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In this paper, we provide variation of constants formulae for linear (forward) stochastic Volterra integral equations (SVIEs, for short) and linear Type-II backward stochastic Volterra integral equations (BSVIEs, for short) in the usual Itô's framework. For these purposes, we define suitable classes of stochastic Volterra kernels and introduce new notions of the products of adapted L 2 -processes. Observing the algebraic properties of the products, we obtain the variation of constants formulae by means of the corresponding resolvent. Our framework includes SVIEs with singular kernels such as fractional stochastic differential equations. Also, our results can be applied to general classes of SVIEs and BSVIEs with infinitely many iterated stochastic integrals. The duality principle between generalized SVIEs and generalized BSVIEs is also proved. [ABSTRACT FROM AUTHOR]

We consider the problem of reconstructing a measurable function over a Locally Compact Abelian group G from random measurements. The results presented herein are partially inspired by the concept of alias-free sampling. Here, the sampling and interpolation operation is modelled as an approximate convolution operator with respect to a stochastic integral defined with an appropriately chosen random measure. In particular, this includes the case where the random sampling points are chosen accordingly to a Poisson random point process. We provide sufficient conditions that guarantee an approximate reconstruction through a sampling process that is similar to alias-free random sampling. These results are applied to the problem of approximating the inverse Radon transform of a function. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). First, the well-posedness of BDSVIEs in the sense of introduced M-solutions is established. Second, a comparison theorem of BDSVIEs is proved. As an application of the comparison theorem, we derive the existence of solutions of BDSVIEs with continuous coefficients. Third, a duality principle between linear forward doubly stochastic Volterra integral equations (FDSVIEs, for short) and BDSVIEs is obtained. Moreover, by virtue of the duality principle, a maximum principle of Pontryagin type is established for an optimal control problem of FDSVIEs. [ABSTRACT FROM AUTHOR]

Operator scaling Gaussian random fields, as anisotropic generalizations of self-similar fields, know an increasing interest for theoretical studies in the literature. However, up to now, they were only defined through stochastic integrals, without explicit covariance functions. In this paper we exhibit explicit covariance functions, as anisotropic generalizations of fractional Brownian fields ones, and define corresponding Operator scaling Gaussian random fields. This allows us to propose a fast and exact method of simulation in dimension 2 based on the circulant embedding matrix method, following ideas of Stein [34] for fractional Brownian surfaces syntheses. This is a first piece of work to popularize these models in anisotropic spatial data modeling. [ABSTRACT FROM AUTHOR]

We consider the Wiener integral with respect to a d -parameter Hermite process with Hurst multi-index H = (H 1 ,.. , H d) ∈ (1 2 , 1) d and we analyze the limit behavior in distribution of this object when the components of H tend to 1 and/or 1 2. As examples, we focus on the solution to the stochastic heat equation with additive Hermite noise and to the Hermite Ornstein-Uhlenbeck process. [ABSTRACT FROM AUTHOR]

Highlights • Population structure can have a significant impact on the outcome of evolution. • Using techniques from physics we approximate evolutionary dynamics on graphs. • Existing pair approximation models are derived directly from the master equation. • Two new models are proposed to approximate the dynamics of individual nodes. • These models facilitate analysis of how initial conditions impact the process. Abstract Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the evolutionary process. However, for more complicated heterogeneous structures, computationally intensive methods are required such as individual-based stochastic simulations. By adapting methods from statistical physics, including moment closure techniques, we first show how to derive existing homogenised pair approximation models and the exact neutral drift model. We then develop node-level approximations to stochastic evolutionary processes on arbitrarily complex structured populations represented by finite graphs, which can capture the different dynamics for individual nodes in the population. Using these approximations, we evaluate the fixation probability of invading mutants for given initial conditions, where the dynamics follow standard evolutionary processes such as the invasion process. Comparisons with the output of stochastic simulations reveal the effectiveness of our approximations in describing the stochastic processes and in predicting the probability of fixation of mutants on a wide range of graphs. Construction of these models facilitates a systematic analysis and is valuable for a greater understanding of the influence of population structure on evolutionary processes. [ABSTRACT FROM AUTHOR]

This article presents a construction of the concept of stochastic integration in Riemannian manifolds from a purely functional-analytic point of view. We show that there are infinitely many such integrals, and that any two of them are related by a simple formula. We also find that the Stratonovich and Itô integrals known to probability theorists are two instances of the general concept constructed herein. [ABSTRACT FROM AUTHOR]

First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems is a significant task to be solved in many science and engineering fields, but remains still an open challenge. The present paper develops a novel approach, termed 'fractional moments-based mixture distribution', to address such challenge. This approach is implemented by capturing the extreme value distribution (EVD) of the system response with the concepts of fractional moment and mixture distribution. In our context, the fractional moment itself is by definition a high-dimensional integral with a complicated integrand. To efficiently compute the fractional moments, a parallel adaptive sampling scheme that allows for sample size extension is developed using the refined Latinized stratified sampling (RLSS). In this manner, both variance reduction and parallel computing are possible for evaluating the fractional moments. From the knowledge of low-order fractional moments, the EVD of interest is then expected to be reconstructed. Based on introducing an extended inverse Gaussian distribution and a log extended skew-normal distribution, one flexible mixture distribution model is proposed, where its fractional moments are derived in analytic form. By fitting a set of fractional moments, the EVD can be recovered via the proposed mixture model. Accordingly, the first-passage probabilities under different thresholds can be obtained from the recovered EVD straightforwardly. The performance of the proposed method is verified by three examples consisting of two test examples and one engineering problem. • A fractional moments-based mixture distribution method is proposed to capture the EVD • A parallel adaptive sampling scheme is proposed to estimate the fractional moments • Two traditional distributions are improved to adapt to heavy tails • A new mixture distribution with improved flexibility is proposed to recover the EVD • Three examples demonstrate the efficiency and accuracy of the proposed method [ABSTRACT FROM AUTHOR]

The paper is devoted to properties of Aumann and Itô set-valued stochastic integrals, defined as some set-valued random variables. In particular the problem of integrable boundedness of the generalized Itô set-valued stochastic integrals is considered. Unfortunately, Itô set-valued stochastic integrals, defined by E.J. Jung and J.H. Kim in the paper [5] , are not in general integrably bounded (see [8,15] ). Therefore, in the present paper we consider generalized Itô set-valued stochastic integrals (see [10,11] ) defined for absolutely summable and countable subsets of the space I L 2 ( I R + × Ω , Σ I F , I R d × m ) of all square integrable I F -nonanticipative matrix-valued stochastic processes. Such integrals are integrably bounded and possess properties needed in the theory of set-valued stochastic equations. [ABSTRACT FROM AUTHOR]

Let A and B be almost commuting (i.e., the commutator AB − BA belongs to trace class) self-adjoint operators. We construct a functional calculus φ ↦ φ ( A , B ) for functions φ in the Besov class B ∞ , 1 1 ( R 2 ) . This functional calculus is linear, the operators φ ( A , B ) and ψ ( A , B ) almost commute for φ , ψ ∈ B ∞ , 1 1 ( R 2 ) , and φ ( A , B ) = u ( A ) v ( B ) whenever φ ( s , t ) = u ( s ) v ( t ) . We extend the Helton–Howe trace formula for arbitrary functions in B ∞ , 1 1 ( R 2 ) . The main tool is triple operator integrals with integrands in Haagerup-like tensor products of L ∞ spaces. [ABSTRACT FROM AUTHOR]

In recent works, beginning with [76] , several stochastic geophysical fluid dynamics (SGFD) models have been derived from variational principles. In this paper, we introduce a new framework for parametrization schemes (PS) in GFD. We derive a class of rough geophysical fluid dynamics (RGFD) models as critical points of rough action functionals using the theory of controlled rough paths. These RGFD models characterize Lagrangian trajectories in fluid dynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps. We formulate three constrained variational approaches for the derivation of these models. The first is the Clebsch formulation, in which the constraints are imposed as rough advection laws. The second is the Hamilton-Pontryagin formulation, in which the constraints are imposed as right-invariant rough vector fields. And the third is the Euler–Poincaré formulation, in which the variations are constrained. These constrained rough variational principles lead directly to the Lie–Poisson Hamiltonian formulation of fluid dynamics on GRP. The GRP framework preserves the geometric structure of fluid dynamics obtained by using Lie group reduction to pass from Lagrangian to Eulerian variational principles, yielding a rough formulation of the Kelvin circulation theorem. The rough formulation enhances its stochastic counterpart developed in [76] , and extended to semimartingales in [109]. For example, the rough-path variational approach includes non-Markovian perturbations of the Lagrangian fluid trajectories. In particular, memory effects can be introduced through a judicious choice of the rough path (e.g. a realization of a fractional Brownian motion). In the particular case when the rough path is a realization of a semimartingale, we recover the SGFD models in [76,109]. However, by eliminating the need for stochastic variational tools, we retain a pathwise interpretation of the Lagrangian trajectories. In contrast, the Lagrangian trajectories in the stochastic framework are described by stochastic integrals, which do not have a pathwise interpretation. Thus, the rough path formulation restores this property. [ABSTRACT FROM AUTHOR]

The paper deals with set-valued stochastic integrals which are defined as a random multifunctions. Integrable boundedness of a such integral is one of the most important features in potential applications. Unfortunately, up to now there were no correct proofs of such property. Surprisingly, also a negative answer to this problem is still not explained correctly. Hence the problem seems to be still undetermined. We shall show that in general the answer is negative. We shall provide several relatively simple examples both in convex and nonconvex-valued case. As a consequence, we will show that under some restrictions on the class of selections of the integrand the integrable boundedness of set-valued Itô's integral is equivalent to its single valuedness. [ABSTRACT FROM AUTHOR]

For most functionals for which pathwise stochastic integration with respect to Brownian motion is defined, sample Brownian paths for which the integral exists are very hard to construct. There exist on the unit interval, functions ω that can be uniformly approximated by sequences of continuous piece-linear functions (ωn) such that each ωn is encoded by a finite binary string of high Kolmogorov-Chaitin complexity. Such functions ω are called complex oscillations. Their set has Wiener measure 1 and they are fully characterised by infinite binary strings of high complexity. In this paper we study stochastic integration from the point of view of complex oscillations. We prove that, under some computability properties on integrands, pathwise stochastic integrals exist for any complex oscillation. We prove also that Itô's lemma holds for each complex oscillation. Thus constructing a continuous function satisfying Itô's lemma is reduced to constructing an infinite binary string of high Kolmogorov-Chaitin complexity. [ABSTRACT FROM AUTHOR]

The purpose of this paper is the estimation of the self-similarity index of the Rosenblatt process by using the Whittle estimator. Via chaos expansion into multiple stochastic integrals, we establish a non-central limit theorem satisfied by this estimator. We illustrate our results by numerical simulations. [ABSTRACT FROM AUTHOR]

Abstract: We present a new approach to a concept of a set-valued stochastic integral with respect to semimartingales. Such an integral, called set-valued stochastic up-trajectory integral, is compatible with the decomposition of the semimartingale. Some properties of this integral are stated. We show applicability of the new integral in set-valued stochastic integral equations driven by multidimensional semimartingales. The uniqueness theorem is presented. Then we extend the notion of the set-valued stochastic up-trajectory integral to definition of a fuzzy stochastic up-trajectory integral with respect to semimartingales. A result on uniqueness of a solution to fuzzy stochastic integral equations incorporating the new fuzzy stochastic up-trajectory integral driven by the multidimensional semimartingale is stated. [Copyright &y& Elsevier]

This paper presents a new approach to modeling wave propagation in random, linearly elastic materials, namely by means of Fourier integral operators (FIOs). The FIO representation of the solution to the equations of motion can be used to identify the elastic parameters of the underlying media, as well as their statistical hyperparameters in the randomly perturbed case. A stochastic version of the FIO representation can be used for damage detection. Hypothesis tests are proposed and validated, which are capable of distinguishing between an undamaged and a damaged material, even in the presence of random material parameters. The paper presents both the theoretical fundamentals as well as a numerical experiment, in which the applicability of the proposed method is demonstrated. [ABSTRACT FROM AUTHOR]

Abstract: General conditions for normalized, time-scaled stochastic integrals of independently scattered Levy random measures to converge to a limit are described. The idea is to provide general conditions to bypass the use of characteristic functions, which can sometimes have tedious calculations, to simplify and shorten the proofs of convergence of infinitely divisible processes, which have previously been done on a case by case basis. It is of particular interest to study both small and large scale asymptotics, since there are many applications, such as modeling internet traffic. The purpose of this paper is to generalize previous results to the greater class of all stochastically continuous (or, more generally, separable in probability) infinitely divisible processes with no Gaussian part and to expand the results to the multi-dimensional case. [Copyright &y& Elsevier]

Abstract: In this paper we use a definition of the fractional stochastic integral given by Carmona et al. (2003) in [19] and develop a simple approximation method to study quasi-linear stochastic differential equations by fractional Brownian motion. We also propose a stochastic process, namely fractional semimartingale, to model for the noise driving in some financial models. [Copyright &y& Elsevier]

Abstract: This paper concerns the random fluctuation theory of a one dimensional elliptic equation with highly oscillatory random coefficient. Theoretical studies show that the rescaled random corrector converges in distribution to a stochastic integral with respect to Brownian motion when the random coefficient has short-range correlation. When the random coefficient has long-range correlation, it was shown for a large class of random processes that the random corrector converged to a stochastic integral with respect to fractional Brownian motion. In this paper, we construct a class of random coefficients for which the random corrector converges to a non-Gaussian limit. More precisely, for this class of random coefficients with long-range correlation, the properly rescaled corrector converges in distribution to a stochastic integral with respect to a Rosenblatt process. [Copyright &y& Elsevier]

Abstract: A Langevin equation with multiplicative white noise and its corresponding Fokker–Planck equation are considered in this work. From the Fokker–Planck equation a transformation into the Wiener process is provided for different orders of prescription in discretization rule for the stochastic integrals. A few applications are also discussed. [Copyright &y& Elsevier]

Abstract: We give a strong approximation of Rosenblatt process via transport processes and we give the rate of convergence. [Copyright &y& Elsevier]

Abstract: The present paper is devoted to properties of set-valued stochastic integrals defined as some special type of set-valued random variables. In particular, it is shown that if the probability base is separable or probability measure is nonatomic then defined set-valued stochastic integrals can be represented by a sequence of Itôʼs integrals of nonanticipative selectors of integrated set-valued processes. Immediately from Michaelʼs continuous selection theorem it follows that the indefinite set-valued stochastic integrals possess some continuous selections. The problem of integrably boundedness of set-valued stochastic integrals is considered. Some remarks dealing with stochastic differential inclusions are also given. [Copyright &y& Elsevier]

Abstract: We consider limiting real interpolation spaces defined by using powers of iterated logarithms and show their description by means of the J-functional. Our results allow to complement some estimates on approximation of stochastic integrals. [Copyright &y& Elsevier]

Abstract: We use a method developed in Carmona et al. (2003) to study the fractional geometric mean-reversion processes. Our obtained results hold for any . [Copyright &y& Elsevier]

Abstract: In this paper, we shall firstly illustrate why we should consider integral of a stochastic process with respect to a set-valued square integrable martingale. Secondly, we shall prove the representation theorem of set-valued square integrable martingale. Thirdly, we shall give the definition of stochastic integral of a stochastic process with respect to a set-valued square integrable martingale and the representation theorem of this kind of integrals. Finally, we shall prove that the stochastic integral is a set-valued sub-martingale. [Copyright &y& Elsevier]

Abstract: We construct a family of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener–Itô integral with respect to the fractional Brownian motion. We assume that and we prove our approximation result for the integrands f in a rather general class. [Copyright &y& Elsevier]

Abstract: Given an increasing function and where , we characterize the property , and give conditions for and for , both in terms of integrability properties of . These results are applied to the approximation of stochastic integrals. [Copyright &y& Elsevier]

Abstract: We derive a change of variable formula for non-anticipative functionals defined on the space of -valued right-continuous paths with left limits. The functionals are only required to possess certain directional derivatives, which may be computed pathwise. Our results lead to functional extensions of the Itô formula for a large class of stochastic processes, including semimartingales and Dirichlet processes. In particular, we show the stability of the class of semimartingales under certain functional transformations. [Copyright &y& Elsevier]

Abstract: R-Boundedness is a randomized boundedness condition for sets of operators which in recent years has found many applications in the maximal regularity theory of evolution equations, stochastic evolution equations, spectral theory and vector-valued harmonic analysis. However, in some situations additional geometric properties such as Pisier''s property (α) are required to guaranty the R-boundedness of a relevant set of operators. In this paper we show that a weaker property called semi-R-boundedness can be used to avoid these geometric assumptions in the context of Schauder decompositions and the -calculus. Furthermore, we give weaker conditions for stochastic integrability of certain convolutions. [Copyright &y& Elsevier]

Abstract: We develop a theory of Malliavin calculus for Banach space-valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener–Itô isometry to Banach spaces. In the white noise case we obtain two sided -estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener–Itô chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer''s inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces. [Copyright &y& Elsevier]

Abstract: By using Malliavin calculus and multiple Wiener–Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in the Watanabe spaces. [Copyright &y& Elsevier]

Abstract: We introduce two types of the Stratonovich stochastic integrals for two-parameter processes, and investigate the relationship of these Stratonovich integrals and various types of Skorohod integrals with respect to a fractional Brownian sheet. By using this relationship, we derive a differentiation formula in the Stratonovich sense for fractional Brownian sheet through Itô formula. Also the relationship between the two types of the Stratonovich integrals will be obtained and used to derive a differentiation formula in the Stratonovich sense. In this case, our proof is based on the repeated applications of differentiation formulas in the Stratonovich form for one-parameter Gaussian processes. [Copyright &y& Elsevier]

Abstract: We prove a smoothing property and the irreducibility of transition semigroups corresponding to a class of semilinear stochastic equations on a separable Hilbert space H. Existence and uniqueness of invariant measures are discussed as well. [Copyright &y& Elsevier]

Abstract: We study in this paper the extremal behavior of stochastic integrals of Legendre polynomial transforms with respect to Brownian motion. As the main results, we obtain the exact tail behavior of the supremum of these integrals taken over intervals with fixed, and the limiting distribution of the supremum on intervals as . We show further how this limit distribution is connected to the asymptotic of the maximally selected quasi-likelihood procedure that is used to detect changes at an unknown time in polynomial regression models. In an application to global near-surface temperatures, we demonstrate that the limit results presented in this paper perform well for real data sets. [Copyright &y& Elsevier]

Bijectivization refines the Yang-Baxter equation into a pair of local Markov moves which randomly update the configuration of the vertex model. Employing this approach, we introduce new Yang-Baxter random fields of Young diagrams based on spin q -Whittaker and spin Hall-Littlewood symmetric functions. We match certain scalar Markovian marginals of these fields with (1) the stochastic six vertex model; (2) the stochastic higher spin six vertex model; and (3) a new vertex model with pushing which generalizes the q -Hahn PushTASEP introduced recently in [27]. Our matchings include models with two-sided stationary initial data, and we obtain Fredholm determinantal expressions for the q -Laplace transforms of the height functions of all these models. Moreover, we also discover difference operators acting diagonally on spin q -Whittaker or (stable) spin Hall-Littlewood symmetric functions. [ABSTRACT FROM AUTHOR]

Abstract: Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let be a fractional Brownian sheet with Hurst parameters , and a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in , and four types of stochastic surface integrals: , , , , , . As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters . Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process. [Copyright &y& Elsevier]

Abstract: We study a white-noise driven integrate-and-fire (IF) neuron with a time-dependent threshold. We give analytical expressions for mean and variance of the interspike interval assuming that the modification of the threshold value is small. It is shown that the variability of the interval can become both smaller or larger than in the case of constant threshold depending on the decay rate of threshold. We also show that the relative variability is minimal for a certain finite decay rate of the threshold. Furthermore, for slow threshold decay the leaky IF model shows a minimum in the coefficient of variation whenever the firing rate of the neuron matches the decay rate of the threshold. This novel effect can be seen if the firing rate is changed by varying the noise intensity or the mean input current. [Copyright &y& Elsevier]

We give a necessary and sufficient condition for the second quantization operator Γ(h) of a bounded operator h on L2(R+), or for its differential second quantization operator λ(h), to have a representation as a quantum stochastic integral. This condition is exactly that h writes as the sum of a Hilbert–Schmidt operator and a multiplication operator. We then explore several extensions of this result. We also examine the famous counterexample due to Journe´ and Meyer and explain its representability defect. [Copyright &y& Elsevier]

In this note we shall prove that the stochastic integral with respect to a semimartingale can be defined by Riemann''s approach. However in this approach we use non-uniform meshes instead of the usual uniform meshes. [Copyright &y& Elsevier]

The objects under investigation are the stochastic integrals with respect to free Le´vy processes. We define such integrals for square-integrable integrands, as well as for a certain general class of bounded integrands. Using the product form of the Itoˆ formula, we prove the full functional Itoˆ formula in this context. [Copyright &y& Elsevier]

• A unified scheme is proposed to solve complex-valued ratio distribution without closed-form solution. • A unified formula is derived by reducing the problem into multi-dimensional integrals. • Fast sparse-grid quadrature scheme is utilized to improve the efficiency of numerical integrals. • The unified method enables uncertainty quantification for FRFs and TFs. • The unified scheme is applicable to statistical inference for arbitrary ratio random variables. Complex-valued ratio distributions arises in many real applications such as statistical inference for frequency response functions (FRFs) and transmissibility functions (TFs) in structural health monitoring. As a sequel to our previous study, a unified scheme to solving complex ratio random variables is proposed in this study for the case when it is highly non-trivial or impossible to discover a closed-form solution such as the complex-valued t ratio distribution. Based on the probability transformation principle in the complex-valued domain, a unified formula is derived by reducing the concerned problem into multi-dimensional integrals, which can be solved by advanced numerical techniques. A fast sparse-grid quadrature (SGQ) scheme by constructing multivariate quadrature formulas using the combinations of tensor products of suitable one-dimensional formulas is utilized to improve the computational efficiency by avoiding the problem of curse of integral dimensionality. The unified methodology enables the efficient calculation of the probability density function (PDF) of a ratio random variable with its denominator and nominator specified by arbitrary probability distributions including Gaussian or non-Gaussian ratio random variables, correlated or independent random variables, bounded or unbounded ratio random variables. The unified scheme is applied to uncertainty quantification for raw FRFs and TFs without any post-processing such as averaging, smoothing and windowing, and the efficiency of the proposed scheme is verified by using the vibration test field data from a simply supported beam and from the Alamosa Canyon Bridge. [ABSTRACT FROM AUTHOR]

• Stochastic Harmonic Function convergence to the Gaussian for ensemble processes shown Reduction of Stochastic Harmonic Function dimensions via random frequency dependency. • Dependent and independent random frequency intervals are compared. • Dependent frequency scheme reduces dimensionality and improves Gaussian convergence. • Large scale numerical simulations carried out to validate the proposed methodology. Two significant developments pertaining to the application of the Stochastic Harmonic Function representation of stochastic processes are presented. Together, they allow for Gaussian records to be simulated within the bounds of the representation with the fewest number of random variables. Specifically, independent random frequencies that form a staple component of the Stochastic Harmonic Function are replaced by dependent random frequencies, along with a specific scheme for choosing frequency interval widths. Numerical examples demonstrating spectrum reconstruction accuracy and estimated PDF convergence to the Gaussian are presented to support the work. [ABSTRACT FROM AUTHOR]

We derive estimates for the integral modulus of continuity of probability densities of infinitely divisible distributions. The paper is split into two parts. The first part deals with general infinitely divisible distributions. The second part is concerned with densities of random integrals with respect to a Lévy process. We will see major differences between integrals over compact and non-compact intervals. [ABSTRACT FROM AUTHOR]