and stochastic discounting models in risk treatment operations

The paper considers a thinning of renewal processes in the development of a stochastic model for the present value of a random cash flow under random timing and constant force of interest. The development of this model is based on two random sums of positive, independent and identically distributed random variables. Applications of the proposed stochastic discounting model in the area of risk frequency reduction operations are also provided by the paper.

The concept of probability is central to some models of Earth surface systems. For instance, in models of hillslopes some of the variables involved are best expressed in probabilistic terms. The uplift which may alter base level and the storms which activate a number of slope processes tend to occur rather erratically; they have a chance-like character and may be represented as stochastic events. By applying rules of stochastic occurrence, the effects of chance can be built explicitly into models, not just of hillslopes, but of most Earth surface systems in which chance seems to play a significant role. Stochastic models, rather than giving an “exact” set of predictions about system change in the manner of deterministic models, focus attention on a set of possible outcomes, each of which has a different probability of occurring in reality. The basis of all stochastic (and statistical) models is probability theory. For this reason, the concepts of probability theory and their simple applications to Earth surface systems will be discussed before more complex and substantial stochastic models, both deductive and inductive, and before statistical models, are considered.

Stochastic Pi Calculus in Financial Markets: An Alternate Approach to High Frequency Trading

Fractional Brownian Motions, Fractional Noises and Applications

Discrete and nonnegative random sums are generally considered as stochastic models with important applications in many practical disciplines. The present paper concentrates on establishing applications in the systemics of some discrete and nonnegative random sums. More precisely, it is shown that discrete and non-negative Poisson binomial random sums can contribute to the description, analysis and implementation of fundamental operations of systems arising in many important practical disciplines. It is also shown that such random sums are useful for providing essential support to systemic thinking. The paper interprets discrete and nonnegative Poisson binomial random sums as strong analytical tools for handling situations in several areas of systemics.

While mean-field models of cellular operations have identified dominant processes at the macroscopic scale, stochastic models may provide further insight into mechanisms at the molecular scale. In order to identify plausible stochastic models, quantitative comparisons between the models and the experimental data are required. The data for these systems have small sample sizes and time-evolving distributions. The aim of this study is to identify appropriate distance metrics for the quantitative comparison of stochastic model outputs and time-evolving stochastic measurements of a system. We identify distance metrics with features suitable for driving parameter inference, model comparison, and model validation, constrained by data from multiple experimental protocols. In this study, stochastic model outputs are compared to synthetic data across three scales: that of the data at the points the system is sampled during the time course of each type of experiment; a combined distance across the time course of each experiment; and a combined distance across all the experiments. Two broad categories of comparators at each point were considered, based on the empirical cumulative distribution function (ECDF) of the data and of the model outputs: discrete based measures such as the Kolmogorov–Smirnov distance, and integrated measures such as the Wasserstein-1 distance between the ECDFs. It was found that the discrete based measures were highly sensitive to parameter changes near the synthetic data parameters, but were largely insensitive otherwise, whereas the integrated distances had smoother transitions as the parameters approached the true values. The integrated measures were also found to be robust to noise added to the synthetic data, replicating experimental error. The characteristics of the identified distances provides the basis for the design of an algorithm suitable for fitting stochastic models to real world stochastic data.

Stochastic integrals, random sums, random contractions, and selfdecomposable random variables constitute fundamental concepts of probability theory with significant applications in several areas of systemics. The main results of the paper are a characterization of a selfdecomposable distribution and a formulation of a Poisson random sum of power contractions. These results are established by incorporating a type of stochastic integral for a continuous in probability, homogeneous stochastic process with independent increments, and the same type of stochastic integral for a compound Poisson stochastic process with positive jumps. Interpretations of the results in treatment of risks threatening various systems are also provided.

Weak Convergence Rate of SDE with Stochastic Integration by Trapezoidal Rule

A probabilistic damage tolerance concept for welded joints. Part 1: data base and stochastic modelling

1. Basic Probability 2. Discrete Random Variables 3. Special Discrete Random Variables 4. Markov Chains 5. Continuous Random Variables 6. Special Continuous Random Variables 7. Markov Counting and Queuing Processes 8. The Distribution of Sums of Random Variables 9. Selected Systems Models 10. Reliability Models Appendix. References. Answers to Selected Odd-Numbered Exercises.

This chapter is a discussion of basic concepts in probability and applied statistics, beginning with the notion of probability and stochastic processes.Experimental design matters are discussed, including the concept of a baseline, an experimental protocol, a population of observational units, samples and subsamples, all with illustrations taken from various projects. The role of baseline variables is discussed, together with various important distinctions, such as the distinction between functions and relationships. Randomization is a procedure specified in the protocol, and the treatment assignment is an immediate post-baseline outcome. The distinction between observational unit and experimental unit is a consequence of the randomization protocol.Every biological population evolves in time; only the current population is accessible today. If we aim to make statements about values for future or past units, the stochastic model must be defined on the entire population, past present and future. Thus, in essentially all biological applications, the population is infinite, while the accessible population is finite. A sample is an ordered subset of the population, and an accessible sample is an ordered subset of the current population. The implications of this reasoning are discussed in the context of stochastic models for clinical trials.The chapter concludes with a discussion of the interpretation of variability for parameter estimates.

Stochastic Day-ahead operation of cascaded hydropower systems with Bayesian neural network-based scenario generation: A Portland general electric system study

In this chapter we give a brief overview of the basic mathematical concepts used for pricing financial products. Modern pricing theory is mainly based on describing the ups and downs of market prices via stochastic processes. Therefore, we start with the definition and characteristics of stochastic processes in Section 2.1. Usually the price-changes of the most basic financial instruments are modelled in terms of a so-called stochastic differential equation (SDE). The price of a financial derivative is considered to be a function of the basic financial instruments’ prices. One of the central tools in stochastic calculus which can be applied to determine the stochastic differential equation for the prices of these derivatives, Itô’s lemma, is described in Section 2.4. This concept is closely related to the stochastic integral, which is defined in Section 2.3 to be the mean square limit of some random sums, carefully put together so that the resulting limit is a (local) martingale. And in fact, martingales are one of the most important elements needed for the evaluation of financial instruments. They are defined in Section 2.5 and are used to describe the prices of financial derivatives in terms of conditional expectations. The Feynman-Kac formula provides a partial differential equation (PDE) that corresponds to such conditional expectations as described in Section 2.6. The advantage of dealing with partial differential equations is that they can be solved numerically. Also, PDE methods come naturally to applied mathematicians and physicists. This approach led Black and Scholes [BS73] to their famous equation for pricing European options. Their model will be discussed in Section 3.5. By contrast, martingales and Itô (stochastic) calculus come naturally to probabilists, and this approach led Merton [Mer74] to his option pricing formula.KeywordsStochastic ProcessStochastic Differential EquationConditional ExpectationWiener ProcessLocal MartingaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

New parallel stochastic global integral and local differential equation modeling and inversion

This paper investigates the impact of uncoordinated, coordinated, and smart charging of plug-in hybrid electric vehicles (PHEVs) on the optimal operation of microgrids (MGs) incorporating the dynamic line rating (DLR) security constraint. The DLR constraint, particularly in the islanding mode, influences the ampacity of MG feeders, when distribution lines reach their maximum capacity. To overcome any line outage or contingency situation, smart PHEVs are utilized to help improve the grid security. However, using PHEVs can cause higher power losses and feeder overloading issues. To address these concerns, a reconfiguration technique is employed in this paper. A heuristic algorithm, known as the collective decision-based optimization algorithm, is utilized to overcome the non-convexity and nonlinearity of the problem. The unscented transform technique is employed to model DLR uncertainties caused by solar radiation, load demand, and weather temperature, as well as PHEVs' uncertainties caused by varying charging strategies, numbers of PHEVs being charged, charging start time, and charging duration. Moreover, a deep learning gated recurrent unit technique is designed to forecast renewable power output for mitigating the uncertainties in renewable energy components. A modified IEEE 33-bus test network is deployed to evaluate the efficiency and performance of the proposed model.