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In the paper we consider the problem of estimating parameters entering the drift of a fractional Ornstein-Uhlenbeck type process in the non-ergodic case, when the underlying stochastic integral is of Young type. We consider the sampling scheme that the process is observed continuously on $[0,T]$ and $T\to\infty$. For known Hurst parameter $H\in(0.5, 1)$, i.e. the long range dependent case, we construct a least-squares type estimator and establish strong consistency. Furthermore, we prove a second order limit theorem which provides asymptotic normality for the parameters of the periodic function with a rate depending on $H$ and a non-central Cauchy limit result for the mean reverting parameter with exponential rate. For the special case that the periodicity parameter is the weight of a periodic function, which integrates to zero over the period, we can even improve the rate to $\sqrt{T}$.

In this paper, a new combination method is being proposed for the numerical solution of nonlinear Ito–Volterra stochastic integral equations based on Adomian decomposition method (ADM), Triangular function (TF) approximation, quadrature methods and Ito stochastic integration formula. The presented method is developed in two steps. First, we apply ADM to the main equation then for calculating the components, we used TF-approximation together with quadrature and Ito stochastic integration formula. Some theorems related to error and convergence analysis of the suggested method are also stated. Finally, several examples confirm the applicability, efficiency and accuracy of the method, along with comparisons.

In this paper, we study the distribution function of the time of explosion of a stochastic differential equation modeling the length of the dominant crack due to fatigue. The main novelty is that initial condition is regarded as an anticipating random variable and the stochastic integral is in the forward sense. Under suitable conditions, we use the substitution formula from Russo and Vallois to find the local solution of this equation. Then, we find the law of blow up time by proving some results on barrier crossing probabilities of Brownian bridge.

In this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients.

In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wave...

We consider two final value problem of a stochastic strongly damped wave equation driven by white noise (Section 3) and fractional noise (Section 4). We show that a stochastic integral in the solut...

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 unifor...

Here we are concerning with two problems of a coupled system of random and stochastic nonlinear differential equations with two coupled systems of nonlinear nonlocal random and stochastic integral conditions. The existence of solutions will be studied. The sufficient condition for the uniqueness of the solution will be given. The continuous dependence of the unique solution on the nonlocal conditions will be proved.

We prove an existence and uniqueness result for the obstacle problem for quasilinear stochastic integral-partial differential equations. Our method is based on the probabilistic interpretation of the solution using backward doubly SDEs with jumps., Comment: arXiv admin note: text overlap with arXiv:1010.2307 by other authors

This paper proposes a stochastic three species food-chain model with harvesting and distributed delays. Some criteria for the global dynamics of all positive solutions, including the existence of global positive solutions, stochastic boundedness, extinction, global asymptotic stability in the mean, and the probability distribution, are established by using the stochastic integral inequalities, Lyapunov function method, and the inequality estimation technique. Furthermore, the effects of harvesting are discussed, the optimal harvesting strategy and the maximum of expectation of sustainable yield (MESY for short) are obtained. Finally, numerical examples are carried out to illustrate our main results.

This paper deals with a class of backward stochastic differential equation driven by two mutually independent fractional Brownian motions. We essentially establish existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.

The mathematical theory of functional integration was developed in the fifties to be applied to problems in probability and quantum field theory. We’re not interested in the formal theory, but in its more practical form which can actually be used to generate numbers. In particular, it leads to the probabilistic collocation method (PCM) and other techniques for high-dimensional model representation (HDMR), which we intend to apply to eddy-current nondestructive evaluation (NDE), using the volume-integral code, VIC-3D®, as our vehicle.

Our aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

The aim of this paper is to provide a modeling of capital transfer between a portfolio consisted by two assets. For this purpose we use the Arrhenius Equation, which is a modeling tool for the specific modeling. We provide a stochastic differential equation of the Arrhenuis equation. We consider a unique uncertainty factor for this purpose, which arises from a generalization of It$\hat{o}$ stochastic integral. The stochastic integral established in this paper, may become a tool of substitution in any application of the It$\hat{o}$ stochastic integral in Finance.

In this paper, the optimal harvesting of a stochastic delay tri-trophic food-chain model with Levy jumps is considered. We introduce two kinds of environmental perturbations in this model. One is called white noise which is continuous and is described by a stochastic integral with respect to the standard Brownian motion. And the other one is jumping noise which is modeled by a Levy process. Under some mild assumptions, the critical values between extinction and persistent in the mean of each species are established. The sufficient and necessary criteria for the existence of optimal harvesting policy are established and the optimal harvesting effort and the maximum of sustainable yield are also obtained. We utilize the ergodic method to discuss the optimal harvesting problem. The results show that white noises and Levy noises significantly affect the optimal harvesting policy while time delays is harmless for the optimal harvesting strategy in some cases. At last, some numerical examples are introduced to show the validity of our results.

We consider processes of the form $\mu(t)=\mu((0,t])$, where $\mu$ is a $\sigma$-additive in probability stochastic set function. Convergence of a random Fourier series to $\mu(t)$ is proved, and t...

We investigate bilateral set-valued stochastic integral equations and these equations combine widening and narrrowing set-valued stochastic integral equations studied in literature. An existence and uniqueness theorem is established using approximate solutions. In addition stability of the solution with respect to small changes of the initial state and coefficients is established, also we provide a result on boundedness of the solution, and an estimate on a distance between the exact solution and the approximate solution is given. Finally some implications for deterministic set-valued integral equations are presented.

We consider multivalued stochastic integral equations driven by semimartingales. Such equations are formulated in two different forms, i.e., using multivalued stochastic up-trajectory and trajectory integrals, which are not equivalent. By the successive approximations method, we show the existence of a unique solution to each equation under a condition much weaker than the Lipschitz one. We indicate that the solutions are stable under small changes of the equation data. The results have immediate implications for solutions to single-valued stochastic integral equations driven by semimartingales.

We derive conditional Edgeworth-type expansions for Skorohod and Ito integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain conditional Stein approximation bounds for multiple stochastic integrals and quadratic Brownian functionals.

Let XH = {XH(t), t ∈ ℝ+} be a subfractional Brownian motion in ℝd. We provide a sufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of XH, we establish Chung’s law of the iterated logarithm for XH.

The finding an efficient way to the approximate solutions of the stochastic integral equations is an essential requirement. In this paper we discuss the convergence analysis of the two-dimensional Haar wavelet functions (2D-HWFs) method for solve 2D linear stochastic Volterra integral equation. The illustrative examples are included to demonstrate the validity and applicability of this numerical method.

In this paper, we introduce stochastic g-fractional integrals of order $$ \alpha $$, containing stochastic fractional integrals (Hafiz in Stoch Anal Appl 22:507–523, 2004), stochastic integral (Shaked and Shanthikumar in Adv Appl Prob 20:427–446, 1988) and stochastic pseudo integrals (Agahi in Stat Probab Lett 124:41–48, 2017). We determine the upper and lower bounds of stochastic g-fractional integrals for convex stochastic processes, generalizing some previous results in Kotrys (Aequat Math 83:143–151, 2012), Agahi (Aequat Math 90:765–772, 2016; 2017) and Agahi and Babakhani (Aequat Math 90:1035–1043, 2016).

In this paper, two-dimensional modified block-pulse functions (2D-MBPFs) method is introduced for approximate solution of 2D-linear stochastic Volterra-Fredholm integral equations so the ordinary and stochastic operrational matrices of integration are utilized to reduce the computation of such equations into some algebraic equations. Convergence analysis of this method is discussed. Finally an illustrative example is given to show the accuracy of the proposed method so the results of it is compared with the block-pulse functions (BPFs) method.

In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a H\"ormander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the $A_2$-conjecture. The results are applied to obtain $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on $\mathbb{R}^d$ and smooth and angular domains., Comment: Minor typos corrected. Published in Analysis & PDE

We present an explicit solution triplet ( Y , Z , K ) to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a compensated Poisson random measure. The process Y is expressed by an integral whose kernel is explicitly given. The processes Z and K are expressed by Hida–Malliavin derivatives involving Y .

In this article, radial basis functions (RBFs) and quadrature rules have been employed to estimate the solution of two-dimensional (2D) stochastic integral equations on hypercube areas. The main advantage of the suggested approach is that this algorithm can be easily implemented to estimate the solution of multidimensional stochastic integral equations defined on irregular domains. Also, it is established that the convergence order is proportional to h X , D l , where h X , D denotes fill distance parameter. Finally, to reveal accuracy, efficiency and applicability of our scheme two test problems are included.

We prove some existence and uniqueness results for a nonlinear stochastic integral equation using fixed-point theory methods to ensure the convergence of the successive approximations to the unique random solution.

In the paper we study properties of Aumann's stochastic integral driven by a two-parameter increasing process and set-valued Ito's integral with respect to the two-parameter martingale. Both types of integrals are understood as set-valued processes. Next, the existence, uniqueness and convergence properties of solutions to set-valued stochastic integral equations with respect to such integrators are investigated. We present new types of such equations that generalize those studied earlier. The results obtained in the paper present a set-valued counterpart dealing with this topic known both in single-valued deterministic and stochastic cases.

In this paper, we have studied space-time Brownian motion and its applications to mixed type stochastic integral equations. Approximate solutions of mixed stochastic integral equations have been obtained by using two-dimensional (2D) second kind Chebyshev wavelets (CWs). Furthermore, some examples have been presented to justify the efficiency of 2D second kind CWs.

This chapter defines the stochastic integral with respect to a volatility modulated Volterra process, by appealing to techniques from the Malliavin calculus. Indeed, the stochastic integral with respect to a volatility modulated Volterra process can be defined for a reasonably large class of (anticipative) stochastic integrand processes by a Skorohod stochastic integral and a classical Lebesgue integral over time. In both the integrals, an operator acting on the integrand process appears where the kernel function in the volatility modulated Volterra process enters explicitly. In the second term, the Malliavin derivative occurs. We derive a series of basic properties of the integral, before applying it to, among other things, volatility modulated Volterra process-driven Ornstein-Uhlenbeck processes, which again become of volatility modulated Volterra type.

In this work we first introduce quasi-infinitely divisible (QID) random measures and formulate spectral representations. Then, we introduce QID stochastic integrals and present integrability conditions and continuity properties. Further, we introduce QID stochastic 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 $X+Y\stackrel{d}{=}Z$ with $Y$ independent of $X$. The class of QID processes is strictly larger than the class of ID processes. We provide spectral representations and L\'{e}vy-Khintchine formulations for potentially all QID processes. Finally, we prove that QID random measures are dense in the space of random measures under convergence in distribution. Throughout this work we present many examples., Comment: New result added: density result for QID random measures. Comments are very welcome!

In Chap. 7, we saw that using random time change, any continuous semimartingale can be transformed into a amenable semimartingale, and then one can have a growth estimate on the stochastic integral similar to the one satisfied by integrals w.r.t. Brownian motion.

The stochastic integral representations (martingale representations) of square integrable processes are well-studied problems in applied probability with broad applications in finance. Yet finding explicit expression is not easy and typically done through the Clack-Ocone formula with the advanced machinery of Malliavin calculus. To find an alternative, Shiryaev and Yor (Teor Veroyatnost i Primenen 48(2):375–385, 2003) introduced a relatively simple method using Ito’s formula to develop representations for extrema of Brownian motion. In this paper, we extend their work to provide representations of functionals of time-homogeneous diffusion processes based on the Ito’s formula.

In this paper we introduce a variant of Burkholder's martingale transform associated with two martingales with respect to different filtrations. Even though the classical martingale techniques cannot be applied, we show that the discussed transformation still satisfies some expected $\mathrm{L}^p$ estimates. Then we apply the obtained inequalities to general-dilation twisted paraproducts, particular instances of which have already appeared in the literature. As another application we construct stochastic integrals $\int_{0}^{t}H_s d(X_s Y_s)$ associated with certain continuous-time martingales $(X_t)_{t\geq 0}$ and $(Y_t)_{t\geq 0}$. The process $(X_t Y_t)_{t\geq 0}$ is shown to be a "good integrator", although it is not necessarily a semimartingale, or even adapted to any convenient filtration., 18 pages; v3: accepted for publication, references updated, research support acknowledged

We present alternative definitions of the stochastic integral introduced by Ayew and Kuo and of the Hitsuda-Skorokhod integral extended to domains in $L^p$-spaces, $p \geq 1$. Our approach is motivated by the S-transform characterization of the Hitsuda-Skorokhod integral and based on simple processes of stochastic exponential type. We prove that the new stochastic integral extends the mentioned stochastic integrals above and we outline their connection.

We consider large systems of stochastic interacting particles through discontinuous kernels which has vision geometrical constrains. We rigorously derive a Vlasov–Fokker–Planck type of kinetic mean-field equation from the corresponding stochastic integral inclusion system. More specifically, we construct a global-in-time weak solution to the stochastic integral inclusion system and derive the kinetic equation with the discontinuous kernels and the inhomogeneous noise strength by employing the 1-Wasserstein distance.

Stochastic disturbances, such as Brown noises and Poisson noises, are ubiquitous in the real world. It is necessary to consider these noises' effect for dynamical behaviors of complex networks. This paper addresses the synchronization problem for complex networks perturbed by Brown noises and Poisson noises. By utilizing the martingale theory such as the predictable projection operator and the dual predictable projection operator, we transform the expectation of the stochastic integral with respect to the Poisson process into the expectation of Lebesgue integral. Then, based on this, this paper presents a global synchronization criterion for complex networks perturbed by Brown noises and Poisson noises. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed approach.

The goal of this paper is to generalize most of the moment formulae obtained in [12] . More precisely, we consider a general point process μ, and show that the quantities relevant to our problem are the so-called Papangelou intensities. When the Papangelou intensities of μ are well-defined, we show some general formulae to recover the moment of order n of the stochastic integral of the point process. We will use these extended results to introduce a divergence operator and study a random transformation of the point process.

In this paper we establish the existence of an operator \(J_q^\mathrm{an}(F)\) for appropriate functionals of the form $$\begin{aligned} F(x)=f(\langle {\theta _1,x}\rangle ,\ldots ,\langle {\theta _n,x}\rangle ) \end{aligned}$$ where \(\langle {\theta _j,x}\rangle \) denotes the Paley–Wiener–Zygmund stochastic integral with \(x\) in \(C_{a,b}[0,T]\) and \(\{\theta _1,\ldots ,\theta _n\}\) is an orthonormal set of functions in \(L_{a,b}^2[0,T]\)

We study the connections of two different pathwise hedging approaches. These approaches are Bender-Sottinen-Valkeila (BSV) by Bender et al. (2008, Pricing by hedging and no-arbitrage beyond semimartingales, finance and stochastics, 12(4), pp. 441–468.) and Cont and Fournie (CF) by Cont and Fournie (2010, Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, 259(4), pp. 1043–1072; in press, Functional Ito calculus and stochastic integral representation of martingales, Annals of probability). We prove that both approaches give the same pathwise hedges, whenever both of the strategies exist. We also prove BSV-type robust replication result for CF strategies.

We show that a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand $g$ can have any prescribed distribution, moreover, we give both necessary and sufficient conditions when random variables can be represented in this form. We also prove that any random variable is a value of such integral in some improper sense. We discuss some applications of these results, in particular, to fractional Black--Scholes model of financial market.

This paper is devoted to the investigation of random dynamics of the stochastic Boussinesq equations driven by Lévy noise. Some fundamental properties of a subordinator Lévy process and the stochastic integral with respect to a Lévy process are discussed, and then the existence, uniqueness, regularity, and the random dynamical system generated by the stochastic Boussinesq equations are established. Finally, some discussions on the global weak solution of the stochastic Boussinesq equations driven by general Lévy noise are also presented.

In this paper, we shall apply Ito formula to establish several moment inequalities for stochastic integrals. These will demonstrate the powerfulness of the Itof ormula.

We study mean-variance hedging under portfolio constraints in a general semimartingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictable way. To obtain the existence of a solution, we first establish the closedness in L 2 of the space of all gains from trade (i.e. the terminal values of stochastic integrals with respect to the price process of the underlying assets). This is a first main contribution which enables us to tackle the problem in a systematic and unified way. In addition, using the closedness allows us to explain and generalise in a systematic way the convex duality results obtained previously by other authors via ad-hoc methods in specific frameworks.

Nakao's stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. Ito's formula in terms of the extended stochastic integrals is obtained., Comment: An additional note has been added between the Acknowledgments part and the References part on page 11

We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands. We show that the integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition for regular functionals of an irregular path with non-vanishing quadratic variation, as a unique sum of a pathwise integral and a component with zero quadratic variation., Comment: To appear in: Journal de Mathematiques Pures et Appliquees

In this paper we give necessary and sufficient conditions for a cylindrical continuous local martingale to be the stochastic integral with respect to a cylindrical Brownian motion. In particular we consider the class of cylindrical martingales with closed operator-generated covariations. We also prove that for every cylindrical continuous local martingale $M$ there exists a time change $\tau$ such that $M\circ \tau$ is Brownian representable., Comment: Minor revision