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The goals of this paper are to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. For a class of anticipating linear stochastic differential equations, we prove the existence and uniqueness of solutions using two approaches: (1) Ayed–Kuo differential formula using an ansatz, and (2) a braiding technique by interpreting the integral in the Skorokhod sense. We establish a Freidlin–Wentzell type large deviations result for the solution of such equations. In addition, we prove large deviation results for small noise where the initial conditions are random. [ABSTRACT FROM AUTHOR]

Let Φ′ denotes the strong dual of a nuclear space Φ. In this article, we introduce sufficient conditions for the convergence uniform on compacts in probability for a sequence of Φ′-valued processes with continuous or càdlàg paths. We illustrate the usefulness of our results by considering two applications to stochastic analysis. First, we introduce a topology on the space of Φ′-valued semimartingales which are good integrators and show that this topology is complete and that the stochastic integral mapping is continuous on the integrators. Second, we introduce sufficient conditions for the convergence uniform on compacts in probability of the solutions to a sequence of stochastic evolution equations; we do this both in the case of linear equations with semimartingale noise and nonlinear equations with multiplicative cylindrical martingale-valued measure noise. [ABSTRACT FROM AUTHOR]

In this study, the stochastic distributed‐order time‐fractional version of the fourth‐order delay sub‐diffusion equation is defined by employing the Caputo fractional derivative. The orthonormal discrete Legendre polynomials, as a well‐known family of discrete polynomials basis functions, are used to develop a numerical method to solve this equation. To employ these polynomials in constructing the expressed approach, the operational matrices of the classical integration, differentiation (ordinary, fractional and distributed‐order fractional), and stochastic integration of these polynomials are extracted. The established method turns solving the introduced stochastic‐fractional equation into solving a more simple linear algebraic system of equations. In fact, by representing the unknown solution in terms of the introduced polynomials and employing the extracted matrices, this system is obtained. The accuracy of the developed algorithm is numerically checked by solving two examples. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonparametric local linear estimator for the drift function of stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H ∈ (1 / 4 , 1). The drift function is one-sided dissipative Lipschitz that ensures the ergodic property of the SDE. We derive the strong consistency of the proposed estimator with proper bandwidth selectors associated with the determined Hurst parameter H. The main tools are the ergodic theorem, Malliavin calculus, and a maximum inequality for It o ̂ –Skorohod integrals. [ABSTRACT FROM AUTHOR]

This paper introduces a Nearly Unstable INteger‐valued AutoRegressive Conditional Heteroscedastic (NU‐INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU‐INARCH process weakly converges to a Cox–Ingersoll–Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type‐I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID‐19 in the United Kingdom. [ABSTRACT FROM AUTHOR]

The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 , by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of p -th moment. A numerical example is given to illustrate our stability results. [ABSTRACT FROM AUTHOR]

We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication − a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication. [ABSTRACT FROM AUTHOR]

Let (V t) t ∈ [ 0 , L ] be a stochastic process with quadratic variation on a probability space (Ω , F , P) and Q (∋ 0) a dense subset of [ 0 , L ] , where [ 0 , L ] is regarded as the infinite interval [ 0 , ∞) when L = ∞. First, we introduce the L 0 (Ω) -module D Q (V) of V-differentiable noncausal processes on Q and V-derivative operator D V , Q = d Q d Q V defined on D Q (V) , which enjoys the modularity: D V , Q (α X + β Y) = α D V , Q X + β D V , Q Y for any X , Y ∈ D Q (V) and α , β ∈ L 0 (Ω). Second, we show that the class Q V , Q = { X ∈ D Q (V) | d [ X ] Q , t d [ V ] Q , t = | d Q X t d Q V t | 2 } forms an L 0 (Ω) -module, where [ ] Q , t stands for the quadratic variation on Q. As a result, we have the isometry: 〈 X , Y 〉 Q , t = 〈 D V , Q X , D V , Q Y 〉 L 2 ([ 0 , t ] , [ V ]) for any X , Y ∈ Q V , Q , where 〈 , 〉 Q , t stands for the quadratic covariation on Q. Finally, we present universal properties and examples of the stochastic integral I with D V , Q ∘ I = i d D (I) . This result is essentially used for solving the identification problem from the stochastic Fourier coefficients. [ABSTRACT FROM AUTHOR]

Iterated stochastic integrals of nonrandom integrands are constructed in the two-dimensional case. They are applied to the velocity fluctuations in a two-dimensional flow and mean kinetic energy of the velocity fluctuations is discussed.

Shafer and Vovk introduce in their book [8] the notion of instant enforcement and instantly blockable properties. However, they do not associate these notions with any outer measure, unlike what Vovk did in the case of sets of "typical" price paths. In this paper an outer measure on the space [0,+∞) x Ω is introduced, which assigns zero value exactly to those sets (properties) of pairs of time t and an elementary event ω which are instantly blockable. Next, for a slightly modified measure, Itô's isometry and BDG inequalities are proved, and then they are used to define an Itô-type integral. Additionally, few properties are proved for the quadratic variation of model-free continuous martingales, which hold with instant enforcement. [ABSTRACT FROM AUTHOR]

In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures. [ABSTRACT FROM AUTHOR]

This paper deals with the expectation of monomials with respect to the stochastic area integral A1,2(t,t+h)=∫t+ht∫stdW1(r)dW2(s)-∫t+ht∫stdW2(r)dW1(s) and the increments of two Wiener processes, ΔWi(t,t+h)=Wi(t+h)-Wi(t), i=1,2. In a monomial, if the exponent of one of the Wiener increments or the stochastic area integral is an odd number, then the expectation of the monomial is zero. However, if the exponent of any of them is an even number, then the expectation is nonzero and its exact value is not known in general. In the present paper, we derive formulae to give the value in general. As an application of the formulae, we will utilize the formulae for a careful stability analysis on a Magnus-type Milstein method. As another application, we will give some mixed moments of the increments of Wiener processes and stochastic double integrals. [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]

This paper consider the Cauchy problem for the Schrödinger equation coupled with the stochastic Benjamin-Ono equation. A priori estimates for the stochastic integral and the nonlinear terms corresponding to the coupling system are achieved by using Fourier transform restriction method introduced by Bourgain. It is shown that the Cauchy problem is locally well-posed as the initial data in appropriate Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this article, we propose a way to study sample path properties of processes indexed by a general poset (T , ≼). This framework encompasses large classes of vector spaces, manifolds and continuous R -trees. We define a Wiener-type integral Y t = ∫ ≼ t f d X for all t ∈ T , a deterministic function f : T → R and a set-indexed Lévy process X. Bounds for the Hölder regularity of Y are given which indicate how the regularities of f and X contributes to that of Y. This work is a continuation of Herbin and Richard (2016) and extends those of Jaffard (1999) and Balança and Herbin (2012). [ABSTRACT FROM AUTHOR]

Evidence of global warming induced from the increasing concentration of greenhouse gases in the atmosphere suggests more frequent warm days and heat waves. The concept of an extreme heat event (EHE), defined locally based on exceedance of a suitable local threshold, enables us to capture the notion of a period of persistent extremely high temperatures. Modeling for extreme heat events is customarily implemented using time series of temperatures collected at a set of locations. Since spatial dependence is anticipated in the occurrence of EHE's, a joint model for the time series, incorporating spatial dependence is needed. Recent work by Schliep et al. (J R Stat Soc Ser A Stat Soc 184(3):1070–1092, 2021) develops a space-time model based on a point-referenced collection of temperature time series that enables the prediction of both the incidence and characteristics of EHE's occurring at any location in a study region. The contribution here is to introduce a formal definition of the notion of the spatial extent of an extreme heat event and then to employ output from the Schliep et al. (J R Stat Soc Ser A Stat Soc 184(3):1070–1092, 2021) modeling work to illustrate the notion. For a specified region and a given day, the definition takes the form of a block average of indicator functions over the region. Our risk assessment examines extents for the Comunidad Autónoma de Aragón in northeastern Spain. We calculate daily, seasonal and decadal averages of the extents for two subregions in this comunidad. We generalize our definition to capture extents of persistence of extreme heat and make comparisons across decades to reveal evidence of increasing extent over time. [ABSTRACT FROM AUTHOR]

The object of research is the process of mathematical modelling of a multidimensional random signal, which in the structure of its generation is the sum of a large number of random impulses that occur at random times. Examples of stochastic signals of this type can be, in particular, electroencephalographic and cardiographic signals, photoplethysmography signals, resource consumption processes (electricity, gas, water consumption), radar signals, vibrations of bearings of electric machines and others. A common mathematical model (especially in the multidimensional case) of this type of signal is a linear random process that allows the signal to be represented as the sum of a large number of stochastically independent random impulses that occur at Poisson moments. If the impulses are stochastically dependent (or the moments of time of their appearance are not Poisson), then the mathematical model is a conditional linear random process. The definition and analysis of the probabilistic properties of such processes for the multidimensional case have not been conducted. The paper defines a multidimensional conditional linear random process, each component of which is represented as a stochastic integral of a random kernel driven by a process with independent increments. Expressions for the characteristic function and moment functions of the specified process are obtained. The approach used was to use the mathematical apparatus of conditional characteristic functions, as well as the known representation of an infinitely divisible characteristic function of a linear random process as a functional of a process with independent increments. The obtained results provide a possibility for theoretical analysis of probabilistic properties of multichannel stochastic signals, the mathematical model of which is a multidimensional conditional linear random process. Justification of their properties of stationarity or cyclostationarity, which are the consequence of corresponding properties of the kernel and process with independent increments, can be carried out. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to demonstrate the moment inequality theorems of the stochastic process. More specifically, we want to establish some stochastic moment inequalities in the stochastic process by applying the Ito formula and the Gronwall-type inequalities as well as introduce a new proofs of some parts of the Burkholder-Davis-Gundy inequality and induce inverse inequality. [ABSTRACT FROM AUTHOR]

We study the questions of the stochastic integral representation of stochastically non-smooth Brownian functionals, which are interesting from the point of view of their practical application in the problem of the European option. In particular, we generalize the Clark-Ocone formula to the case when the functional is not stochastically smooth, but its conditional mathematical expectation is stochastically differentiable, and we establish a method for finding its integrand. Moreover, we consider such functionals that do not satisfy even weakened conditions, that is, non-smooth path-dependent functionals whose conditional mathematical expectations are also not stochastically differentiable. [ABSTRACT FROM AUTHOR]

Let SH be a sub-fractional Brownian motion with index 12

We consider extreme value analysis for independent but nonidentically distributed observations. In particular, the observations do not share the same extreme value index. Assuming continuously changing extreme value indices, we provide a nonparametric estimate for the functional extreme value index. Besides estimating the extreme value index locally, we also provide a global estimator for the trend and its joint asymptotic theory. The asymptotic theory for the global estimator can be used for testing a prespecified parametric trend in the extreme value indices. In particular, it can be applied to test whether the extreme value index remains at a constant level across all observations. [ABSTRACT FROM AUTHOR]

Stochastic integrals, random sums, random contractions, and selfdecomposable random variables constitute fundamental concepts of probability theory with significant applications in several areas of systemics. The main results of the paper are a characterization of a selfdecomposable distribution and a formulation of a Poisson random sum of power contractions. These results are established by incorporating a type of stochastic integral for a continuous in probability, homogeneous stochastic process with independent increments, and the same type of stochastic integral for a compound Poisson stochastic process with positive jumps. Interpretations of the results in treatment of risks threatening various systems are also provided. [ABSTRACT FROM AUTHOR]

Proactivity constitutes a structural factor of decision making in many practical disciplines. It is generally adopted that stochastic discounting models substantially contribute to the development and implementation of proactive operations. The paper concentrates on the formulation of a stochastic discounting model by incorporating a stochastic integral and a positive random variable. Moreover, the paper provides interpretation of the stochastic discounting model in proactive environments. [ABSTRACT FROM AUTHOR]

Random contractions are generally recognized as strong analytical tools of probability theory with significant practical applications in a wide variety of disciplines. The paper formulates a uniform random contraction of a stochastic integral. Properties and applications in strategic decision making and other practical disciplines of the formulated uniform random contraction are also established by the paper. [ABSTRACT FROM AUTHOR]

This work is divided in three parts. First, we introduce quasi-infinitely divisible (QID) random measures and formulate spectral representations. Second, we introduce QID stochastic integrals and present integrability conditions, continuity properties and spectral representations. Finally, we introduce QID processes, i.e. stochastic processes with QID finite dimensional distributions. For example, a process X is QID if there exist two ID processes Y and Z such that X + Y = d Z with Y independent of X. The class of QID processes is strictly larger than the class of ID processes. We provide spectral representations and Lévy–Khintchine formulations for potentially all QID processes. Many examples are presented. [ABSTRACT FROM AUTHOR]

We are concerned with a noncausal approach to the numerical evaluation of the stochastic integral ∫ f d W t with respect to Brownian motion. Viewed as a special case of the numerical solution (in strong sense) of the SDE, it may be believed that the precision level of such an approximation scheme that uses only a finite number of increments Δ k W = W (t k + 1) - W (t k) of Brownian motion, would not exceed the order O (1 n) where n is the number of steps for discretization. We present in this note a simple but not trivial example showing that this belief is not correct. The discussion is developed on the basis of the noncausal theory of stochastic calculus introduced by the author. [ABSTRACT FROM AUTHOR]