Learning Whenever Learning is Possible: Universal Learning under General Stochastic Processes

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.

We give a functional integral representation of the semigroup generated by the spin-boson Hamiltonian by making use of a Poisson point process and a Euclidean field. We present a method of constructing Gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities. Using these tools we then show existence and uniqueness of the ground state of the spin-boson, and analyze ground state properties. In particular, we prove super-exponential decay of the number of bosons, Gaussian decay of the field operators, derive expressions for the positive integer, fractional and exponential moments of the field operator, and discuss the field fluctuations in the ground state.

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

Training rats in a particularly difficult olfactory-discrimination task results in acquisition of high skill to perform the task superbly, termed "rule learning" or "learning set." Such complex learning results in enhanced intrinsic neuronal excitability of piriform cortex pyramidal neurons, and in their excitatory synaptic interconnections. These changes, while subserving memory maintenance, must be counterbalanced by modifications that prevent overspreading of activity and uncontrolled synaptic strengthening. Indeed, we have previously shown that the average amplitude of GABAA-mediated miniature IPSCs (mIPSCs) in these neurons is enhanced for several days after learning, an enhancement mediated via a postsynaptic mechanism. To unravel the molecular mechanism of this long-term inhibition enhancement, we tested the role of key second-messenger systems in maintaining such long-lasting modulation. The calcium/calmodulin-dependent kinase II (CaMKII) blocker, KN93, significantly reduced the average mIPSC amplitude in neurons from trained rats only to the average pretraining level. A similar effect was obtained by the CaMKII peptide inhibitor, tatCN21. Such reduction resulted from decreased single-channel conductance and not in the number of activated channels. The PKC inhibitor, GF109203X, reduced the average mIPSC amplitude in neurons from naive, pseudo-trained, and trained animals, and the difference between the trained and control groups remained. Such reduction resulted from a decrease in the number of activated channels. The PKA inhibitor H89 dihydrochloride did not affect the average mIPSC amplitude in neurons from any of the three groups. We conclude that learning-induced enhancement of GABAA-mediated synaptic inhibition is maintained by persistent CaMKII activation.

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.