Remarks on strongly convex stochastic processes

Strongly convex stochastic processes are introduced. Some well-known results concerning convex functions, like the Hermite–Hadamard inequality, Jensen inequality, Kuhn theorem and Bernstein–Doetsch theorem are extended to strongly convex stochastic processes.     

Remarks on some associated Laguerre integral results

Motivated essentially by their possible need in a fairly large number of physical and chemical contexts, Mavromatis and Alassar [1] derived several associated Laguerre integral results by eliminating an unnecessary constraint used in an earlier paper on the subject by Mavromatis [2]. The main object of the present sequel to these recent works is to investigate and apply much more general families of integral formulas, involving products of two or more Laguerre polynomials, which have been considered in the mathematical literature rather extensively.     

Remarks on stochastic permanence of population models

This paper examines the asymptotic behavior of the stochastic Lotka–Volterra model under Markovian switching. We show that the stochastic Lotka–Volterra model is stochastically permanent. Moreover, we give another type of stochastic permanence, and consider the relationship of two types of stochastic permanence under white noise perturbation.     

Refinements of mean-square stochastic integral inequalities on convex stochastic processes

Recently, in the class of convex stochastic processes, Kotrys (Aequat Math 83:143–151, 2012; Aequat Math 86:91–98, 2013) proposed upper and lower bounds of mean-square stochastic integrals by using Hermite–Hadamard inequality. This paper shows that these bounds can be refined. Our results extend and refine the corresponding ones in the literature. Finally, an open problem for further investigations is given.     

Remarks on frequency domain methods for Volterra integral equations

The equation u(t) = ? ∫₀^tA(t ? τ) g(u(τ)) dτ + h(t), t ? 0 is studied on a Hilbert space H. A(t) is a family of bounded linear operators and g can be unbounded and nonlinear. Stability and asymptotic stability of solutions are studied. Frequency domain conditions are statements about the Laplace transform of A. An extension of the frequency domain method of Popov for H = R¹ is given. Here it is assumed that g is the gradient of a functional G. The frequency domain conditions are related to monotonicity and convexity conditions on A thus connecting Popov's result with work of Levin and London on equations in R¹. A second result is given in which g is not assumed to be a gradient. This extends a result of Levin in R¹. The ideas are illustrated by an example of a nonlinear partial differential functional equation.     

Remarks on non-trivial solutions of a Volterra integral equation

In this paper non-linear integral equations describing shock wave phenomena are presented. Some necessary and sufficient conditions for the existence of non-trivial solutions to equations of this type are given.     

Remarks on two-parameter stochastic differential equations in Hilbert space

An existence theorem for the solution of a two-parameter stochastic Goursat problem in Hilbert space is proved. In doing this we assume that the equation contains a principal linear term with an operator that is in general unbounded and two-parameter white noise.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 64–71, 1986.     

Backward stochastic Volterra integral equations on Markov chains

This paper studies Backward Stochastic Volterra Integral Equations (BSVIEs) driven by finite state, continuous time Markov chains. First, the existence and uniqueness of the solutions to two types of BSVIEs are established. Second, some scalar and vector comparison theorems are given. Finally, the applications of BSVIEs to a linear-quadratic optimal control problem and time-inconsistent coherent risk measures are presented.     

Remarks on a semilinear heat equation with integral boundary conditions

For a semilinear heat equation we consider a nonlocal boundary problem. On the basis of the solution of a Dirichlet problem for a parabolic equation and Volterra integral equation we establish the well-posedness for the nonlocal problem, which generalizes some recent results.     

Remarks on the solvability of a convolution integral equation on a finite interval

We present some results on the solvability of an integral equation of the second kind with a difference kernel on a finite interval, construct a counterexample to an assertion, earlier believed to have been proved, on the solvability of this equation, and pose an open problem.     

Abstract stochastic integral equation involving a vector generalized Stochastic integral

Translated from Matematicheskie Zametki, Vol. 49, No. 3, pp. 153–155, March, 1991.     

Solutions of stochastic differential-functional equations via bounded stochastic integral contractors

In this paper we shall use the bounded stochastic integral contractors to investigate the existence and uniqueness of the solution of a general stochastic differential-functional equation $$d\varphi (t) = F(D(\varphi _t ),dt)$$ where, ? _t ={?(t?s): ??s?0}, D:C([?τ,0],R ^d )→R ^m ,F(x,t) is ad-dimensional continuousC-semimartingale with spatial parameterx∈R ^m, and the integral involved here is a nonlinear stochastic integral.     

Comonotonic stochastic processes and generalized mean-square stochastic integral with applications

The concept of monotonic stochastic processes was introduced by Skowroński [Aequationes mathematicae 44 (1992) 249–258]. In this paper, we intro     

Remarks on non-linear noise excitability of some stochastic heat equations

We consider nonlinear parabolic SPDEs of the form ${\partial }_{t}u=\Delta u+\lambda \sigma \left(u\right)\dot{w}$ on the interval $(0,L)$, where $\dot{w}$ denotes space–time white noise, $\sigma$ is Lipschitz continuous. Under Dirichlet boundary conditions and a linear growth condition on $\sigma$, we show that the expected $L^2$-energy is of order $exp[\text{const}\times {\lambda }^4]$ as $\lambda \to \infty$. This significantly improves a recent result of Khoshnevisan and Kim. Our method is very different from theirs and it allows us to arrive at the same conclusion for the same equation but with Neumann boundary condition. This improves over another result in Khoshnevisan and Kim.     

Linear Volterra backward stochastic integral equations

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 $Y$.