Spin-boson model through a Poisson-driven stochastic process

Charge transfer rate constants were calculated for the carotenoid-porphyrin-C60 (CPC60) molecular triad dissolved in explicit tetrahydrofuran. The calculation was based on mapping the all-atom anharmonic Hamiltonian of this system onto the spin-boson Hamiltonian. The mapping was based on discretizing the spectral density from the time correlation function of the donor-acceptor potential energy gap, as obtained from all-atom molecular dynamics simulations. Different spin-boson Hamiltonians were constructed for each of the possible transitions between the three excited electronic states in two different triad conformations. The rate constants of three possible transitions were calculated via the quantum-mechanically exact Fermi's golden rule (FGR), as well as a progression of more approximate expressions that lead to the classical Marcus expression. The advantage of the spin-boson approach is that once the mapping is established, the quantum-mechanically exact FGR and the hierarchy of approximations are known in closed form. The classical Marcus charge transfer rate constants obtained with the spin-boson Hamiltonians were found to reproduce those obtained from all-atom simulations with the linearized semiclassical approximation, thereby confirming the equivalence of the two approaches for this system. Within the spin-boson Hamiltonian, we also found that the quantum-mechanically exact FGR rate constants were significantly enhanced compared to the classical Marcus theory rate constants for two out of three transitions in one of the two conformations under consideration. The results confirm that mapping to the spin-boson model can yield accurate predictions for charge transfer rate constants in a system as complex as CPC60 dissolved in tetrahydrofuran.

The ability to price risks and devise optimal investment strategies in the presence of an uncertain random market is the cornerstone of modern finance theory. We first consider the simplest such problem of a so-called European call initially solved by Black and Scholes using Ito stochastic calculus for markets modelled by a log-Brownien stochastic process. A simple and powerful formalism is presented which allows us to generalize the analysis to a large class of stochastic processes, such as ARCH, jump or Levy processes. We also address the case of correlated Gaussian processes, which is shown to be a good description of three different market indices (MATIF, CAC40, FTSE100). Our main result is the introduction of the concept of an optimal strategy in the sense of (functional) minimization of the risk with respect to the portfolio. If the risk may be made to vanish for particular continuous uncorrelated 'quasiGaussian' stochastic processes (including Black and Scholes model), this is no longer the case for more general stochastic processes. The value of the residual risk is obtained and suggests the concept of risk-corrected option prices. In the presence of very large deviations such as in Levy processes, new criteria for rational fixing of the option prices are discussed. We also apply our method to other types of options, `Asian', `American', and discuss new possibilities (`doubledecker'...). The inclusion of transaction costs leads to the appearance of a natural characteristic trading time scale.

SHARPE [12] AND LINTNER [7] have recently proposed models directed at the following questions: (a) What is the appropriate measure of the risk of a capital asset? (b) What is the equilibrium relationship between this measure of the asset's risk and its one-period expected return?' Lintner contends that the measure of risk derived from his model is different and more general than that proposed by Sharpe. In his reply to Lintner, Sharpe [13] agrees that their results are in some ways conflicting and that Lintner's paper supersedes his. This paper will show that in fact there is no conflict between the SharpeLintner models. Properly interpreted they lead to the same measure of the risk of an individual asset and to the same relationship between an asset's risk and its one-period expected return. The apparent conflicts discussed by Sharpe and Lintner are caused by Sharpe's concentration on a special stochastic process for describing returns that is not necessarily implied by his asset pricing model. When applied to the more general stochastic processes that Lintner treats, Sharpe's model leads directly to Lintner's conclusions.

Computer Science has historically been strong on data structures and weak on inference from data, whereas Statistics has historically been weak on data structures and strong on inference from data. One way to draw on the strengths of both disciplines is to develop “inferential methods for data structures”; i.e., methods that update probability distributions on recursively-defined objects such as trees, graphs, grammars and function calls. This is the world of “nonparametric Bayes,” where prior and posterior distributions are allowed to be general stochastic processes. Both statistical and computational considerations lead one to certain classes of stochastic processes, and these tend to have interesting connections to combinatorics. I will give some examples of how this blend of ideas leads to useful models in some applied problem domains, including natural language parsing, computational vision, statistical genetics and protein structural modeling.

This work initiates a general study of learning and generalization without the i.i.d. assumption, starting from first principles. While the standard approach to statistical learning theory is based on assumptions chosen largely for their convenience (e.g., i.i.d. or stationary ergodic), in this work we are interested in developing a theory of learning based only on the most fundamental and natural assumptions implicit in the requirements of the learning problem itself. We specifically study universally consistent function learning, where the objective is to obtain low long-run average loss for any target function, when the data follow a given stochastic process. We are then interested in the question of whether there exist learning rules guaranteed to be universally consistent given only the assumption that universally consistent learning is possible for the given data process. The reasoning that motivates this criterion emanates from a kind of optimist's decision theory, and so we refer to such learning rules as being optimistically universal. We study this question in three natural learning settings: inductive, self-adaptive, and online. Remarkably, as our strongest positive result, we find that optimistically universal learning rules do indeed exist in the self-adaptive learning setting. Establishing this fact requires us to develop new approaches to the design of learning algorithms. Along the way, we also identify concise characterizations of the family of processes under which universally consistent learning is possible in the inductive and self-adaptive settings. We additionally pose a number of enticing open problems, particularly for the online learning setting.

This part is concerned with the applications of the theorems on error estimates for asymptotic expansions in the central limit theorem for general linear stochastic processes of Part I to two important typical models. These are the general random noise process and a class of pulse train processes that occur in stochastic signal theory and in a variety of digital communication schemes. The assumptions on the inherent structure of the general theorems turn out to be simple here.

By means of nested inequalities in semigroup form we give a characterization of the value functions of the starting-stopping problem for general Markov-Feller processes. Next, we consider two versions of constrained problems on the final state or on the final time. The optimal stopping problems have been extensively studied for diffusion processes, or other Markov processes, or for more general stochastic processes.

A simple and efficient methodology to approximate a general non-Gaussian stationary stochastic vector process by a translation process with applications in wind velocity simulation

In this paper, we introduce a new concept of preinvex functions which is called general h-harmonic preinvex for real-valued stochastic processes. Further, we define the multidimensional general h-harmonic preinvex stochastic processes. We prove the Hermite-Hadamard inequality and obtain some important results for these processes. Some previous results are special cases of the results obtained in this paper.

Even the best investors seem to find it hard to do better than the comprehensive common-stock averages, or better on the average than random selection among stocks of comparable variability. Examination of historical samples of percentage changes in a stock's price show that, when these relative price changes are properly adjusted for expected dividends paid out, they are more or less indistinguishable from white noise, or, at the least, their expected percentage movements constitute a driftless random walk (or random walk with mean drift specifiable in terms of an interest factor appropriate to the stock's variability or riskiness). The present contribution shows that such observable patterns can be deduced rigorously from a model which hypothesizes that a stock's present price is set at the expected discounted value of its future dividends, where the future dividends are supposed to be random variables generated according to any general (but known) stochastic process. This fundamental theorem follows by an easy superposition applied to the 1965 Samuelson theorem that properly anticipated futures prices fluctuate randomly -- i.e., constitute a martingale sequence, or a generalized martingale with specifiable mean drift. Examples demonstrate that even when the economy is not free to wander randomly, intelligent speculation is able to whiten the spectrum of observed stock-price changes. A subset of investors might have better information or modes of analysis and get above average gains in the random-walk model; and the model's underlying probabilities could be shaped by fundamentalists' economic forces.

This special issue collects contributions from the participants of the “Information in Dynamical Systems and Complex Systems” workshop, which cover a wide range of important problems and new approaches that lie in the intersection of information theory and dynamical systems. The contributions include theoretical characterization and understanding of the different types of information flow and causality in general stochastic processes, inference and identification of coupling structure and parameters of system dynamics, rigorous coarse-grain modeling of network dynamical systems, and exact statistical testing of fundamental information-theoretic quantities such as the mutual information. The collective efforts reported here in reflect a modern perspective of the intimate connection between dynamical systems and information flow, leading to the promise of better understanding and modeling of natural complex systems and better/optimal design of engineering systems.

This paper contains a survey over the mathematical foundations, properties and potential applications of copulas in insurance and finance. Special emphasis is put on relationships between copulas and correlation as well as dependence measures, parametric families of copulas, Archimedian copulas (in particular in higher dimensions), tail dependence and general stochastic processes.

The life cycle/permanent income hypothesis suggests that optimization by consumers should cause marginal utility to approximate a random walk. As a consequence, purchases of nondurable goods should also approximately follow a random walk and purchases of durable goods should approximately follow a white noise process. The univariate processes for nondurables and durables both appear to be random walks, which would confirm the LC/PIH for nondurables and reject it for durables. Adjustment costs and nonseparability between durables and nondurables are then added to the specification of utility. The resulting more general stochastic process is estimated and the data tend to support the existence of nonseparabilty, but not adjustment costs. While forecasts of durables are not fully informationally efficient, they do about as well as do forecasts of nondurables. The behavior of durable consumption purchases is shown to be consistent with the life cycle/permanent income hypothesis. Copyright 1989 by MIT Press.

This volume presents recent developments in the area of Lévy-type processes and more general stochastic processes that behave locally like a Lévy process. Although written in a survey style, quite a f

We extend Kyle's (1985) model of insider trading to the case where noise trading volatility follows a general stochastic process. We determine conditions under which, in equilibrium, price impact and price volatility are both stochastic, driven by shocks to uninformed volume even though the fundamental value is constant. The volatility of price volatility appears 'excessive' because insiders choose to trade more aggressively (and thus more information is revealed) when uninformed volume is higher and price impact is lower. This generates a positive relation between price volatility and trading volume, giving rise to an endogenous subordinate stochastic process for prices.