Stochastic Integrals

K. Ito invented his famous stochastic calculus on Brownian motion in the 1940s. In the 1960s and 1970s, the “Strasbourg school,” headed by P.A. Meyer, developed a modern theory of martingales, the general theory of stochastic processes, and stochastic calculus on semimartingales. It turned out soon that semimartingales constitute the largest class of right continuous adapted integrators with respect to which stochastic integrals of simple predictable integrands satisfy the theorem of dominated convergence in probability. Stochastic calculus on semimartingales not only became an important tool for modern probability theory and stochastic processes but also has broad applications to many branches of mathematics, physics, engineering, and mathematical finance.

We model a stream of cash flows as an optional stochastic process, and value the cash flows by using a continuous and strictly positive linear functional. By applying a representation theorem from the general theory of stochastic processes we are able to study this valuation principle, as well as properties of the stochastic discount factor it implies. This approach to valuation is useful in the non-presence of a financial market, as is often the case when valuing cash flows arising from insurance contracts and in the application of real options.

Continuous-Time Public Good Contribution Under Uncertainty

d1. Filip Lundberg's works on risk theory were all written at a time when no general theory of stochastic processes existed, and when collective reinsurance methods, in the present day sense of the word, were entirely unknown to insurance companies. In both respects his ideas were far ahead of his time, and his works deserve to be generally recognized as pioneering works of fundamental importance.

On the basis of the statistical theory of branching processes, the kinetics of esterification of adipic acid (AA) with pentaerythritol are reinvestigated, and extended to the system in which trimethylol ethane (TME)(2,2-dihydroxymethylpropanol) replaces pentaerythritol (PE). The mathematical model is expressed in terms of eleven (TME) or fifteen (PE) simultaneous ordinary differential equations, or equivalently, in terms of two simultaneous partial differential equations. This model was shown previously to become asymptotically exact as the degree of cyclisation goes to zero, e.g., by extrapolation to zero volume. The substitution effect is allowed for, and it is found that each pre-existent ester link carried by a PE unit lowers the free energy of activation for forming a further such link by about 350 cal./mole. A small correction is also incorporated for the effect of the shrinkage on the reactant concentrations. This model allows extremely close fittings to be made to experimental rate curves on the assumption that the basic esterification reaction is first order in hydroxyls and second order in carboxyls (as postulated by Flory long ago). The model and this assumption are also supported by measurements of gel points as function of mole ratio [COOH/OH] and dilution by solvent. The model leads to the conclusion that in AA/PE near the gel point, about one in ten of the ester-links present was formed intramolecularly. (The proportion is higher if the system is diluted with solvent.) This high degree of cyclisation agrees quantitatively with that calculated from random-flight statistics. Together with the substitution effect, it seems to account for the lowering of the apparent reaction order to about 2·5 observed by some previous investigators of polyesterification reactions. The reactions have been followed continuously using measurements of pressure of the water evolved by means of a transducer and recorder. This method is superior to the classical procedure of intermittent titration of samples, with which it has been shown to give good agreement.The model based on the theory of branching processes has been briefly compared to the general stochastic theory of ring-chain competition processes by Whittle.

This chapter is a self-contained introduction to the theory of stochastic integration with respect to continuous semimartingales. Apart from some simple facts on the predictable σ-field (which are included with proofs in Section 1.2), we do not need any other result from the general theory of processes. The Doob—Meyer decomposition of the square of a continuous (local) martingale is given by constructing the quadratic variation process directly. Once this is done, the stochastic integral is defined and its properties are obtained in a natural fashion, using arguments common in the theory of the Lebesgue integral, namely, the monotone class theorem and dominated convergence theorem. We first give the definition of the stochastic integral with respect to Brownian motion and then deal with the case of continuous semimartingales in stages. In the last section, we define the integral with respect to a general (r.c.l.l.) semimartingale. However, this time we state without proof some key results, including the Doob—Meyer decomposition. We show that these results, three to be precise, enable us to define the integral in the general case and derive the properties of the integral following the route adopted earlier.KeywordsBrownian MotionDominate Convergence TheoremQuadratic VariationStochastic IntegrationLocal 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.

General Process Theory (GPT) is a new (non-Whiteheadian) process ontology. According to GPT the domains of scientific inquiry and everyday practice consist of configurations of ‘goings-on’ or ‘dynamics’ that can be technically defined as concrete, dynamic, non-particular individuals called general processes. The paper offers a brief introduction to GPT in order to provide ontological foundations for research programs such as interactivism that centrally rely on the notions of ‘process,’ ‘interaction,’ and ‘emergence.’ I begin with an analysis of our common sense concept of activities, which plays a crucial heuristic role in the development of the notion of a general process. General processes are not individuated in terms of their location but in terms of ‘what they do,’ i.e., in terms of their dynamic relationships in the basic sense of one process being part of another. The formal framework of GPT is thus an extensional mereology, albeit a non-classical theory with a non-transitive part-relation. After a brief sketch of basic notions and strategies of the GPT-framework I show how the latter may be applied to distinguish between causal, mechanistic, functional, self-maintaining, and recursively self-maintaining interactions, all of which involve ‘emergent phenomena’ in various senses of the term.

IntroductionSince the financial crisis of 2008, the theory of financial innovation has been a focus at a time of re-evaluation and re-conceptualization. However, little has been done to evaluate the current state of research considering the increasing complexity of financial innovation. This paper examines the hypothesis of a general theory that encompasses increasing complexities in the financial innovation process.MethodsThe paper begins with an overview of the definitions, the features, and the classification schemes of financial innovation. Additionally, the paper reviews the existing literature on the main objects of study in financial innovation and groups the findings under four main concepts. A conceptual analysis is presented that evaluates current approaches to the study of the financial innovation process and the difficulties inherent in constructing a single general theory. The paper proposes a framework based on a meta-theory of financial innovation as a better approach to understanding the inherent complexities and diversities affecting financial innovations.Discussion(1) Financial innovations present diversities and complexities that make it infeasible to build a unifying general theory to explain their development. (2) The current state of research on financial innovation theories is limited and requires additional input. (3) A meta-theory that identifies, classifies, and connects theories of development for financial innovations is better suited to explaining the complexity of financial innovation processes.

On March 29, 1900, at the Sorbonne University in Paris, Luis Jean-Baptiste Alphonse Bachelier, at the time a 30-year-old postgraduate student, successfully defended a dissertation titled Theorie de la Speculation. In this work Bachelier singlehandedly introduced the idea that in an efficient market, the absence of unexploited profit opportunities means that the mathematical expectation of the speculator’s gain is zero, and hence, the current price of an asset is the best predictor of its future price. He showed how stock prices followed a stochastic process, developed the mathematics of Brownian motion to model such a process and came up with a formula for pricing options that is not that far from the Nobel Prize-winning solution to the same problem proposed by Fischer Black, Myron Scholes, and Robert Merton in 1973. This was all well before Einstein used essentially the same diffusion equation to model the Brownian motion in 1905, before Andrey Markov began working on continuous-time processes in 1906, and before Andrey Kolmogorov published his 1931 paper laying the foundations of the general theory of Markov processes.

In the modern world, most gross product is created within Enterprise firms, project programs, state agencies, transnational corporations and their divisions, as well as various associations and compositions of the above entities. Enterprises, being, on the one hand, complex, and, on the other hand, widespread systems, are the subject matter of cybernetics, system theory, operations research, management sciences and many other fields of knowledge. However, the complexity of the system obstructs the development of mathematically rigorous foundations for Enterprise control. Moreover, methods of operations research and related sciences, which are widely used in practice, provide optimization of the constituents of an Enterprise, without modeling it as a whole system. But the optimization of parts does not lead to the optimality of the whole, and, also, the absence of top-down and holistic mathematical models of Enterprise contradicts the principle of holism and the system approach. The approach in this book looks first at Enterprise Systems and their essential aspects as complex sociotechnical systems composed of integrated sets of structural and process models (Chapters 1 and 2). A uniform description of all the heterogeneous fields of the modern Enterprise (marketing, sales, manufacturing, HR, finance, etc.) is then made, and the Enterprise Control Problem is posed as a top-down and holistic mathematical optimization problem (Chapter 3). Original models and methods of contract theory (Chapter 4), technology management (Chapter 5), human behavior and human capital (Chapter 6) and complex activity and resource planning (Chapter 7) are developed to solve the problem. Structural processes and mathematical models constitute an Optimal Enterprise Control Framework (Chapter 8) that provides a practical solution to the Enterprise Control Problem. This book is a resource for postgraduate and doctoral students, postdoctoral researchers and professors with research interests in the following fields of science: Fundamental Complex Systems study, Complex Systems Engineering, Enterprise Systems Engineering Applications of Operations Research, Optimization, Probability and Stochastic processes to Management Science, Economics and Business Theory of the Firm Business and Management – general, strategy/leadership, organization management, operations management and management information systems Theory of Business Processes, Business Processes Improvement and Reengineering

Compensated pulsed alternators (compulsators) with electromagnetic distribution shields are usually described from a constructive point of view. Generally, the mathematical theory for modelling compulsators is not well developed. The purpose of this paper is to present a theoretical investigation of compulsators based on the general dynamic theory of transient processes in synchronous machines. Particular emphasis is given to the developed general relations between mass and power characteristics. Results show that a significant frequency dependence of mass-power characteristics exists for compulsators operating at typical values for frequency and number of current pulses. This work also treats autonomous power energy systems with compulsators as components of modules with two and four pulse power channels.

The theoretical advancement in the field of modern nonlinear continuum mechanics is discussed. The paper includes elements of the mathematical apparatus, development of the foundations of a General tensor theory of mechanical processes and their representations, including generalization of concepts of objective derivatives and integrals, concepts of tensor measures of stresses and deformations, new approaches in theory of the resistance of solids to deformation.

This article presents a new model of inference processes in the social perceiver, along with the basis of the model in prior experimental results and in the ACT* cognitive theory. The two stages of the model involve different types of processing: interpretive versus procedural. In the first stage, an inference is performed by general, domain-independent processes guided by a learned inference rule that is selected from long-term memory. In the second, procedural stage, after much experience with a particular inferential domain, an inference process is selected without explicit search, based on a pattern match to the current goal and knowledge state. The inference becomes automatic. Existing experimental evidence supports predictions of the model in both the attribution and attitude domains. Insights and findings of current approaches to inference, including information integration and judgmental heuristics, can be encompassed within the new model in a natural way. Directions for future research applying the model are discussed. Social inferences—those involving attributions, other beliefs, and attitudes—are intrinsic components of many processes of interest to social and cognitive psychologists, including person perception, social judgment, causal attribution, language comprehension, and attitude formation and change (cf. Nisbett & Ross, 1980; Schank & Abelson, 1977; Wyer & Carlston, 1979). This article presents a new model of social inference processes, based on a general theory of cognitive structure and processes and on recent results in social psychology. Existing theoretical approaches to social inference fall into three general categories (Hastie, 1983): Information Integration Theory (N. H. Anderson, 1981), judgmental heuristics (Nisbett & Ross, 1980; Tversky & Kahneman, 1973), and information-processing models (Wyer & Srull, 1980). The relationship of the new model to these current approaches is discussed later in the article. However, current

A general theory of surface seperation processes

SummaryAssuming duality and certain analytic conditions on the potential density u(x,y), we show that any harmonic function is a constant and the invariant measure, if one exists, is unique.We will assume that Xt is a standard process in duality with another standard process X t ′ relative to some fixed reference measure m on the state space E. The reader is refered to [l, VI, Sec 1] for the precise definition of duality and to either [1] or [2] for other usual definitions. We will also assume that both X and X′ are transient; for any compact K, TKo θt → ∞ as t → ∞ a.s.. However, only the transience of X is used in Proposition 1 and only that of X′ is used in the rest of the discussion.For simplicity, any subset A of E and any function f defined on E is automatically assumed to be measurable with respect to the Borel field of E.There is a common potential density u(x,y) which is excessive in x and co-excessive in y and satisfies:Uf(x) = ∫u(x,y)f(y)m(dy) and U′f(y) = ∫m(dx)f(x)u(x,y) for any f ≥ 0, where U and U′ are, repectively, the potential operators of X and X′.A function h ≥ 0 on E is said to be harmonic if for any compact subset K of E, PKch = h.For Brownian motion in Rn with n ≥ 3, any harmonic function is a constant. Our first proposition generalizes this classical result under a general setting, which includes as special cases, not only Brownian motion, but also transient symmetric stable processes and one sided stable processes. Our proof which uses the general theory of Markov processes is quite short, compared with the complicated analytic proof for symmetric stable processes given in [3].