Causal interpretation of stochastic differential equations

We give a causal interpretation of stochastic differential equations (SDEs) by defining the postintervention SDE resulting from an intervention in an SDE. We show that under Lipschitz conditions, the solution to the postintervention SDE is equal to a uniform limit in probability of postintervention structural equation models based on the Euler scheme of the original SDE, thus relating our definition to mainstream causal concepts. We prove that when the driving noise in the SDE is a L\'evy process, the postintervention distribution is identifiable from the generator of the SDE

Statistička teorija uzročnosti, stohastičke diferencijalne jednačine i svojstvo martingalne reprzentacije

One of the important and basic goals of science is to establish cause-e®ect re- lations between events. Many discussions were about the concept of causality and how it can be measured. The concept of Granger's causality (Granger, 1969) is very well known in economy and it can be applied in researches. Granger's de¯nition of causality is based on the idea that the present and the future cannot e®ect the past. About the concept of causality have been discussed for a very long time in all areas of science. In last decade we are dealing with a signi¯cant progress. Today, a concept of causality have a wide application in physical, biological andsocial sciences, history, medicine, especially in epidemiology, economy and etc. The area of research of this Phd dissertation is statistical theory of causality and its application on weak solutions of stochastic di®erential equations and martingale representation property. It have been shown that this concept of causality is equivalent with the concept of weak uniqueness of weak solutions for the stochastic di®erential equations and extremal solutions of the martingale problem. This concept of causality can be characterized with stopping times and its connection with extremal solution of the stopped martingale problem can be proved, as well as with locally unique weak local solutions. The concept of causality can be related to the theory of martingales, too. Namely, this concept can be connected with the preservation of the martingale property, orthogonal martingales, stable subspaces as well as with martingale representation property, which have an application, especially in ¯nancial mathematics

Adapted Wasserstein Distances and Stability in Mathematical Finance

Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she is left with the possibility that Q does not provide an "exact" description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? If we measure proximity with the usual Wasserstein distance (say), the answer is NO. Models which are similar w.r.t. Wasserstein distance may provide dramatically different information on which to base a hedging strategy. Remarkably, this can be overcome by considering a suitable "adapted" version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler \cite{Pf09,PfPi12,PfPi14}. It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.Comment: An author's name had been wrongfully give

Statistička teorija uzročnosti u neprekidnom slučaju

Finding the cause or determining what is the cause and what is the consequence are probably one of the eldest problems of science. Philosophy from the beginning deals with these issues in the most general way, but other sciences also try to solve this kind of problems within their object of interest. Based on the results of Probability theory, Theory of random processes and Statistics, Statistical theory of causality originated as one of mathematical answers to the problem of determining causality in an arbitrary system. After the seminal papers of Granger (1969) and Sims (1972) many authors considered different types of stochastically defined causality. These researches mainly belong to predicting theory. Namely, the question of interest is: whether we can predict with the same accuracy in case of reduction of available information. At first, the researches were focused on discrete time stochastic processes (time series). However, as it is pointed out, there is a need for defining causality for continuous time stochastic processes, because many processes of interest have continuous time parameter. Namely, for financial time series is explained that even though the agents have only perceptions in discrete time, the underlying stochastic process of interest is in continuous time. Thus, the development of continuous time modeling in finance is important motivation for considering causality in continuous time. Also, the observed causality in a discrete time model may depend on the length of interval between each two successive samplings. Mykland (1986) and Florens and Fougères (1996) were the authors of first papers in which we can find definitions of causality in continuous time, given in terms of σ-algebras, i.e. natural filtrations of stochastic processes. Also, in Gill and Petrović (1987) and in Petrović (1996) definition of causality was given in continuous time, but in term of Hilbert spaces, i.e. L2-framework. Recently, there have been several papers which deal with these themes. The field of research in this dissertation is consideration of some causality concepts that are generalizations of Granger causality adopted for stochastic processes with continuous time. Also, same relationships of developed concept of causality and already existed related theories (adopted distributions) are considered. This dissertation, beside Preface and References with 86 items, consists of four chapters 1. Theory of random processes - basic notions; 2. Theory of causality - review of known results; 3. Generalization of Granger causality for stochastic processes with continuous time; 4. Causality and adopted distribution of stochastic processes. Chapter 1 is a brief overview of notions of theory of probability and stochastic processes that will be use later. Some known concepts of causality, both in discrete and in continuous case, are presented in Chapter 2. We followed the chronological development of the Statistical theory of causality, and special attention is given to the concepts that have contributed to our researches. Chapter 3 presents some generalizations of Granger causality adopted for stochastic processes with continuous time. Our original results, related to properties of developed concepts of causality, are given there. Specially we focused our attention to integration of stoping times into considered concept of causality, to invariance of causality under convergence and to relationship between causality and markovianity. Finally, in Chapter 4, we give connections between considered concept of causality and concept of adapted distribution of stochastic processes (specific concept of equivalence of stochastic processes), which introduced mathematicians from Model theory, Kiesler and Hoove

Challenges in Statistical Theory: Complex Data Structures and Algorithmic Optimization

Technological developments have created a constant incoming stream of complex new data structures that need analysis. Modern statistics therefore means mathematically sophisticated new statistical theory that generates or supports innovative data-analytic methodologies for complex data structures. Inherent in many of these methodologies are challenging numerical optimization methods. The proposed workshop intends to bring together experts from mathematical statistics as well as statisticians involved in serious modern applications and computing. The primary goal of this meeting was to advance the mathematical and methodological underpinnings of modern statistics for complex data. Particular focus was given to the advancement of theory and methods under non-stationarity and complex dependence structures including (multivariate) financial time series, scientific data analysis in neurosciences and bio-physics, estimation under shape constraints, and highdimensional discrimination/classification

Stochastic Parameterization: A Rigorous Approach to Stochastic Three-Dimensional Primitive Equations

The atmosphere is a strongly nonlinear and infinite-dimensional dynamical system acting on a multitude of different time and space scales. A possible problem of numerical weather prediction and climate modeling using deterministic parameterization of subscale and unresolved processes is the incomplete consideration of scale interactions. A stochastic treatment of these parameterizations bears the potential to improve the simulations and to provide a better understanding of the scale interactions of the simulated atmospheric variables. The scientific community that is dealing with stochastic meteorological models can be divided into two groups: the first one uses pragmatic approaches to improve existing complex models. The second group pursues a mathematical rigorous way to develop stochastic models, which is currently limited to conceptual models. The overall objective of this work is to narrow the gap between pragmatic approaches and the mathematical rigorous methods. Using conceptual climate models, we point out that a stochastic formulation must not be chosen arbitrarily but has to be derived based on the physics of the system at hand. Equally important is a rigorous numerical implementation of the resulting stochastic model. The dynamics of sub grid and unresolved processes are often described by time continuous stochastic processes, which cannot be treated with deterministic numerical schemes. We show that a stochastic formulation of the three-dimensional primitive equations fits in the mathematical framework of abstract stochastic fluid models. This allows us to utilize recent results regarding existence and uniqueness of solutions of such systems. Based on these theoretical results we propose a Galerkin scheme for the discretization of spatial and stochastic dimensions. Using the framework of mild solutions of stochastic partial differential equations we are able to prove quantitative error bounds and strong mean square convergence. Under additional assumptions we show the convergence of a numerical scheme which combines the Galerkin approximation with a temporal discretization.Stochastische Parametrisierung: Ein Rigoroser Ansatz für die Stochastischen Drei-Dimensionalen Primitiven Gleichungen Die Atmosphäre ist ein von starken Nichtlinearitäten geprägtes, unendlich-linebreak dimensionales dynamisches System, dessen Variablen auf einer Vielzahl verschiedener Raum- und Zeitskalen interagieren. Ein potentielles Problem von Modellen zur numerischen Wettervorhersage und Klimamodellierung, die auf deterministischen Parametrisierungen subskaliger Prozesse beruhen, ist die unzureichende Behandlung der Interaktion zwischen diesen Prozessen und den Modellvariablen. Eine stochastische Beschreibung dieser Parametrisierungen hat das Potential die Qualität der Simulationen zu verbessern und das Verständnis der Skalen-Interaktion atmosphärischer Variablen zu vertiefen. Die wissenschaftlich Gemeinschaft, die sich mit stochastischen meteorologischen Modellen beschäftigt, kann grob in zwei Gruppen unterteilt werden: die erste Gruppe ist bemüht durch pragmatische Ansätze bestehende, komplexe Modelle zu erweitern. Die zweite Gruppe verfolgt einen mathematisch rigorosen Weg, um stochastische Modelle zu entwickeln. Dies ist jedoch aufgrund der mathematischen Komplexität bisher auf konzeptionelle Modelle beschränkt. Das generelle Ziel der vorliegenden Arbeit ist es, die Kluft zwischen den pragmatischen und mathematisch rigorosen Ansätzen zu verringern. Die Diskussion zweier konzeptioneller Klimamodelle verdeutlicht, dass eine stochastische Formulierung nicht willkürlich gewählt werden darf, sondern aus der Physik des betrachteten Systems abgeleitet werden muss. Ebenso unabdingbar ist eine rigorose numerische Implementierung des resultierenden stochastischen Modells. Diesem Aspekt wird besondere Bedeutung zu Teil, da dynamische subskalige Prozesse oftmals durch zeitabhängige stochastische Prozesse beschrieben werden, die sich nicht mit deterministischen numerischen Methoden behandeln lassen. Wir zeigen auf, dass eine stochastische Formulierung der dreidimensionalen primitiven Gleichungen im mathematischen Rahmen abstrakter stochastischer Fluidmodelle behandelt werden kann. Dies ermöglicht die Anwendung kürzlich gewonnener Erkenntnisse bezüglich Existenz und Eindeutigkeit von Lösungen. Wir stellen einen auf dieser theoretischen Grundlage basierenden Galerkin Ansatz zur Diskretisierung der räumlichen und stochastischen Dimensionen vor. Mit Hilfe sogenannter milder Lösungen der stochastischen partiellen Differentialgleichungen leiten wir quantitative Schranken der Diskretisierungsfehler her und zeigen die starke Konvergenz des mittleren quadratischen Fehlers. Unter zusätzlichen Annahmen leiten wir die Konvergenz eines numerischen Verfahrens her, das den Galerkin Ansatz um eine zeitliche Diskretisierung erweitert

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described