Stochastics

The concept of a Morse decomposition consisting of nonautonomous sets is reviewed for linear cocycle mappings w.r.t. to past, future and all-time convergences. In each case, the set of accumulation points of the finite-time Lyapunov exponents corresponding to points in a nonautonomous set is shown to be an interval. For a finest Morse decomposition, the Morse spectrum is defined as the union of all of the above accumulation point intervals over the different nonautonomous sets in such a finest Morse decomposition. In addition, Morse spectrum is shown to be independent of which finest Morse decomposition is used, when more than one exists.

In our paper, we analyze, based on a new rating methodology, 105 enterprises from Saxony with respect to their ability to meet their financial obligations. It is based on classical financial-statement approach, a direct inclusion of risk and a stochastic simulation model of enterprise development. The results show that the method used is superior to presently used approaches and that it extends our knowledge of enterprise development. On and above its Basel-II applicability, it is a tool to analyze individual development strategies of firms.

Während meines Studiums der Wirtschaftsinformatik an der Hochschule Harz hatte ich in den Jahren 2002 bis 2005 das große Glück, als Tutor für verschiedene Statistik-Vorlesungen von Dr. Walter Strube tätig werden zu dürfen. Zu Beginn meiner Tutorentätigkeit konnte ich eine über die Jahre von Tutor zu Tutor weiterentwickelte Aufgabensammlung in Form einer ,,Lose-Blatt-Sammlung" von meiner direkten Vorgängerin, Frau Weinert, übernehmen. Um den Studierenden die Arbeit mit den Übungsaufgaben zu erleichtern, habe ich damals für sämtliche Aufgaben Musterlösungen und Erläuterungen verfasst und den Teilnehmern meiner Tutorien in Form von Handouts zur Verfügung gestellt. In den vergangenen sieben Jahren bin ich immer wieder von Studierenden nach dieser Sammlung von Aufgaben mit Musterlösungen gefragt worden, so dass ich mich dazu entschlossen habe, sie noch einmal zu überarbeiten, mit einigen weiteren Erläuterungen zu versehen und über den GRIN-Verlag sowie verschiedene Webseiten als kostenfreien Download zur Verfügung zu stellen. Ich hoffe, dass sie so vielleicht auch für den einen oder anderen Studierenden außerhalb unserer Hochschule von Nutzen sein kann.

Some known results from statistical thermophysics as well as from hydrology are revisited from a different perspective trying: (a) to unify the notion of entropy in thermodynamic and statistical/stochastic approaches of complex hydrological systems and (b) to show the power of entropy and the principle of maximum entropy in inference, both deductive and inductive. The capability for deductive reasoning is illustrated by deriving the law of phase change transition of water (Clausius-Clapeyron) from scratch by maximizing entropy in a formal probabilistic frame. However, such deductive reasoning cannot work in more complex hydrological systems with diverse elements, yet the entropy maximization framework can help in inductive inference, necessarily based on data. Several examples of this type are provided in an attempt to link statistical thermophysics with hydrology with a unifying view of entropy.

For a large class of vanilla contingent claims, we establish an explicit Föllmer-Schweizer decomposition when the underlying is an exponential of an additive process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electricity market are performed.

Im Rahmen des größtenteils digitalen "Corona-Semesters" entstand am Fachbereich Wirtschaftswissenschaften der Hochschule Harz diese kurze Einführung in R und RStudio für die Studierenden im berufsbegleitenden Bachelor-Studiengang Betriebswirtschaftslehre.

We consider the parameter estimation problem for the Ornstein-Uhlenbeck process X driven by a fractional Ornstein-Uhlenbeck process V , i.e. the pair of processes defined by the non-Markovian continuous-time long-memory dynamics dX t = −θX t dt + dV t ; t 0, with dV t = −ρV t dt + dB H t ; t 0, where θ > 0 and ρ > 0 are unknown parameters, and B H is a fractional Brownian motion of Hurst index H ∈ (1 2 , 1). We study the strong consistency as well as the asymptotic normality of the joint least squares estimator θ T , ρ T of the pair (θ, ρ), based either on continuous or discrete observations of {X s ; s ∈ [0, T ]} as the horizon T increases to +∞. Both cases qualify formally as partial-observation questions since V is unobserved. In the latter case, several discretization options are considered. Our proofs of asymptotic normality based on discrete data, rely on increasingly strict restrictions on the sampling frequency as one reduces the extent of sources of observation. The strategy for proving the asymptotic properties is to study the case of continuous-time observations using the Malliavin calculus, and then to exploit the fact that each discrete-data estimator can be considered as a perturbation of the continuous one in a mathematically precise way, despite the fact that the implementation of the discrete-time estimators is distant from the continuous estimator. In this sense, we contend that the continuous-time estimator cannot be implemented in practice in any naïve way, and serves only as a mathematical tool in the study of the discrete-time estimators' asymptotics.

The ratio P (S n = x)/P (Z n = x) is investigated for three cases: (a) when S n is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and Z n is Poisson rv; (b) when S n is a statistic of 2-runs and Z n is negative binomial rv; and (c) when S n is statistic of N (1, 1)-events and Z n is a binomial r.v. We also consider the approximation of P (S n x) by Poisson distribution with parameter depending on x.

The present paper introduces the conceptual framework for an artificial system for visual creativity addressing the idea of niche creativity that is domain specific and non-anthropocentric in its conceptual approach. We think that the visual creative output of the system reflects the artificial medium and the specific artificial processes engaged in its production and, therefore, it is an expression of the idea of embodied creativity with the proposed system offering in this sense an example of digital embodiment of creativity. Although our approach to artificial creativity is non-anthropocentric, the system design is inspired by processes in the natural world that lead to the production of new and useful structures in both living and non-living systems with human creative cognition being included among these processes. The main problem raised by this abstract approach to artificial creativity in visual arts is the compatibility of its artistic production with human aesthetics, the ultimate goal of the proposed system being to produce visual output that would aesthetically engage human visual perception.

For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index α ∈ (0, 2) and weak dependence conditions. The limiting process consists of an α-stable Lévy process and an extremal process. We also describe the dependence between these two components of the limit. The convergence takes place in the space of R 2-valued càdlàg functions on [0, 1], with the Skorohod weak M 1 topology. We further show that this topology in general can not be replaced by the stronger (standard) M 1 topology.

The periodic KdV equation u_t=u_xxx+β uu_x arises from a Hamiltonian system with infinite-dimensional phase space L^2(T). Bourgain has shown that there exists a Gibbs measure ν on balls {ϕ :Φ^2_L^2≤ N} in the phase space such that the Cauchy problem for KdV is well posed on the support of ν, and ν is invariant under the KdV flow. This paper shows that ν satisfies a logarithmic Sobolev inequality. The stationary points of the Hamiltonian on spheres are found in terms of elliptic functions, and they are shown to be linearly stable. The paper also presents logarithmic Sobolev inequalities for the modified periodic KdV equation and the cubic nonlinear Schrödinger equation, for small values of N.

This paper analyzes customers' impatience in Markovian queueing system with multiple working vacations and Bernoulli schedule vacation interruption, where customers' impatience is due to the servers' vacation. During the working vacation period, if there are customers in the queue, the vacation can be interrupted at a service completion instant and the server begins a regular busy period with probability 1 − or continues the vacation with probability . We obtain the probability generating functions of the stationary state probabilities and deduce the explicit expressions of the system sizes when the server is in a normal service period and in a Bernoulli schedule vacation interruption, respectively. Various performance measures such as the mean system size, the proportion of customers served, the rate of abandonment due to impatience, and the mean sojourn time of a customer served are derived. We obtain the stochastic decomposition structures of the queue length and waiting time. Finally, some numerical results to show the impact of model parameters on performance measures of the system are presented.

In this paper we establish the absolute continuity of the law of the solution of stochastic differential equations driven by generalised grey noise under the Hörmander condition. We then derive an upper bound and show the smoothness of the density. Finally we study the stability problem of the solutions.

We show that all local martingales with respect to the initially enlarged natural filtration of a vector of multivariate point processes can be weakly represented up to the minimum among the explosion times of the components. We also prove that a strong representation holds if any multivariate point process of the vector has almost surely infinite explosion time and discrete mark’s space. Then we provide a condition under which the components of the multidimensional local martingale driving the strong representation are pairwise orthogonal.

In this paper we study the Föllmer-Schweizer decomposition of a square integrable random variable ξ with respect to a given semimartingale S under restricted information. Thanks to the relationship between this decomposition and that of the projection of ξ with respect to the given information flow, we characterize the integrand appearing in the Föllmer-Schweizer decomposition under partial information in the general case where ξ is not necessarily adapted to the available information level. For partially observable Markovian models where the dynamics of S depends on an unobservable stochastic factor X, we show how to compute the decomposition by means of filtering problems involving functions defined on an infinite-dimensional space. Moreover, in the case of a partially observed jump-diffusion model where X is described by a pure jump process taking values in a finite dimensional space, we compute explicitly the integrand in the Föllmer-Schweizer decomposition by working with finite dimensional filters.

A partially observable control problem for an R d -valued jump process with counting observations is studied. The state and the observations may be strongly dependent and, in particular, the two processes may jump together. An equivalent separated problem is introduced and the existence of an optimal control for the separated problem is obtained in the class of relaxed and generalized controls. Equivalence between the initial problem and the relaxed generalized separated control problem is discussed.

In this paper we study the Föllmer-Schweizer decomposition of a square integrable random variable ξ with respect to a given semimartingale S under restricted information. Thanks to the relationship between this decomposition and that of the projection of ξ with respect to the given information flow, we characterize the integrand appearing in the Föllmer-Schweizer decomposition under partial information in the general case where ξ is not necessarily adapted to the available information level. For partially observable Markovian models where the dynamics of S depends on an unobservable stochastic factor X, we show how to compute the decomposition by means of filtering problems involving functions defined on an infinite-dimensional space. Moreover, in the case of a partially observed jump-diffusion model where X is described by a pure jump process taking values in a finite dimensional space, we compute explicitly the integrand in the Föllmer-Schweizer decomposition by working with finit...

In this paper use the Stein method to characterize the M-Wright distribution M1 /3 and its symmetrization. The Stein operator is associated with the general Airy equation and the corresponding Stein equation is nothing but a general inhomogeneous Airy equation.

We revisit the Dynkin game problem in a general framework and relax some assumptions. The payoffs and the criterion are expressed in terms of families of random variables indexed by stopping times. We construct two nonnegative supermartingales families J and J whose finitness is equivalent to the Mokobodski's condition. Under some weak right-regularity assumption on the payoff families, the game is shown to be fair and J − J is shown to be the common value function. Existence of saddle points is derived under some weak additional assumptions. All the results are written in terms of random variables and are proven by using only classical results of probability theory.

We study a particular class of stationary random closed sets in R d called Poisson kcylinder models (short: P-k-CM's) for k = 1,. .. , d − 1. We show that all P-k-CM's are weakly mixing and possess long-range correlations. Further, we derive necessary and sufficient conditions in terms of the directional distribution of the cylinders under which the corresponding P-k-CM is mixing. Regarding the P-(d − 1)-CM as union of "thick hyperplanes" which generates a stationary process of polytopes we prove that the distribution of the polytope containing the origin does not depend on the thickness of the hyperplanes.

In this paper we give a survey on some basic ideas related to random utility, extreme value theory and multinomial logit models. These ideas are well known within the field of spatial economics, but do not appear to be common knowledge to researchers in probability theory. The purpose of the paper is to try to bridge this gap.

The case of SPSA algorithms with two trial simultaneous perturbations is discussed. The better asymptotic convergence of the algorithm, depending on the properties of bounding of additional noise in observations is proved. The Lyapunov function with degree from 1 to 2 is considered. The results of are in the base of the research.

We obtain a functional central limit theorem (CLT) for sums of the form , where is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties, is a certain centralizing constant and are arbitrary polynomials taking on positive integer values on positive integers, i.e. polynomials satisfying , where is the set of natural numbers. For polynomial ’s this CLT generalizes [The Annals of Probability, 2014, Vol. 42, No.2, 649–688] which allows only linear ’s to have the same polynomial degree. We also prove that exists and provide necessary and sufficient conditions for its positivity, which is equivalent to the statement that the weak limit of is not zero almost surely. Finally, we study independence properties of the increments of the limiting process. Our proofs require studying asymptotic densities of special subsets of , which is done in a separate section. As in...

Sustaining efficiency and stability by properly controlling the equity to asset ratio is one of the most important and difficult challenges in bank management. Due to unexpected and abrupt decline of asset values, a bank must closely monitor its net worth as well as market conditions, and one of its important concerns is when to raise more capital so as not to violate capital adequacy requirements. In this paper, we model the tradeoff between avoiding costs of delay and premature capital raising, and solve the corresponding optimal stopping problem. In order to model defaults in a bank's loan/credit business portfolios, we represent its net worth by Lévy processes, and solve explicitly for the double exponential jump diffusion process and for a general spectrally negative Lévy process.

We construct a stochastic flow generated by an SDE with its drift being a function of bounded variation and its noise being a stable process with exponent from (1,2). It is proved that the flow is non-coalescing and Sobolev differentiable with respect to initial data. The representation for the derivative is given. t 0 a ′ (ϕ s (x))ds .

We develop a stochastic calculus of divergence type with respect to the fractional Brownian sheet (fBs) with any Hurst parameters in (0, 1) and beyond the fractional scale. We define stochastic integration in the extended Skorohod sense, and derive Itô and Tanaka formulas. In the case of Gaussian fields that are more irregular than fBs for any Hurst parameters, we are able to complete the same program for those Gaussian fields that are almost-surely uniformly continuous.

Let Z H = {Z H (t), t ∈ R N } be a real-valued N-parameter harmonizable fractional stable sheet with index H = (H 1 ,. .. , H N) ∈ (0, 1) N. We establish a random wavelet series expansion for Z H which is almost surely convergent in all the Hölder spaces C γ ([−M, M] N), where M > 0 and γ ∈ (0, min{H 1 ,. .. , H N }) are arbitrary. One of the main ingredients for proving the latter result is the LePage representation for a rotationally invariant stable random measure. Also, let X = {X(t), t ∈ R N } be an R d-valued harmonizable fractional stable sheet whose components are independent copies of Z H. By making essential use of the regularity of its local times, we prove that, on an event of positive probability, the formula for the Hausdorff dimension of the inverse image X −1 (F) holds for all Borel sets F ⊆ R d. This is referred to as a uniform Hausdorff dimension result for the inverse images.

Die in der Zeitschrift veröffentlichten Beiträge sind urheberrechtlich geschützt. Alle Rechte, insbesondere das der Übersetzung in fremde Sprachen, vorbehalten. Kein Teil dieser Zeitschrift darf ohne schriftliche Genehmigung des Verlages in irgendeiner Form-durch Fotokopie, Mikrofilm oder ein anderes Verfahrenreproduziert oder in eine von Maschinen, insbesondere Datenverarbeitungsanlagen, verwendbare Sprache übertragen werden. Auch die Rechte der Wiedergabe durch Vortrag, Funk-und Fernsehsendung, im Magnettonverfahren oder auf ähnlichem Wege bleiben vorbehalten. Fotokopien für den persönlichen oder sonstigen eigenen Gebrauch dürfen nur von einzelnen Beiträgen oder Teilen daraus als Einzelkopie hergestellt werden. Jede im Bereich eines gewerblichen Unternehmens hergestellte oder genützte Kopie dient gewerblichen Zwecken gem. § 54 (2) UrhG und verpflichtet zur Gebührenzahlung an die VG Wort, Abteilung Wissenschaft, Goethestr. 49, 80336 München, von der die einzelnen Zahlungsmodalitäten zu erfragen sind.

Die in der Zeitschrift veröffentlichten Beiträge sind urheberrechtlich geschützt. Alle Rechte, insbesondere das der Übersetzung in fremde Sprachen, vorbehalten. Kein Teil dieser Zeitschrift darf ohne schriftliche Genehmigung des Verlages in irgendeiner Form-durch Fotokopie, Mikrofilm oder ein anderes Verfahrenreproduziert oder in eine von Maschinen, insbesondere Datenverarbeitungsanlagen, verwendbare Sprache übertragen werden. Auch die Rechte der Wiedergabe durch Vortrag, Funk-und Fernsehsendung, im Magnettonverfahren oder auf ähnlichem Wege bleiben vorbehalten. Fotokopien für den persönlichen oder sonstigen eigenen Gebrauch dürfen nur von einzelnen Beiträgen oder Teilen daraus als Einzelkopie hergestellt werden. Jede im Bereich eines gewerblichen Unternehmens hergestellte oder genützte Kopie dient gewerblichen Zwecken gem. § 54 (2) UrhG und verpflichtet zur Gebührenzahlung an die VG Wort, Abteilung Wissenschaft, Goethestr. 49, 80336 München, von der die einzelnen Zahlungsmodalitäten zu erfragen sind.

Let (Ω, B) be a measurable space, An ⊂ B a sub-σ-field and µn a random probability measure on (Ω, B), n ≥ 1. In various frameworks, one looks for a probability P on B such that µn is a regular conditional distribution for P given An for all n. Conditions for such a P to exist are given. The conditions are quite simple when (Ω, B) is a compact Hausdorff space equipped with the Borel or the Baire σ-field (as well as under similar assumptions). Applications to Gibbs measures and Bayesian statistics are given as well. 1. The problem Let (Ω, B) be a measurable space and P the collection of all probability measures on B. For B ∈ B and any map µ : Ω → P, we let µ(B) denote the function on Ω given by ω → µ(ω)(B). We also let σ(µ) = σ µ(B) : B ∈ B. Let P ∈ P and A ⊂ B a sub-σ-field. A regular conditional distribution (r.c.d.), for P given A, is a map µ : Ω → P such that µ(B) is a version of E P I B | A for all B ∈ B. For a r.c.d. to exist, it suffices that P is perfect and B countably generated. This note originates from the following question. Given a sub-σ-field A ⊂ B and a map µ : Ω → P such that σ(µ) ⊂ A, under what conditions is there P ∈ P such that µ is a r.c.d. for P given A ? Such a question is easily answered. Once stated, however, it grows quickly into the following new question. Suppose we are given {A n , µ n : n ∈ I}, where A n ⊂ B is a sub-σ-field, µ n : Ω → P is a map such that σ(µ n) ⊂ A n , and I = {1, 2,. . .} or I = {1,. .. , k} for some k ≥ 1. Under what conditions is there P ∈ P such that µ n is a r.c.d. for P given A n for all n ∈ I ? If such a P exists, the µ n are said to be consistent. We aim to give conditions for the µ n to be consistent; see Theorems 6 and 7. Throughout, M denotes the (possibly empty) set M = {P ∈ P : µ n is a r.c.d. for P given A n for all n ∈ I}. 2. Motivations Reasonable conditions for consistency, if available, are of potential interest. As a first (heuristic) example suppose that, for each n ∈ I, expert n declares his/her opinions on a certain phenomenon conditionally on his/her information A n. This produces a collection of random probability measures µ n : Ω → P such that σ(µ n) ⊂ A n. In this framework, most literature focus on how to summarize the

This paper consists of two independent parts. In the first one, we contribute to the study of the class (Σ). For instance, we provide a new way to characterize stochastic processes of this class. We also present some new properties and solve the Bachelier equation. In the second part, we study the class of stochastic processes Σ(H). This class was introduced in [6] where from tools of the theory of martingales with respect to a signed measure of [19], the authors provide a general framework and methods for dealing with processes of this class. In this work, after developing some new properties, we embed a non-atomic measure ν in X, a process of the class Σ(H). More precisely, we find a stopping time T < ∞ such that the law of X T is ν.

This article deals with IDT processes, i.e. processes which are infinitely divisible with respect to time. Given an IDT process (X t , t ≥ 0), there exists a unique (in law) Lévy process (L t ; t ≥ 0) which has the same one-dimensional marginals distributions, i.e for any t ≥ 0 fixed, we have X t (law) = L t. Such processes are said to be associated. The main objective of this work is to exhibit numerous examples of IDT processes and their associated Lévy processes. To this end, we take up ideas of the monograph Peacocks and associated martingales from F. Hirsch, C. Profeta, B. Roynette and M. Yor (Lévy, Sato and Gaussian sheet methods) and apply them in the framework of IDT processes. This gives a new interesting outlook to the study of processes whose only one-dimensional marginals are known. Also, we give an integrated weak Itô type formula for IDT processes (in the same spirit as the one for Gaussian processes) and some links between IDT processes and selfdecomposability. The last sections are devoted to the study of some extensions of the notion of IDT processes in the weak sense as well as in the multiparameter sense. In particular, a new approach for multiparameter IDT processes is introduced and studied. Main examples of this kind of processes are the R N +-parameter Lévy process and the Lévy's R M-parameter Brownian motion. These results give some better understanding of IDT processes, and may be seen as some continuation of the works of K. Es-Sebaiy and Y. Ouknine [How rich is the class of processes which are infinitely divisible with respect to time ? ] and R. Mansuy [On processes which are infinitely divisible with respect to time].

We present two examples of loss of the predictable representation property for semi-martingales by enlargement of the reference filtration. First of all we show that the predictable representation property for a semi-martingale X does not transfer from the reference filtration F to a larger filtration G if the information starts growing up to a positive time. Then we study the case G = F ∨ H when there exists a second special semi-martingale Y enjoying the predictable representation property with respect to H. We establish conditions under which the triplet (X, Y, [X, Y ]) enjoys the predictable representation property with respect to G.

Let X and Y be an m-dimensional F-semi-martingale and an n-dimensional H-semi-martingale respectively on the same probability space (Ω, F, P), both enjoying the strong predictable representation property. We propose a martingale representation result for the square-integrable (P, G)-martingales, where G = F ∨ H. As a first application we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of d i=1 F i , where, fixed i ∈ (1,. .. , d), F i is the reference filtration of a real martingale M i , which enjoys the (P, F i) predictable representation property. A second application falls into the framework of credit risk modeling and in particular into the study of the progressive enlargement of the market filtration by a default time. More precisely, when the risky asset price is a multidimensional semi-martingale enjoying the strong predictable representation property and the default time satisfies the density hypothesis, we present a new proof of the analogous of the classical Kusuoka's theorem.

We consider stochastic difference equation x n+1 = x n 1 − hf (x n) + √ hg(x n)ξ n+1 , n = 0, 1,. .. , x 0 ∈ R 1 , where functions f and g are nonlinear and bounded, random variables ξ i are independent and h > 0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution x n ≡ 0. We also show, that for some natural choices of the nonlinearities f and g, the rate of decay of x n is approximately polynomial: we find α > 0 such that x n decays faster than n −α+ε but slower than n −α−ε for any ε > 0. It also turns out that if g(x) decays faster than f (x) as x → 0, the polynomial rate of decay can be established exactly, x n n α → const. On the other hand, if the coefficient by the noise does not decay fast enough, the approximate decay rate is the best possible result.

Using the martingale approach we find sufficient conditions for exponential boundedness of first passage times over a level for ergodic first order autoregressive sequences (AR ). Further, we prove a martingale identity to be used in obtaining explicit bounds for the expectation of first passage times.