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In this paper we obtain an Ito differential representation for a class of singular stochastic Volterra integral equations. As an application, we investigate the rate of convergence in the small time central limit theorem for the solution.

In this paper, we use the Kolmogorov distance to investigate the Smoluchowski–Kramers approximation for stochastic differential equations. We obtain an explicit Berry–Esseen error bound for the rate of convergence. Our main tools are the techniques of Malliavin calculus.

In this paper, we investigate the tail distribution of a fundamental class of one-dimensional diffusion processes. Based on the techniques of the Malliavin calculus, we obtain explicit estimates for tail distributions. Some examples and applications are given to illustrate the theory.

We consider a class of semi-linear stochastic partial differential equations with fractional noise. The finite-time blowup of positive solutions to this class has been investigated in Dozzi et al. (2014). The aim of this note is to provide a lower bound for the blowup probability of solutions.

In this article, we investigate the density of the solution to a class of stochastic functional differential equations by means of Malliavin calculus. Our aim is to provide upper and lower Gaussian estimates for the density.

In this paper, we investigate the distribution of the integral of maximum processes which arises naturally in the recent studies. We first establish the estimates for the Malliavin derivative of maximum processes. Then, we obtain the lower and upper bounds for the tail probability of the integral of these maximum processes. Some examples are given to illustrate the theory.

In this paper, we consider a fundamental class of stochastic differential equations with time delays. Our aim is to investigate the weak convergence with respect to delay parameter of the solutions. Based on the techniques of Malliavin calculus, we obtain an explicit estimate for the rate of convergence. An application to the Carathéodory approximation scheme of stochastic differential equations is provided as well.

In this paper, based on techniques of Malliavin calculus, we obtain an explicit bound for tail probabilities of a general class of additive functionals. Applications to fractional Brownian motion and Cox–Ingersoll–Ross process are given to illustrate the theory.

In this paper we consider a generalized Ait-Sahalia interest rate model. We first extend the space of admissible parameters that ensures the existence of a unique positive solution to the model. Then, we provide an explicit estimate for tail probabilities of solutions.

In this note, we investigate the continuity in law with respect to the Hurst index of the exponential functional of the fractional Brownian motion. Based on the techniques of Malliavin's calculus, we provide an explicit bound on the Kolmogorov distance between two functionals with different Hurst indexes., Comment: To appear in Comptes Rendus Mathematique, 7 pages

In this article, we consider a general class of linear advanced differential equations, and obtain explicitly sufficient conditions of convergence and exponential convergence to zero. A necessary condition is provided as well.

We investigate the multivariate central limit theorem for nonlinear statistics by means of Stein's method and Slepian's smart path interpolation method. Based on certain difference operators in theory of concentration inequalities, we obtain two explicit bounds for the rate of convergence. Applications to Rademacher functionals, the runs and quadratic forms are provided as well., Comment: To appear in Acta Mathematica Hungarica, 23 pages

In this paper we show a connection between Levin-Nohel integro-differential equations and ordinary functional differential equations. Based on this connection, we obtain several new conditions for the stability of the solution, including the famous 3/2 stability criterion.

In this article, we consider a class of stochastic Volterra integro-differential equations with infinite delay and impulsive effects, driven by fractional Brownian motion with the Hurst index in a Hilbert space. The cases of Lipschitz and bounded impulses are studied separately. The existence and uniqueness of mild solutions are proved by using different fixed-point theorems. An example is given to illustrate the theory.

We use a method developed in Carmona et al. (2003) [2] to study the fractional geometric mean-reversion processes. Our obtained results hold for any H ∈ ( 1 4 , 1 ) .

The aim of this paper is to provide a semimartingale approximation of a fractional stochastic integration. This result leads us to approximate the fractional Black–Scholes model by a model driven by semimartingales, and a European option pricing formula is found.

Let \(\Omega \) be a bounded domain in the plane, representing an elastic body. Let \(\Gamma_0 \) be a portion of the boundary \(\Gamma\) of \(\Omega \), \(\Gamma_0 \) being assumed to be paralled to the \(x\)-axis. It is proposed to determine the stress field in \(\Omega \) from the displacements and surface stresses given on \(\Gamma_0 \). Under the assumption of plane stress, it is shown that \(u_x + u_y\) is a harmonic function. An Airy stress function is introduced, from which the stress field is computed.

A particular discontinuous Galerkin finite element formulation for the simulation of Kirchhoff plates is presented. It is rotation-free and utilises standard C 0 Lagrange finite element basis functions, with the required continuity imposed in a weak sense across element boundaries. The implications of the scheme in terms of coercivity and convergence of the Galerkin problem in various norms are studied, with the formulation shown to be stable for any positive value of a penalty parameter. A priori error estimates are supported by a range of numerical examples. Properties of the approach for the important eigenvalue problems of plate buckling and vibration are also examined through numerical examples. Based on the results of the analysis and numerical examples, it is concluded that the formulation is robust, accurate and relatively simple.

The authors consider a long elastic cylinder of constant section Ω partly immersed in a liquid of constant density. It is assumed that the body is in the state of plane strain and that Ω has a boundary consisting of two piecewise smooth arcs assumed to meet each other at two, and only two, points on a horizontal line. The upper arc Γ, which is exposed to air, is assumed known, while the lower arc ϒ, assumed to be totally immersed in a liquid of constant density, is unknown and is to be determined. Under the conditions that the displacements and surface stresses on a subarc of Γ are known and that the lower arc ϒ is subjected to a known constant hydrostatic pressure, the authors prove a uniqueness theorem and in the case of existence of a solution, show the existence of a sequence of regularized solutions converging to the exact solution.

In this paper we study a Jacobi equation driven by fractional Brownian motion with Hurst index $H\in (\frac{1}{2},1).$ We first prove the existence and uniqueness of the solution. Then we investigate Malliavin differentiability and smoothness of the density of the solution. Finally, we point out that the solution can be approximated by semimartingales.

In this paper we use the fractional stochastic integral given by Carmona et al. (2003, “Stochastic Integration With Respect to Fractional Brownian Motion,” Ann. I.H.P. Probab. Stat., 39(1), pp. 27–68) to study a delayed logistic equation driven by fractional Brownian motion which is a generalization of the classical delayed logistic equation. We introduce an approximate method to find the explicit expression for the solution. Our proposed method can also be applied to the other models and to illustrate this, two models in physiology are discussed.

The aim of this paper is to introduce an approximation approach to fractional stochastic integration. Based on our obtained result, we find explicit solution of some fractional stochastic differential equations and study the ruin probability in the asset liability management (ALM) model.

The aim of this paper is to investigate the exponential stability for linear Levin-Nohel integro-differential equations with time-varying delays. To the best of our knowledge, the exponential stability for such equations has not yet been discussed. In addition, since we do not require that the kernel and delay are continuous, our results improve those obtained in Becker and Burton [Proc. R. Soc. Edinburgh, Sect. A: Math. 136, 245-275 (2006)]; Dung [J. Math. Phys. 54, 082705 (2013)]; and Jin and Luo [Comput. Math. Appl. 57(7), 1080-1088 (2009)].

In this paper we investigate the properties of a delayed bistable system under the effect of multiplicative noise. We first prove the existence and uniqueness of the positive solution and show that its moments are uniformly bounded. Then, we study stochastic dynamics of the solution in long time, the lower and upper bounds for the paths and an estimate for the average value are provided.

In this paper, we provide new necessary and sufficient conditions of the asymptotic stability for a class of quasilinear differential equations with several delays and oscillating coefficients. Our results are established by means of fixed point theory and improve those obtained in [J. R. Graef, C. Qian and B. Zhang, Asymptotic behavior of solutions of differential equations with variable delays, Proc. London Math. Soc.81 (2000) 72–92; B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal.63 (2005) e233–e242].

For the mixed Levin-Nohel integro-differential equation, we obtain new necessary and sufficient conditions of asymptotic stability. These results improve those obtained by Becker and Burton [“Stability, fixed points and inverse of delays,” Proc. - R. Soc. Edinburgh, Sect. A 136, 245–275 (2006)]10.1017/S0308210500004546 and Jin and Luo [“Stability of an integro-differential equation,” Comput. Math. Appl. 57(7), 1080–1088 (2009)]10.1016/j.camwa.2009.01.006 when b(t) = 0 and supplement the 32-stability theorem when a(t, s) = 0. In addition, the case of the equations with several delays is discussed as well.