Statistical causality, martingale problems and local uniqueness

The purpose of this dissertation is to establish, in a certain infinite dimensional setting, some existence and uniqueness results pertaining to the type of exponential martingale problem previously considered by Strook and Varadhan in the finite dimensional setting. The setting is among those considered by K. Ito in his work on stochastic differential equations in infinite dimensional spaces. Briefly, the martingale problem takes the following form: to show the existence and uniqueness of a measure m on the space S of continuous functions defined on the nonnegative real lien with values in some state space, having the property that exp(f(t,w)) is an m-martingale, f being a function of type, with t denoting a time parameter, and w denoting a generic element of S. The measure m is then called a solution of the martingale problem. In this dissertation, the state spaces under consideration are subspaces of the space of tempered distributions on the real line. The existence of solutions is established under certain continuity conditions by constructing a family of measures satisfying a Prokhorov criterion for relative compactness with respect to weak convergence, and then showing that a weak limit point of that family has the desired martingale property. Secondly, under somewhat stronger assumptions, a uniqueness result for a previously constructed solution is proved in a manner similar to that in which the uniqueness of a related stochastic differential equation is established. Finally, the relation of the martingale problem to certain infinite dimensional stochastic integrals and stochastic equations is pointed out.

Models of causation and the semantics of causal verbs

SummaryThis paper aims at displaying a synthetic view of the historical development and the current research concerning causal relationships, starting from the Aristotelian doctrine of causes, following with the main philosophical streams until the middle of the twentieth century, and commenting on the present intensive research work in the statistical domain. The philosophical survey dwells upon various concepts of cause, and some attempts towards picking out spurious causes. Concerning statistical modelling, factorial models and directed acyclic graphs are examined and compared. Special attention is devoted to randomization and pseudo‐randomization (for observational studies) in view of avoiding the effect of possible confounders. An outline of the most common problems and pitfalls, encountered in modelling empirical data, closes the paper, with a warning to be very cautious in modelling and inferring conditional independence between variables.

This chapter starts with a few landmarks throughout the history of philosophy, presenting the reader with various types of causes. For the discussion of the various meanings of causality developed from a theoretical standpoint, I take a transdisciplinary perspective based on the idea that causality is a semiotic tool, about which philosophy has much to say, and also empirical studies, for instance Piaget’s genetic epistemology, science education, or social psychology, stressing the everyday use of causality. The transdisciplinary perspective allows displaying a variety of semiotic roles played by causality in our attempts to explain, model, or simply make sense of the world and of our own actions. This perspective stresses the diversity of meanings encapsulated in the concept of cause, and shows relevant links between the development of cognition, the making of science, and the definitions of causality. From the selected landmarks and the routes chosen for our map of causality, it becomes possible to address the question of how causality is related to the newly introduced concept of semiotic catalysis, and to make a few suggestions for the philosophical definition of the semiotic catalysis.

Objectives: To apply three of the most common epidemiologic causal models for glaucoma and explain the issues pertaining to the use of them. Methods: Three causal models will be explained: (1) counterfactual model, (2) graphical model, and (3) sufficient-component causal model. Using literature search and experts' opinions these models will be applied in glaucoma. Results: Introducing the counterfactual model, as one of the most important quantitative causal models and formalizes the concepts of cause and effect, this article emphasizes the limitations with the use of this model for glaucoma. Explaining causal diagram, as a useful qualitative tool that provides a lucid depiction of the assumptions concerning a causal pathway, this article provides a working causal diagram for glaucoma based on the evidence in the literature and experts' comments. In addition, the limitations in developing appropriate causal diagrams for glaucoma are discussed. Finally, the sufficient-component causal models and their importance in portraying the mechanics of causal interactions will be discussed. Conclusion: Paucity of evidence is the main issue with the use of causal models for glaucoma. This article calls for convergent review of different theories and different pieces of evidence that can enable us to apply different causal models for glaucoma.

Let $A$ be a pseudo-differential operator with negative definite symbol $q$. In this paper we establish a sufficient condition such that the well-posedness of the $(A,C_c^{\infty }(\mathbb{R} ^d))$-martingale problem implies that the unique solution to the martingale problem is a Feller process. This provides a proof of a former claim by van Casteren. As an application we prove new existence and uniqueness results for Lévy-driven stochastic differential equations and stable-like processes with unbounded coefficients.

Linear equations and Ornstein-Uhlenbeck processes; strong existence, uniqueness, and nonexplosion criteria; weak solutions and local martingale problems; well-posedness and measurability; pathwise uniqueness and functional solution; weak existence and continuity; transformation of SDEs; strong Markov and Feller properties

Probabilistic approximation for a porous medium equation

The Partial Directed Coherence (PDC) and its generalized formulation (gPDC) are popular tools for investigating, in the frequency domain, the concept of Granger causality among multivariate (MV) time series. PDC and gPDC are formalized in terms of the coefficients of an MV autoregressive (MVAR) model which describes only the lagged effects among the time series and forsakes instantaneous effects. However, instantaneous effects are known to affect linear parametric modeling, and are likely to occur in experimental time series. In this study, we investigate the impact on the assessment of frequency domain causality of excluding instantaneous effects from the model underlying PDC evaluation. Moreover, we propose the utilization of an extended MVAR model including both instantaneous and lagged effects. This model is used to assess PDC either in accordance with the definition of Granger causality when considering only lagged effects (iPDC), or with an extended form of causality, when we consider both instantaneous and lagged effects (ePDC). The approach is first evaluated on three theoretical examples of MVAR processes, which show that the presence of instantaneous correlations may produce misleading profiles of PDC and gPDC, while ePDC and iPDC derived from the extended model provide here a correct interpretation of extended and lagged causality. It is then applied to representative examples of cardiorespiratory and EEG MV time series. They suggest that ePDC and iPDC are better interpretable than PDC and gPDC in terms of the known cardiovascular and neural physiologies.

Consider A Closed E of the Real Line R and a (Measureble) Function Defined on R and Strictly Concave on Every Component of R/E. The Present Paper Deals with the Following Marticgale Problem: Find a Continuous Local Martingale (X,F), Given on a Family () of Probability Spaces With for every such that x is stopped at the debut of e, and the process is a local martingale up to We show that this martingale problem possesses a unique solution. This solution is a strong Markov continuous local martingale. Furthermore, it is a pure continuous local martingale and therefore satisfies the previsible representation property. Conversely, for every strong Markov continuous local martingale (X,F) we can find a closed subset E of R and a function f strictly concave on every component of R\E such that (X, F) is the unique solution of the above martingale problem. In the particular case that E and f(X) we recover the theorem of P. Levy on the martingale characterization of the Brownian motion.

In this chapter, we survey some of the concepts in the theory of stochastic processes which will be used in the sequel. The material is intended to be only a discussion of some of the main ideas which will be needed and to serve as a convenient reference. It is not intended to be complete in any sense. Additional details, motivation and background can be found in the references. The reader familiar with the basic facts concerning stochastic differential equations can skip the chapter and refer to the results as needed. Section 1 gives some definitions and facts concerning martingales. In Section 2, we define stochastic integrals with respect to a Wiener process and state Itô's Formula, the fundamental tool in the stochastic calculus. Section 3 uses the stochastic integrals to define stochastic differential equations (SDE) of the diffusion type, and obtains bounds on their solutions which will be used later in the study of properties of the solution processes, as well as in the proofs of tightness of sequences of solutions to such equations. In Section 4, we discuss three standard methods for obtaining existence and uniqueness of solutions to SDE's. These sections give some flavor of a few of the basic ideas. But they just touch the surface of the subject. Sections 5 and 6 concern an alternative and very useful method for verifying whether a process satisfies an SDE of the diffusion type. The so-called "martingale problem" method which is discussed in these sections is perhaps the most useful approach to showing that the limit of a weakly convergent sequence of processes is, indeed, a solution to a SDE.

We prove uniqueness of a martingale problem with boundary conditions on a simplex associated to a differential operator with an unbounded drift. We show that the solution of the martingale problem remains absorbed at the boundary once it attains it, and that, after hitting the boundary, it performs a diffusion on a lower dimensional simplex, similar to the original one. We also prove that in the diffusive time scale condensing zero-range processes evolve as this absorbed diffusion.

In a previous paper the authors studied a Fleming—Viot process with house-of-cards mutation and an unbounded haploid selection intensity function. Results included existence and uniqueness of solutions of an appropriate martingale problem, existence, uniqueness, and reversibility of stationary distributions, and a weak limit theorem for a corresponding sequence of Wright—Fisher models. In the present paper we extend these results to the diploid setting. The existence and uniqueness results carry over fairly easily, but the limit theorem is more difficult and requires new ideas.

Evolutionary equations with coefficients perturbed by diffusion processes are considered. It is proved that the solutions of these equations converge weakly in distribution, as a small parameter tends to zero, to a unique solution of a martingale problem that corresponds to an evolutionary stochastic equation in the case where the powers of a small parameter are inconsistent.

Well-posedness for some non-linear SDEs and related PDE on the Wasserstein space