This thesis focuses on stochastic processes and some of their properties are investigated which are necessary to determine the tools, the extremal index and the extremogram. Both mathematical tools measure extremal dependency within random time series. Two different models are introduced and related properties are discussed. The probability function of the Agent based model is surveyed explicitly and strong stationarity is proven. Data sets for both processes are simulated and clustering of the data is investigated with two different methods. Finally an estimation of the extremogram is used to interpret dependency of extremes within the data.

We introduce a class of one-dimensional continuous reflected backward stochastic Volterra integral equations driven by Brownian motion, where the reflection keeps the solution above a given stochastic process (lower obstacle). We prove existence and uniqueness by a fixed point argument and derive a comparison result. Moreover, we show how the solution of our problem is related to a time-inconsistent optimal stopping problem and derive an optimal strategy. 

The purpose of this paper is twofold. First, we extend the well-known relation between optimal stopping and randomized stopping of a given stochastic process to a situation where the available information flow is a filtration with no a priori assumed relation to the filtration of the process. We call these problems optimal stopping and randomized stopping with general information. we introduce a special singular stochastic control problem with general information and show that this is also equivalent to the partial information optimal stopping and randomized stopping problems. Then we show that the solution of this singular control problem can be expressed in terms of partial information variational inequalities.

We consider the problem of optimal control of a mean-field stochasticdifferential equation (SDE) under model uncertainty. The model uncertaintyis represented by ambiguity about the law LðXðtÞÞ of the stateX(t) at time t. For example, it could be the law LPðXðtÞÞ of X(t) withrespect to the given, underlying probability measure P. This is the classicalcase when there is no model uncertainty. But it could also be thelaw LQðXðtÞÞ with respect to some other probability measure Q or,more generally, any random measure lðtÞ on R with total mass 1. Werepresent this model uncertainty control problem as a stochastic differentialgame of a mean-field related type SDE with two players. Thecontrol of one of the players, representing the uncertainty of the lawof the state, is a measure-valued stochastic process lðtÞ and the controlof the other player is a classical real-valued stochastic process u(t).This optimal control problem with respect to random probability processeslðtÞ in a non-Markovian setting is a new type of stochastic controlproblems that has not been studied before. By constructing a newHilbert space M of measures, we obtain a sufficient and a necessarymaximum principles for Nash equilibria for such games in the generalnonzero-sum case, and for saddle points in zero-sum games. As anapplication we find an explicit solution of the problem of optimal consumptionunder model uncertainty of a cash flow described by amean-field related type SDE.

Landstinget i Kronobergs län utförde 2006 en enkätundersökning bland elever i årskurserna 5, 8 och gymnasiets årskurs 2. Datamaterialet för gymnasieeleverna har studerats i detta arbete där den psykiska ohälsan och alkoholkonsumtion varit i fokus.

Det statistiska arbetet är uppdelat i tre delar: Metodbeskrivning, hantering av programmet SPSS samt en undersökning av gymnasieelevers psykiska ohälsa och alkoholkonsumtion. I metodbeskrivningen har metoderna dikotomisering, logistisk regression, Wald, faktorinteraktion och kategorisering beskrivits. För hantering av stora datamaterial samt för att kunna ta fram statistiska samband rörande psykisk ohälsa och alkoholkonsumtion har programmet SPSS använts och illustrerats i arbetet. De förklarande variablerna som varit mest inflytelserika mot den psykiska ohälsan var ’Tycker du att du är frisk?’, ’Har du någonsin använt läkemedel tillsammans med alkohol i berusningssyfte’ samt ’Trivs du med livet?’. När undersökningen för gymnasieelevers alkoholkonsumtion studerades var de gemensamma signifikanta faktorerna för både killar och tjejer ’Skolkar du?’, ’Känner du dig ofta irriterad eller på dåligt humör?’ och ’Har någon vuxen behandlat dig fysiskt illa?’ där samtliga förklarande faktorer bidrog till en ökad sannolikhet att konsumera mycket alkohol. Då undersökning av variabeln symtom, variabeln oro samt deras interaktion studerades för killar respektive för tjejer mot den psykiska ohälsan kunde man se att variabeln symtom för killar var inflytelserik samt interaktionen mellan oro och symtom. För tjejerna var däremot variabeln oro inflytelserik samt variabeln symtom men inte deras interaktion.

In this paper, a sample estimator of the tangency portfolio (TP) weights is con-sidered. The focus is on the situation where the number of observations is smaller than the number of assets in the portfolio and the returns are i.i.d. normally distributed. Under these as-sumptions, the sample covariance matrix follows a singular Wishart distribution and, therefore, the regular inverse cannot be taken. In the paper, bounds and approximations for the first two moments of the estimated TP weights are derived, as well as exact results are obtained when the population covariance matrix is equal to the identity matrix, employing the Moore-Penrose inverse. Moreover, exact moments based on the reflexive generalized inverse are provided. The properties of the bounds are investigated in a simulation study, where they are compared to the sample moments. The difference between the moments based on the reflexive generalized inverse and the sample moments based on the Moore-Penrose inverse is also studied.

In this paper, we suggest a unit root test for a system of equations using a spectral variance decomposition method based on the Maximal Overlap Discrete Wavelet Transform. We obtain the limiting distribution of the test statistic and study its small sample properties using Monte Carlo simulations. We find that, for multiple time series of small lengths, the wavelet-based method is robust to size distortions in the presence of cross-sectional dependence. The wavelet-based test is also more powerful than the Cross-sectionally Augmented Im et al. unit root test (Pesaran, M. H. 2007. "A Simple Panel Unit Root Test in the Presence of Cross-section Dependence." Journal of Applied Econometrics 22 (2): 265-312.) for time series with between 20 and 100 observations, using systems of 5 and 10 equations. We demonstrate the usefulness of the test through an application on evaluating the Purchasing Power Parity theory for the Group of 7 countries and find support for the theory, whereas the test by Pesaran (Pesaran, M. H. 2007. "A Simple Panel Unit Root Test in the Presence of Cross-section Dependence." Journal of Applied Econometrics 22 (2): 265-312.) finds no such support. © 2019 Walter de Gruyter GmbH, Berlin/Boston.

The multiple time series and ridge regression techniques are proposed for modeling and analyzing a scaled real life (or a simulated) data as a SUR model with VAR(p) disturbances. The regression coefficients are estimated via the generalized least squares method if collinearity is weak and otherwise the regression coefficients are estimated by the generalized ridge regression method. Small sample likelihood ratio test statistic and model selection criteria are employed for selecting the smallest possible lag order for the VAR process. Moreover, Monte Carlo simulations (1000 replications) are conducted to examine the properties of some new and some of the existing ridge parameters in rectifying the collinearity problem in SUR models with VAR(2) disturbances via the trace(MSE) and condition number criteria. Two data sets are analyzed to illustrate the findings of the article.

This paper utilizes Wavelet based methodology to estimate and test for trends and granger causality in temperature andprecipitation. We use quarterly data from Sweden for the period 1884 up to 2011. The analysis suggests that temperatureand precipitation in Sweden currently have a positive trend in 2011. Thus the recent lower levels of the variables 2009-2010are estimated to be temporary fluctuations or deviations from the trend. Moreover, in the short run there are feedbackeffects between the variables and over longer periods, 4-8 years, temperature granger cause precipitation.

In this paper we outline a framework for forecasting using maximal overlap discrete wavelet transform (MODWT) based multiresoulution analysis (MRA). This framework has been applied for forecasting the tourism arrival series from Denmark to Norway. We compare forecasted values obtained from modeling the data in the time domain with the forecasted values from the wavelet domain using the traditional Box-Jenkins methodology. In both cases, diagnostic tests have been conducted to insure the specification of the model. The results have shown that the wavelet based forecasts outperforms the traditional Box-Jenkins approach in term of forecasts accuracy.

In this thesis we generalize results by Smoluchowski [43], Chandrasekhar[6], Kramers, and Nelson [30]. Their aim is to construct Brownian motion as a limit of stochastic processes with differentiable sample paths by exploiting a scaling limit which is a particular type of averaging studied by Papanicolao [35]. Their construction of Brownian motion differs from the one given by Einstein since it constitutes a dynamical theory of Brownian motion. Nelson sets off by studying scaled standard Ornstein-Uhlenbeck processes. Physically these describe classical point particles subject to a deterministic friction and an external random force of White Noise type, which models perpetuous collisions with surrounding(water) molecules. Nelson also studies the case when the particles are subject to an additional deterministic nonlinear force. The present thesis generalizes the work of Chandrasekhar in that it deals with finite dimensional α-stable Lévy processes with 0 < α < 2, and Fractional Brownian motion as driving noises and mathematical techniques like deterministic time change and a Girsanov theorem. We consider uniform convergence almost everywhere and in -sense. In order to pursue the limit we multiply all vector fields in the cotangent space by the scaling parameter including the noise. For α-stable Lévy processes this correspondsto scaling the process in the tangent space, , , according to . Sending β to infinity means sending time to infinity. In doing so the noise evolves with a different speed in time compared to the component processes. For α≠2, α-stable Lévy processes are of pure jump type, therefore the approximation by processes having continuous sample paths constitutes a valuable mathematical tool. α-stable Lévy processes exceed the class studied by Zhang [46]. In another publication related to this thesis we elaborate on including a mean-field term into the globally Lipschitz continuous nonlinear part of the drift while the noise is Brownian motion, whereas Narita [28] studied a linear dissipation containing a mean-field term. Also the classical McKean-Vlasov model is linear in the mean-field. In a result not included in this thesis the scaling result of Narita [29], which concerns another scaling limit of the tangent space process (velocity) towards a stationary distribution, is generalized to α-stable Lévy processes. The stationary distribution derived by Narita is related to the Boltzmann distribution. In the last part of this thesis we study Fractional Brownian motion with a focus on deriving a scaling limit of Smoluchowski-Kramers type. Since Fractional Brownian motion is no semimartingale the underlying theory of stochastic differential equations is rather involved. We choose to use a Girsanov theorem to approach the scaling limit since the exponent in the Girsanov denvsity does not contain the scaling parameter explicitly. We prove that the Girsanov theorem holds with a linear growth condition alone on the drift for 0 < H < 1, where H is the Hurst parameterof the Fractional Brownian motion.

Edward Nelson derived Brownian motion from Ornstein-Uhlenbeck theory by a scaling limit. Previously we extended the scaling limit to an Ornstein-Uhlenbeck process driven by an α-stable Lévy process. In this paper we extend the scaling result to α-stable Lévy processes in the presence of a nonlinear drift, an external field of force in physical terms.

Brownian motion has met growing interest in mathematics, physics and particularly in finance since it was introduced in the beginning of the twentieth century. Stochastic processes generalizing Brownian motion have influenced many research fields theoretically and practically. Moreover, along with more refined techniques in measure theory and functional analysis more stochastic processes were constructed and studied. Lévy processes, with Brownian motionas a special case, have been of major interest in the recent decades. In addition, Lévy processes include a number of other important processes as special cases like Poisson processes and subordinators. They are also related to stable processes.

In this thesis we generalize a result by S. Chandrasekhar [2] and Edward Nelson who gave a detailed proof of this result in his book in 1967 [12]. In Nelson’s first result standard Ornstein-Uhlenbeck processes are studied. Physically this describes free particles performing a random and irregular movement in water caused by collisions with the water molecules. In a further step he introduces a nonlinear drift in the position variable, i.e. he studies the case when these particles are exposed to an external field of force in physical terms.

In this report, we aim to generalize the result of Edward Nelson to the case of α-stable Lévy processes. In other words we replace the driving noise of a standard Ornstein-Uhlenbeck process by an α-stable Lévy noise and introduce a scaling parameter uniformly in front of all vector fields in the cotangent space, even in front of the noise. This corresponds to time being sent to infinity. With Chandrasekhar’s and Nelson’s choice of the diffusion constant the stationary state of the velocity process (which is approached as time tends to infinity) is the Boltzmann distribution of statistical mechanics.The scaling limits we obtain in the absence and presence of a nonlinear drift term by using the scaling property of the characteristic functions and time change, can be extended to other types of processes rather than α-stable Lévy processes.

In future, we will consider to generalize this one dimensional result to Euclidean space of arbitrary finite dimension. A challenging task is to consider the geodesic flow on the cotangent bundle of a Riemannian manifold with scaled drift and scaled Lévy noise. Geometrically the Ornstein-Uhlenbeck process is defined on the tangent bundle of the real line and the driving Lévy noise is defined on the cotangent space.

We derive a Smoluchowski-Kramers type scaling limit for second order stochastic differential equations driven by Fractional Brownian motion.We show a Girsanov theorem for the solution processes with respect to corresponding Fractional Ornstein-Uhlenbeck processes which are Gaussian. This reveals existence of weak solutions as well as a weak scaling limit. Subsequently the results are strengthened.

We establish a scaling limit for autonomous stochastic Newton equations, the solutions are often called nonlinear stochastic oscillators,where the nonlinear drift includes a mean field term of Mckean type and the driving noise is Gaussian. Uniform convergence in  sense is achieved by applying -type estimates and the Gronwall Theorem.The approximation is also called Smoluchowski-Kramers limit and is a particular averaging technique studied by Papanicolaou. It reveals an approximation of diffusions with a mean-field contribution in the drift by diffusions with differentiable trajectories.

We introduce a new formulation of reflected backward stochastic differential equations (BSDEs) and doubly reflected BSDEs associated with irregular obstacles. In the first part of the paper, we consider an extension of the classical optimal stopping problem over a larger set of stopping systems than the set of stopping times (namely, the set of split stopping times), where the payoff process   is irregular and in the case of a general filtration. Split stopping times are a powerful tool for modeling financial contracts and derivatives that depend on multiple conditions or triggers, and for incorporating stochastic processes with jumps and other types of discontinuities. We show that the value family can be aggregated by an optional process v, which is characterized as the Snell envelope of the reward process   over split stopping times. Using this, we prove the existence and uniqueness of a solution Y to irregular reflected BSDEs. In the second part of the paper, motivated by the classical Dynkin game with completely irregular rewards considered by Grigorova et al. (Electron J Probab 23:1–38, 2018), we generalize the previous equations to the case of two reflecting barrier processes.

We study a second order stochastic differential equation which was derived from a non-linear Schrödinger equation with non-linear damping and additive noise to describe the width of related wave solutions. We prove existence of solutions and discuss their long-time behaviour.

We show that a physically motivated trial solution of a dampeddriven non-linear Schr ̈odinger equation does neither encounter collapse norso-called pseudocollapse although the exponent of the non-linearity is crit-ical. This result sheds new light on the accuracy of numerical solutions tothis problem obtained in an earlier paper where the authors claim pseudo-collapse of the trial solution when the variance of the driving noise is below a certain level.

We investigate mathematical tools, Edgeworth series expansion and the saddle point method, which are approximation techniques that help us to estimate the distribution function for the standardized mean of independent identical distributed random variables where we will take into consideration the lattice case. Later on we will describe one important application for these mathematical tools where game developing companies can use them to reduce the amount of time needed to satisfy their standard requests before they approve any game

The mean-field game theory is the study of strategic decision making in very large populations of weakly interacting individuals. Mean-field games have been an active area of research in the last decade due to its increased significance in many scientific fields. The foundations of mean-field theory go back to the theory of statistical and quantum physics. One may describe mean-field games as a type of stochastic differential game for which the interaction between the players is of mean-field type, i.e the players are coupled via their empirical measure. It was proposed by Larsy and Lions and independently by Huang, Malhame, and Caines. Since then, the mean-field games have become a rapidly growing area of research and has been studied by many researchers. However, most of these studies were dedicated to diffusion-type games. The main purpose of this thesis is to extend the theory of mean-field games to jump case in both discrete and continuous state space. Jump processes are a very important tool in many areas of applications. Specifically, when modeling abrupt events appearing in real life. For instance, financial modeling (option pricing and risk management), networks (electricity and Banks) and statistics (for modeling and analyzing spatial data). The thesis consists of two papers and one technical report which will be submitted soon:

In the first publication, we study the mean-field game in a finite state space where the dynamics of the indistinguishable agents is governed by a controlled continuous time Markov chain. We have studied the control problem for a representative agent in the linear quadratic setting. A dynamic programming approach has been used to drive the Hamilton Jacobi Bellman equation, consequently, the optimal strategy has been achieved. The main result is to show that the individual optimal strategies for the mean-field game system represent 1/N-Nash equilibrium for the approximating system of N agents.

As a second article, we generalize the previous results to agents driven by a non-linear pure jump Markov processes in Euclidean space. Mathematically, this means working with linear operators in Banach spaces adapted to the integro-differential operators of jump type and with non-linear partial differential equations instead of working with linear transformations in Euclidean spaces as in the first work. As a by-product, a generalization for the Koopman operator has been presented. In this setting, we studied the control problem in a more general sense, i.e. the cost function is not necessarily of linear quadratic form. We showed that the resulting unique optimal control is of Lipschitz type. Furthermore, a fixed point argument is presented in order to construct the approximate Nash Equilibrium. In addition, we show that the rate of convergence will be of special order as a result of utilizing a non-linear pure jump Markov process.

In a third paper, we develop our approach to treat a more realistic case from a modelling perspective. In this step, we assume that all players are subject to an additional common noise of Brownian type. We especially study the well-posedness and the regularity for a jump version of the stochastic kinetic equation. Finally, we show that the solution of the master equation, which is a type of second order partial differential equation in the space of probability measures, provides an approximate Nash Equilibrium. This paper, unfortunately, has not been completely finished and it is still in preprint form. Hence, we have decided not to enclose it in the thesis. However, an outlook about the paper will be included.

We investigate mean-field games from the point of view of a large number of indistinguishable players, which eventually converges to infinity. The players are weakly coupled via their empirical measure. The dynamics of the states of the individual players is governed by a non-autonomous pure jump type semi group in a Euclidean space, which is not necessarily smoothing. Investigations are conducted in the framework of non-linear Markovian semi groups. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean–Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal an epsilon-NashEquilibrium for the N-player games.

We investigate mean field games from the point of view of a large number of indistinguishable players which eventually converges to in- finity. The players are weakly coupled via their empirical measure. The dynamics of the individual players is governed by pure jump type propagators over a finite space. Investigations are conducted in the framework of non-linear Markov processes. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean-Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal a 1 N -Nash Equilibrium for the approximating system of N players.

This study attempts to improve forecasts of solar energy production (SEP), so that energy trading companies can propose more accurate bids to Nord Pool. The aim ismake solar energy a more lucrative business, and therefore lead to more investments in this green energy form. The model that is introduced is a hidden Markov model (HMM) that we call a Time-dynamic Markov-chain (TDMC). The TDMC is presented in general, but applied to the energy sector SE4 in south of Sweden. A simple linear regression model is used to compare with the performance of the TDMC model. Regarding the mean absolute error (MAE) and the root-mean-square error (RMSE), the TDMC model outperforms a simple linear regression; both when the training data is relatively fresh and also when the training data has not been updated in over 300 days. A paired t-test also shows a non-significant deviation from the true SEP per day, at the 0.05 significance level, when simulating the first two months of 2023 with the TDMC model. The simple linear regression model, however, shows a significant difference from reality, in comparison.

Syftet med studien är att redovisa hur läromedlen Matte Direkt Borgen 5A, Mera favorit matematik 5b och Gleerups matematik 5 presenterade lägesmått i olika elevuppgifter. Studien har även en avsikt att granska hur lägesmått framhävs i olika problemlösningsuppgifter. 

Inom resultatanalysen har variationsteorin används vid analys. Begrepp som har varit centrala inom teorin är lärandeobjekt, kritiska aspekter, variationsmönster, kontrast, generalisering, separation och fusion. Utöver har egenkomponerade frågor används för att besvara frågeställningarna. 

Resultatanalysen är uppdelat utifrån frågeställningarna. Där underrubrikerna namngavs efter läromedlets titlar. Anledningen till det var att bistå med en tydlighet i resultatet. För att se hur läromedlet framhäver de olika matematiska områdena. I varje underrubrik berörs de ingående begrepp inom variationsteorin, med hjälp av framtagna uppgifter som blivit analyserade. 

Avslutningsvis kan studiens resultat bidra till lärarens undervisning om lägesmått eftersom ett vanligt förekommande är att undervisningen är baserad på läromedlet. Den kan även vara behjälplig med hur problemlösningsuppgifter med lägesmått kan presenteras för att få en vetskap om vad som saknas.

Some diseases such as herpes, bovine and human tuberculosis exhibit relapse in which the recovered individuals do not acquit permanent immunity but return to infectious class. Such diseases are modeled by SIRI models. In this paper, we establish the existence of a unique global positive solution for a stochastic epidemic model with relapse and jumps. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we present some numerical results to support the theoretical work.

Since many macroeconomic models are linear, it is not desirable to use themwith an asymmetric dependent variable. In this article, we formulate aunivariate test for symmetry, based on the third central moment and extendit to a multivariate test; the test does not require modelling and it is robustagainst serial correlation, Autoregressive Conditional Heteroscedasticity(ARCH) and nonnormality. In the empirical application of the test it isfound that orthodox theory seem to be supported; consumption expendi-ture on durable goods is found to be symmetric while consumptionexpenditure on nondurable goods is asymmetric for the USA and the UK,with peaks being higher than troughs are deep. Also, the empiricalimportance of the choice between the univariate and the multivariate testfor possibly correlated series is underscored; the results from the twoapproaches  clearly  differ.  Given  the  widespread  practice  of  usingconsumption expenditure on nondurable goods as the dependent variablein linear models for the USA and the UK, our results might be noteworthy.

We give an outlook, how to realize the ideas of complex scaling from [15–17] to phase space path integrals in the framework of White Noise Analysis. The idea of this scaling method goes back to [9]. Therefore we extend the concept complex scaling to scaling with suitable bounded operators.

The concepts of Feynman integrals in white noise analysis are used to construct the Feynman integrand for the harmonic oscillator in momentum space representation as a Hida distribution. Moreover it is shown that in a limit sense, the potential free case fulfills the conservation of momentum.

In this chapter we summarize applications of White Noise Analysis within the framework of Feynman Path Integrals. With more than 30 years of intensive study this field can be seen as one of the most important applications. Of course this chapter cannot give an exhaustive summary of all approaches to different path integrals in White Noise Analysis. Here we give a slice through different attemtps to path integrals in the White Noise framework. 

Mehler's formula is an important tool in Gaussian analysis. In this article, we study two generalizations of Mehler's formula for the Ornstein–Uhlenbeck semigroup, i.e. the semigroup generated by the number operator. The first generalization leads to transformation groups which have as infinitesimal generator a perturbation of the number operator with suitable integral kernel operators, which are well studied in white noise analysis. For the second one, we characterize the complex Hida measures for which a version of Mehler's formula for the Ornstein–Uhlenbeck semigroup can be extended to. We apply this result to the Feynman integrand for a quadratic potential. Here the time independent eigenstates of the considered transformation groups and the time evolution of eigenvalues are provided.

We use an off-lattice discretization of fractional Brownian motion (fBm) and a Metropolis algorithm to determine the asymptotic scaling of this discretized fBm under the influence of an excluded volume as in the Edwards and Domb–Joyce models. We find a good agreement between the Flory index describing the scaling of end-to-end length with a mean field formula proposed earlier for this class of models.

In this article we present recent results in our ongoing study for weakly self-avoiding fractional processes leading to polymer models. In particular we sketch the results for stars and loops. For fractional random walks we give an explicit formula for the spring constants in the bead-spring model. Furthermore recent findings for the scaling properties of a weakly self-avoiding fractional Brownian motion are presented.

Various authors have invoked discretized fractional Brownian motion (fBm) as a model for chain polymers with long-range interaction of monomers along the chain. We show that for these, in contrast to the Brownian case, linear forces are acting between all pairs of constituents, attractive for small Hurst index H and mostly repulsive when H is larger than 1/2. In the second part of this paper, we extend this study to periodic fBm and related models with a view to ring polymers with long range interactions.

We identify the integrand for the Hamiltonian path integral in space representation as a Kondratiev distribution. For this purpose we use methods from white noise analysis to compute also the Green's function of the underlying Schrödinger equation. We show that its generalized expectation solves the Schrödinger equation and that a functional form of the canonical commutation realtions is fulfilled.

We derive solutions of the Ornstein-Uhlenbeck and of a linear Wick-type stochastic differential equation (SDE) driven by grey Brownian motion. The solutions are characterized to be in a suitable distribution space.

In this paper we introduce and study three classes of fractional periodic processes. An application to ring polymers is investigated. We obtain closed analytic expressions for the form factors, the Debye functions and their asymptotic decay. The relation between the end-to-halftime and radius of gyration is computed for these classes of periodic processes. 

In this paper, we study the self-intersection local times of multifractional Brownian motion (mBm) in higher dimensions in the framework of white noise analysis. We show that when a suitable number of kernel functions of self-intersection local times of mBm are truncated then we obtain a Hida distribution. In addition, we present the expansion of the self-intersection local times in terms of Wick powers of white noises. Moreover, we obtain the convergence of the regularized truncated self-intersection local times in the sense of Hida distributions.

In [10] the existence of a diffusion process whose invariant measure is the fractional polymer or Edwards measure for fractional Brownian motion in dimension d ∈ ℕ with Hurst parameter H ∈ (0, 1) fulfilling dH < 1 is shown. This Markov process is constructed via Dirichlet form techniques in infinite-dimensional (Gaussian) analysis. This article uses these results as starting point. In particular, we provide a Fukushima decomposition for the stochastic quantization of the fractional Edwards measure and prove that the constructed process solves a stochastic differential equation in infinite dimension for quasi-all starting points in a probabilistically weak sense. Moreover, the solution process is driven by an Ornstein–Uhlenbeck process taking values in an infinite-dimensional distribution space and is unique, in the sense that the underlying Dirichlet form is Markov unique. The equilibrium measure, which is by construction the fractional Edwards measure, is specified to be an extremal Gibbs state and therefore the constructed stochastic dynamics is time ergodic. The studied stochastic differential equation provides in the language of polymer physics the dynamics of the bonds, i.e. stochastically spoken the noise of the process. An integration leads then to polymer paths. 

In this paper, we investigate the potential for a class of non-Gaussian processes so-called generalized grey Brownian motion. We obtain a closed analytic form for the potential as an integral of the M-Wright functions and the Green function. In particular, we recover the special cases of Brownian motion and fractional Brownian motion. In addition, we give the connection to a fractional partial differential equation and its the fundamental solution.

We present the expansion of the multifractional Brownian motion (mBm) local time in higher dimensions, in terms of Wick powers of white noises (or multiple Wiener integrals). If a suitable number of kernels is subtracted, they exist in the sense of generalized white noise functionals. Moreover, we show the convergence of the regularized truncated local times for mBm in the sense of Hida distributions. 

In this paper, we investigate the representation of a class of non-Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular, the underlying process can be seen as a non-Gaussian extension of the Ornstein–Uhlenbeck process, hence generalizing the representation results of Muravlev, Russian Math. Surveys 66 (2), 2011 as well as Harms and Stefanovits, Stochastic Process. Appl. 129, 2019 to the non-Gaussian case. 

In this article we set up a microscopic model for the spread of an infectious disease based on configuration space analysis. Using the so-called Vlasov scaling we obtained the corresponding mesoscopic (kinetic) equations, describing the density of susceptible and infected individuals (particles) in space. The resulting system of equations can be seen as a generalization to a ‘spatial’ SIS-model. The equations showing up in the limiting system are of the type which is know in literature as Fisher–Kolmogorov–Petrovsky–Piscounov type.

In recent years much about the modeling and understanding of various types of disease spreading and epidemic behavior have been studied. In principle one can distinguish two types of models for disease spread. On the one hand there is the classical SIR-model from Kermack and McKendrick [1] which describes the time evolution of the number of susceptible (S), infected (I) and recovered (R) individuals by a system of ordinary differential equations. This model has been developed and extended exhaustively in the last 90 years. Among those extensions are the introduction of new compartments to model vector-bourne diseases, see e.g. [2], delay equations to model incubation time, e.g. [3], models considering the age and wealth structure etc. Recently, models with fractional derivatives have also been considered[4]. Unfortunately, we are unable to provide a detailed account concerning this subject and refer the interested reader to [5]. A main drawback of the models described above is that they do not provide any information about the spatial spread of a disease. Nevertheless, there have been various approaches to link many different SIR-areas to obtain spatial behavior. In the SIR-model case, an advection-diffusion equation has been identified as the limiting equation,see e.g. [6]. Another approach in incorporating spatial information for the SIR-model may also be found in [7]. Although the SIR-model and all its extensions are very flexible in describing the different aspects of disease dynamics, the modeling assumptions of the disease spread is purely on the macroscopic level. However, for many different diseases the infection mechanism is only known on the microscopic, i.e., particle-to-particle or individ uumto-individuum level. One way to consider both microscopical modeling and spatial resolution is to describe of the disease dynamics by means of an interacting particle system with suitable interaction potentials. Fundamental in this area are dynamics in so-called marked configuration spaces [8]. These techniques together with a proper scaling of the microscopic system,the so-called Vlasov scaling, have been recently used to model the dynamics of cancer cells [9]. In our approach the components of particle configurations consist of susceptible and infected/infective particles that interact with one another. One may also easily incorporate other types of particles to model recovery or short time immunity. Themicroscopic dynamics then results from suitable ”spin-flip”-processes (particle changes the type).This article shows first numerical results for the SIS-system without movement. The SIS-system allows infective/infected particles to recover and become susceptible again. The numerical methods are based on the analysis provided in [10].

We prove that there exists a diffusion process whose invariant measure is the fractional polymer or Edwards measure for fractional Brownian motion, μg , H, H∈ (0 , 1) for dH< 1. The diffusion is constructed in the framework of Dirichlet forms in infinite dimensional (Gaussian) analysis. Moreover, the process is invariant under time translations.

We extend Varadhan’s construction of the Edwards model forpolymers to fractional Brownian loops and fractional Brownian starbursts.We show that, as in the fBm case, the Edwards density under a renormalizaionis an integrable function for the case Hd ≤ 1.

The concepts of phase space Feynman integrals in White Noise Analysis are established. As an example the harmonic oscillator is treated. The approach perfectly reproduces the right physics. I.e., solutions to the Schrodinger equation are obtained and the canonical commutation relations are satisfied. The later can be shown, since we not only construct the integral but rather the Feynman integrand and the corresponding generating functional.

The concepts of Hamiltonian Feynman integrals in white noise analysis are used to realize as the first velocity-dependent potential of the Hamiltonian Feynman integrand for a charged particle in a constant magnetic field in coordinate space as a Hida distribution. For this purpose the velocity-dependent potential gives rise to a generalized Gauss kernel. Besides the propagators, the generating functionals are obtained. 

The concepts of Feynman integrals in white noise analysis areused to realize the Feynman integrand for a charged particle in a constantmagnetic field as a Hida distribution. For this purpose the velocity dependentpotential gives rise to a generalized Gauss kernel.