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The fuzzy stochastic differential equations with Markovian switching are considered. First, under the $ \rm Carath\acute {e}odory $ Carath e ´ odory assumptions, the existence and uniqueness theorem for the aforementioned equations is given by means of stopping time techniques, discretization method, and the Gronwall-Bellman inequality. Then the boundedness of the second moment of the solutions for such equations is established. Furthermore, by using the averaging method and stochastic analysis, as well as the Bihari's inequality, this paper mainly demonstrates the asymptotic relationship between the solutions of the original equations and the corresponding averaged equations in terms of mean square and probability. Finally, the theoretical result of the averaging method is illustrated by an example. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating Freidlin–Wentzell’s large deviation principle for one (spatial) dimensional nonlinear stochastic wave equation ∂ 2 u ε (t , x) ∂ t 2 = ∂ 2 u ε (t , x) ∂ x 2 + ε σ (t , x , u ε (t , x)) W ˙ (t , x) , where W ˙ is white in time and fractional in space with Hurst parameter H ∈ (1 4 , 1 2) . The variational framework and the modified weak convergence criterion proposed by Matoussi et al. (Appl Math Optim 83(2):849–879, 2021) are adopted here. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic processes in the square mean sense, and establish some basic results on the function space of such processes like completeness, convolution and composition theorems. Under the situation that the functions forcing are Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic and verify some suitable assumptions, we establish the existence and uniqueness of square-mean (weighted) pseudo S-asymptotically Bloch type periodic mild solutions of some fractional stochastic integrodifferential equations (driven by fractional Brownian motion). Finally, the most important findings are substantiated with the assistance of an illustration. [ABSTRACT FROM AUTHOR]

This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, H ħ , V : = - ħ - 2 L ħ + V on the lattice ħ Z n , where V is a positive multiplication operator and L ħ is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit ħ → 0 . [ABSTRACT FROM AUTHOR]

In this article, we investigate the following stochastic tempered fractional general diffusion equation driven by tempered fractional Brownian motion (TFBM) ∂ t β , η u (t , x) = L u (t , x) + u (t , x) B ˙ H , λ (t) , (t , x) ∈ R + × R d , where ∂ t β , η denotes the Caputo tempered fractional derivative with order β ∈ (0 , 1) and tempered parameter η > 0, is the infinitesimal generator of some general time-homogeneous symmetric strong Markov process { X t } t ≥ 0 with the corresponding transition semigroup { P t , t ≥ 0 } defined by Ptf(x): = x[f(Xt)], and { B H , λ (t) } t ≥ 0 is a TFBM with Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 which is independent of { X t } t ≥ 0 . First of all, a novel estimate of multiple stochastic integrals with respect to the inverse tempered β-stable subordinator Dβ, η(t) is presented, which greatly contributes to the stochastic analysis. Then after establishing the Fubini Theorem for the stochastic integrals with respect to TFBM BH, λ(t) and the inverse tempered β-stable subordinator Dβ, η(t), we show that the Feynman-Kac representation u (t , x) = E D [ P D β , η (t) f (x) e ∫ 0 t ∫ 0 t δ (s − r) d D β , η (s) d B H , λ (r) ] is the weak solution of the stochastic tempered fractional general diffusion equation driven by TFBM with the initial value f by means of the techniques of Malliavin calculus and convergence analysis. From the Feynman-Kac formula, several kinds of continuity of the solutions with respect to time and tempered parameter η of tempered fractional derivative are proved by using the scaling property of the inverse tempered β-stable subordinator Dβ, η(t) and the novel estimate of stochastic integrals with respect to BH, λ(t) and Dβ, η(t). In particular for the stochastic fractional general diffusion equation driven by fractional Brownian motion (FBM) but in the time fractional derivative case, we can also obtain the sharp upper and lower bounds for the Hölder regularity and the moments of the solution. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence of optimal solutions for control problems involving nonlinear stochastic Schrödinger equations perturbed by a multiplicative fractional Brownian motion. Finite-dimensional approximations and a linearization method are employed and lead to ε-optimal solutions for the considered problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the composition of a homogeneous Poisson process with an independent time-fractional Poisson process. We call this composition the generalized iterated Poisson process (GIPP). The probability law in terms of the fractional Bell polynomials, governing fractional differential equations, and the compound representation of the GIPP are obtained. We give explicit expressions for mean and covariance and study the long-range dependence property of the GIPP. It is also shown that the GIPP is over-dispersed. Some results related to first-passage time distribution and the hitting probability are also examined. We define the compound and the multivariate versions of the GIPP and explore their main characteristics. Further, we consider a surplus model based on the compound version of the iterated Poisson process (IPP) and derive several results related to ruin theory. Its applications using the Poisson–Lindley and the zero-truncated geometric distributions are also provided. Finally, simulated sample paths for the IPP and the GIPP are presented. [ABSTRACT FROM AUTHOR]

In this paper, we establish the functional large deviation principle (LDP) for the Kac–Stroock approximations of a wild class of Gaussian processes constructed by telegraph types of integrals with L 2 -integrands under mild conditions and find the explicit form for their rate functions. Our investigation includes a broad range of kernels, such as those related to Brownian motions, fractional Brownian motions with whole Hurst parameters, and Ornstein–Uhlenbeck processes. Furthermore, we consider a class of non-Markovian stochastic differential equations driven by the Kac–Stroock approximation and establish their Freidlin–Wentzell type LDP. The rate function clearly indicates an interesting phase transition phenomenon as the approximating rate moves from one region to the other. [ABSTRACT FROM AUTHOR]

Lévy processes time-changed by inverse subordinators have been intensively studied in the last years. Their importance in connection with non-local operators and semi-Markov dynamics is well understood, but, in our view, several questions remain open concerning the probabilistic structure of such processes. The time-changed Lévy processes are particularly useful to describe complex systems with fractional and/or anomalous dynamics. The purpose of our work is to analyze the features of the sample paths of such processes, focusing on a martingale-based approach. We introduce the fractional Poisson random measure as the main tool for dealing with the jump component of time-changed càdlàg processes. Further, the fractional random measure is an interesting and novel topic in itself, and thus, it is thoroughly analyzed in the paper. A central role in our analysis is then played by fractional Poisson integrals (involving the aforementioned fractional Poisson measure) which allow a useful description of the random jumps. We investigate these stochastic integrals and the martingale property of their compensated counterpart. Therefore, we are able to obtain a semimartingale representation of time-changed processes analogous to the celebrated Lévy–Itô decomposition. Finally, an approximation scheme of such random processes will be discussed. [ABSTRACT FROM AUTHOR]

The paper investigates fractional differential equations involving the Liouville derivative. Solution to these equations under a boundary condition inside the time interval are derived in explicit form, their uniqueness is established using integral transforms technique. [ABSTRACT FROM AUTHOR]

This article proposes a predictor-corrector scheme for solving the fractional differential equations 0 C D t α y (t) = f (t , y (t)) , α > 0 with non-uniform meshes. We reduce the fractional differential equation into the Volterra integral equation. Detailed error analysis and stability analysis are investigated. The convergent order of this method on non-uniform meshes is still 3 though 0 C D t α y (t) is not smooth at t = 0 . Numerical examples are carried out to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

We investigate the problem of joint statistical estimation of several parameters for a stochastic differential equation driven by an additive fractional Brownian motion. Based on discrete-time observations of the model, we construct an estimator of the Hurst parameter, the diffusion parameter and the drift, which lies in a parametrised family of coercive drift coefficients. Our procedure is based on the assumption that the stationary distribution of the SDE and of its increments permits to identify the parameters of the model. Under this assumption, we prove consistency results and derive a rate of convergence for the estimator. Finally, we show that the identifiability assumption is satisfied in the case of a family of fractional Ornstein–Uhlenbeck processes and illustrate our results with some numerical experiments. [ABSTRACT FROM AUTHOR]

This paper considers a class of mixed slow-fast McKean–Vlasov stochastic differential equations that contain the fractional Brownian motion with Hurst parameter H > 1 / 2 and the standard Brownian motion. Firstly, we prove an existence and uniqueness theorem for the mixed coupled system. Secondly, under suitable assumptions on the coefficients, using the approach of Khasminskii's time discretization, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the mean square sense. [ABSTRACT FROM AUTHOR]

The multivariate Gaussian random fields with matrix-based scaling laws are widely used for inference in statistics and many applied areas. In such contexts, interests are often Hölder regularities of spatial surfaces in any given direction. This article analyzes the almost sure sample function behavior for operator fractional Brownian motion, including multivariate fractional Brownian motion. We obtain the estimations of small ball probability and the strongly locally nondeterministic for operator fractional Brownian motion in any given direction. By applying these estimates, we obtain Chung type laws of the iterated logarithm for operator fractional Brownian motion. Our results show that the precise Hölder regularities of these spatial surfaces are completely determined by the real parts of the eigenvalues of self-similarity exponent and the covariance matrix at time point 1.

This article concentrates in analyzing optimal controls for stochastic integrodifferential equation (SIDE) in Hilbert space. Necessary parameters are imposed to demonstrate the system that follows a unique variation of parameter formula using Leray Schauder Alternative. Subsequently, the existence of optimal control is investigated for the considered Lagrange control problem. The theoretical example with the mechanical example of ethanol fuelled engine are discussed to validate the results obtained.

We consider an ordinary differential equation driven by rough paths, in the T. Lyons sense (Rev Mat Iberoamer. 1998;14(2)215–310), depending on a small parameter and with a negative long-range random coefficient. We establish sufficient conditions under which the solution of this ordinary differential equation converges to the solution of a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H ∈ (1 4 , 1 2) , depends on the asymptotic behavior of the covariance function of the random coefficient. [ABSTRACT FROM AUTHOR]

This paper gives a new property for stochastic processes, called square-mean μ - pseudo-S-asymptotically Bloch-type periodicity. We show how this property is preserved under some operations, such as composition and convolution, for stochastic processes. Our main results extend the classical results on S-asymptotically Bloch-type periodic functions. We also apply our results to some problems involving semilinear stochastic integrodifferential equations in abstract spaces [ABSTRACT FROM AUTHOR]

We consider rough differential equations whose coefficients contain path-dependent bounded variation terms and prove the existence and a priori estimate of solutions. These equations include classical path-dependent stochastic differential equations containing running maximum processes and normal reflection terms. We apply these results to determine the topological support of the solution processes. [ABSTRACT FROM AUTHOR]

This article explores a stochastic volatility model that incorporates fractional Brownian motion (fBm) into the constant elasticity of variance (CEV) framework. We use time series models to estimate the drift and volatility parameters of the model and validate its performance. We also examine the fractional Black-Scholes (BS) equation arising from the CEV model with fBm. To solve this equation numerically, we apply a Chebyshev collocation method and analyze its convergence properties. We demonstrate the effectiveness of the numerical method with an example and apply it to the option pricing problem. [ABSTRACT FROM AUTHOR]

The pricing problem of European option is investigated under the generalized fractional jump-diffusion model. First of all, the generalized fractional jump-diffusion model is proposed, with the assumption that the underlying asset price follows this model, and the explicit solution is derived using the Itô formula. Then, the partial differential equation (PDE) of the European option price is obtained by using the delta-hedging strategy, and the analytical solutions of the European call and put option prices are obtained through the risk-neutral pricing principle. Moreover, the accuracy of closed-form formula for European option pricing is verified by the Monte Carlo simulation. Furthermore, the properties of the pricing formulas are discussed and the impact of main parameters on the option pricing model are analyzed via calculations of Greeks. Finally, the rationality and validity of the established option pricing model are verified by numerical analysis. [ABSTRACT FROM AUTHOR]

In this article, convolution-type fractional derivatives generated by Dickman subordinator and inverse Dickman subordinator are discussed. The Dickman subordinator and its inverse are generalizations of stable and inverse stable subordinators, respectively. The series representations of densities of the Dickman subordinator and inverse Dickman subordinator are also obtained, which could be helpful for computational purposes. Moreover, the space and time-fractional Poisson-Dickman processes, space-fractional Skellam Dickman process and non-homogenous Poisson-Dickman process are introduced and their main properties are studied. [ABSTRACT FROM AUTHOR]

As delays are common, persistent, and ingrained in daily life, it is imperative to take them into account. In this work, we explore the averaging principle for impulsive Atangana–Baleanu fractional stochastic delay differential equations driven by Lévy noise. The link between the averaged equation solutions and the equivalent solutions of the original equations is shown in the sense of mean square. To achieve the intended outcomes, fractional calculus, semigroup properties, and stochastic analysis theory are used. We also provide an example to demonstrate the practicality and relevance of our research. [ABSTRACT FROM AUTHOR]

We propose a nonparametric estimation of random effects from the following fractional diffusions d X j (t) = ψ j X j (t) d t + X j (t) d W H , j (t) , X j (0) = x 0 j , t ≥ 0 , j = 1 , ... , n , where ψ j are random variables and W j , H are fractional Brownian motions with a common known Hurst index H ∈ (0 , 1) . We are concerned with the study of Hermite projection and kernel density estimators for the ψ j 's common density, when the horizon time of observation is fixed or sufficiently large. We corroborate these theoretical results through simulations. An empirical application is made to the real Asian financial data. [ABSTRACT FROM AUTHOR]

The non-Markovian nature of rough volatility makes Monte Carlo methods challenging, and it is in fact a major challenge to develop fast and accurate simulation algorithms. We provide an efficient one for stochastic Volterra processes, based on an extension of Donsker's approximation of Brownian motion to the fractional Brownian case with arbitrary Hurst exponent H ∈ (0 , 1) . Some of the most relevant consequences of this 'rough Donsker (rDonsker) theorem' are functional weak convergence results in Skorokhod space for discrete approximations of a large class of rough stochastic volatility models. This justifies the validity of simple and easy-to-implement Monte Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark hybrid scheme and find remarkable agreement (for a large range of values of H ). Our rDonsker theorem further provides a weak convergence proof for the hybrid scheme itself and allows constructing binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan options. [ABSTRACT FROM AUTHOR]

In this paper, the existence and uniqueness of a specific class of fractional stochastic integro-differential equations considering the stochastic Brownian motion equipped with an appropriate form of a random initial condition is introduced regarding the Hilfer fractional derivative. The proofs of the existence and uniqueness of the solution are presented utilizing sensible constraints upon the deterministic and stochastic coefficients, Schauder's fixed point theorem, and some stochastic theories. Moreover, to get approximations of the exact paths solving such equations we introduce a numerical technique based upon the time-dependent spectral collocation technique considering the shifted Legendre polynomials as a basis. The underlying concept of this technique involves transforming complex equations into a set of algebraic ones by selecting an appropriate set of collocation points within the specified domain where collocation is applied. Herein, the values of the stochastic Brownian motion are calculated using the Mathematica program. For approximating the integrals, the Gauss–Legendre integration scheme is implemented. In addition, we establish the convergence concerning the presented scheme with the error estimate in detail. For this purpose, we present the graphs of maximum errors under the log-log scale. The utilized procedure is leveraged to tackle a variety of stochastic examples encompassing various types to confirm the effectiveness of the obtained theoretical and numerical results. The acquired upshots expose the efficiency and applicability of the presented methodology in the fractional stochastic field.

This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy. [ABSTRACT FROM AUTHOR]

The contributions of this paper are twofold: we define and investigate the properties of a short rate model driven by a general Gaussian Volterra process and, after defining precisely a notion of convexity adjustment, derive explicit formulae for it. [ABSTRACT FROM AUTHOR]

In this article, we focus on the application of the recent notion of time-fractional derivative developed in Sobolev spaces to the study of well-posedness and stability for a time-fractional wave-like equation with superdiffusion and subdiffusion terms. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with multi-scale distribution-dependent stochastic differential equations driven by fractional Brownian motion (with Hurst index H > 1 2 ) and standard Brownian motion, simultaneously. Our aim is to establish a large deviation principle for the multi-scale distribution-dependent stochastic differential equations. This is done via the weak convergence approach and our proof is based heavily on the fractional calculus. [ABSTRACT FROM AUTHOR]

We analyze the p-random Wishart tensor ( p ≥ 2 ) associated with an initial n × d random matrix X n , d whose entries are independent and belong to the Wiener chaos. The order of the Wiener chaos is constant in each row of the matrix, and it may change from one row to another. Therefore, the entries of our starting matrix are not identically distributed (we only assume that their second moments coincide). We prove that, in high-dimensional regime (i.e. when n and d are large enough), the multidimensional random vector corresponding to the p-Wishart tensor is close in distribution to a standard multidimensional Gaussian vector. We also evaluate the Wasserstein distance between the probability distributions of these two random vectors. By using the techniques of the Stein–Malliavin calculus, we show that when the dimensions n and d are large enough and p is fixed, this Wasserstein distance is of order less than n 2 p - 1 d . [ABSTRACT FROM AUTHOR]

We introduce and study a fractional version of the Skellam process of order k by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order k (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of FSPoK which are utilized to establish its long-range dependence property. Later, we consider two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent Lévy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes. [ABSTRACT FROM AUTHOR]

Fractional and bifractional Brownian motions can be defined on a metric space if the associated metric or distance function is conditionally negative definite (or of negative type). This paper introduces several forms of scalar or vector bifractional Brownian motions on various metric spaces and presents their properties. A metric space of particular interest is the arccos-quasi-quadratic metric space over a subset of R d + 1 such as an ellipsoidal surface, an ellipsoid, or a simplex, whose metric is the composition of arccosine and quasi-quadratic functions. Such a metric is not only conditionally negative definite but also a measure definite kernel, and the metric space incorporates several important cases in a unified framework so that it enables us to study (bi, tri, quadri)fractional Brownian motions on various metric spaces in a unified manner. The vector fractional Brownian motion on the arccos-quasi-quadratic metric space enjoys an infinite series expansion in terms of spherical harmonics, and its covariance matrix function admits an ultraspherical polynomial expansion. We establish the property of strong local nondeterminism of fractional and bi(tri, quadri)fractional Brownian motions on the arccos-quasi-quadratic metric space. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic properties for the drift parameter estimators in the fractional Ornstein-Uhlenbeck process with periodic mean function and long range dependence. The Cremér-type moderate deviations, as well as the moderation deviation principle with explicit rate function can be obtained. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the mean value theorem for tempered fractional operators, some properties with monotone functions and the Taylor theorem for Caputo tempered fractional derivative. Furthermore, we present a geometric interpretation of the Riemann-Liouville tempered fractional integral as a shadow on the wall. [ABSTRACT FROM AUTHOR]

We construct an infinite dimensional analysis with respect to non-Gaussian measures of fractional Gamma type which we call fractional Gamma noise measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using generalized Appell polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the fractional Gamma measures. Finally, we gives some new properties of the kernels expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions and we construct the so-called fractional Gamma noise Gel’fand triple. [ABSTRACT FROM AUTHOR]

Intermediate dimensions were recently introduced by Falconer et al. (Math Z 296:813–830, 2020) to interpolate between the Hausdorff and box-counting dimensions. In this paper, we show that for every subset E of the symbolic space, the intermediate dimensions of the projections of E under typical self-affine coding maps are constant and given by formulas in terms of capacities. Moreover, we extend the results to the generalized intermediate dimensions introduced by Banaji (Monatsh Math 202: 465–506, 2023) in several settings, including the orthogonal projections in Euclidean spaces and the images of fractional Brownian motions. [ABSTRACT FROM AUTHOR]

In this work we generalize some results of the Riemann-Liouville fractional calculus for the tempered case, namely, we deal with some boundedness results of Riemann-Liouville tempered fractional integrals on continuous function space and Lebesgue spaces in bounded intervals and on the real line. Moreover, the limit behavior of the Riemann-Liouville tempered fractional integrals approaching to the Riemann-Liouville fractional integrals is considered. [ABSTRACT FROM AUTHOR]

We show that the tail probability of the rough line integral ∫ 0 1 ϕ (X t) d Y t , where (X, Y) is a 2D fractional Brownian motion with Hurst parameter H ∈ (1 / 4 , 1 / 2) and ϕ is a C b ∞ -function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a γ -Weibull tail with any exponent γ > 2 H + 1 . In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian tail even though the underlying vector fields are assumed to be of class C b ∞ . This also demonstrates that the well-known upper tail estimate proved by Cass–Litterer–Lyons in 2013 is essentially sharp. [ABSTRACT FROM AUTHOR]

Let B a , b be a weighted-fractional Brownian motion with Hurst indexes a and b such that a > − 1 and 0 < b < 1 ∧ (1 + a). In this paper, we consider the linear self-repelling diffusion d X t a , b = d B t a , b + (θ ∫ 0 t (X t a , b − X s a , b) ds + ν) dt with X 0 a , b = 0 , where θ > 0 , ν ∈ R are two real parameters. The process is an analogue of the linear self-interacting diffusion (Cranston and Le Jan, Math. Ann.303 (1995), 87-93). We introduce its large time behaviors, and the behavior presents a recursive convergence which is quite different from the asymptotic behavior of stochastic differential equations without interacting drifts. As a related question, we also consider the asymptotic behaviors of the least squares estimations of θ and ν. [ABSTRACT FROM AUTHOR]