Volume 26, Issue 2

We consider the problem of selecting a change of mean which minimizes the variance of Monte Carlo estimators for the expectation of a functional of a continuous Gaussian field, in particular continuous Gaussian processes. Functionals of Gaussian fields have taken up an important position in many fields including statistical physics of disordered systems and mathematical finance (see, for example, [A. Comtet, C. Monthus and M. Yor, Exponential functionals of Brownian motion and disordered systems, J. Appl. Probab. 35 1998, 2, 255–271], [D. Dufresne, The integral of geometric Brownian motion, Adv. in Appl. Probab. 33 2001, 1, 223–241], [N. Privault and W. I. Uy, Monte Carlo computation of the Laplace transform of exponential Brownian functionals, Methodol. Comput. Appl. Probab. 15 2013, 3, 511–524] and [V. R. Fatalov, On the Laplace method for Gaussian measures in a Banach space, Theory Probab. Appl. 58 2014, 2, 216–241]. Naturally, the problem of computing the expectation of such functionals, for example the Laplace transform, is an important issue in such fields. Some examples are considered, which, for particular Gaussian processes, can be related to option pricing.

Recently, S. K. Mahato and P. D. Srivastava [A class of sequence spaces defined by 𝑙-fractional difference operator, preprint 2018, http://arxiv.org/abs/1806.10383] studied 𝑙-fractional difference sequence spaces. In this article, we intend to make a new approach to introduce and study some lambda 𝑙-fractional convergent, lambda 𝑙-fractional null and lambda 𝑙-fractional bounded sequences over 𝑛-normed spaces. Various algebraic and topological properties of these newly formed sequence spaces have been explored, and some inclusion relations concerning these spaces are also established. Finally, some characterizations of the newly formed sequence spaces are given.

In this article, we define a convolution operator and study its boundedness on mixed-norm spaces. In particular, we obtain a well-known result on the boundedness of composition operators given by Avetisyan and Stević in [K. Avetisyan and S. Stević, The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc, Appl. Math. Comput. 213 2009, 2, 304–311]. Also we consider the adjoint 𝒜b,c{\mathcal{A}^{b,c}} for b>0{b>0} of two parameter families of Cesáro averaging operators and prove the boundedness on Besov mixed-norm spaces Bα+(c-1)p,q{B_{\alpha+(c-1)}^{p,q}} for c>1{c>1}.

In this work, we use the fixed-point theorem in double cones to study the existence of multiple positive solutions for an impulsive first-order differential system with integral boundary conditions, when the nonlinearities change sign.

In this paper, we study the images of circles, lines, balls and half-planes under the Möbius transformations.

In this paper, we introduce and study an Ishikawa-type iteration process for the class of generalized hemicontractive mappings in 𝑝-uniformly convex metric spaces, and prove both Δ-convergence and strong convergence theorems for approximating a fixed point of generalized hemicontractive mapping in complete 𝑝-uniformly convex metric spaces. We give a surprising example of this class of mapping that is not a hemicontractive mapping. Our results complement, extend and generalize numerous other recent results in CAT(0) spaces.

Diversities are a generalization of metric spaces which associate a positive real number to every finite subset of the space. In this paper, we introduce ultradiversities which are themselves simultaneously diversities and a sort of generalization of ultrametric spaces. We also give the notion of spherical completeness for ultradiversities based on the balls defined in such spaces. In particular, with the help of nonexpansive mappings defined between ultradiversities, we show that an ultradiversity is spherically complete if and only if it is injective.

In this paper, controllability results for a class of multi-term time-fractional differential systems with state-dependent delay have been studied. The concept of fractional calculus, measure of noncompactness and Mönch fixed-point theorem has been implemented to obtain a new set of controllability results. Finally, an application is given to illustrate the obtained results.

We discuss some properties of integrals associated with the free particle wave packet, ψ⁢(x,t){\psi(x,t)}, which are solutions to the time-dependent Schrödinger equation for a free particle. Some noteworthy discussion is made in relation to integrals which have appeared in the literature. We also obtain formulas for half-integer arguments of the Riemann zeta function.

The objective of this work is to study the local existence, uniqueness, stability and other properties of solutions of iterative mixed integrodifferential equations of fractional order. The Successive Approximation Method is applied for the numerical solution of iterative mixed integrodifferential equations of fractional order.

We study a system of quasilinear eigenvalue problems with Dirichlet boundary conditions on complete compact Riemannian manifolds. In particular, Cheng comparison estimates and the inequality of Faber–Krahn for the first eigenvalue of a (p,q){(p,q)}-Laplacian are recovered. Lastly, we reprove a Cheeger-type estimate for the p -Laplacian, 1<p<∞{1<p<\infty}, from where a lower bound estimate in terms of Cheeger’s constant for the first eigenvalue of a (p,q){(p,q)}-Laplacian is built. As a corollary, the first eigenvalue converges to Cheeger’s constant as p,q→1,1{p,q\to 1,1}.

In this paper, the stability of conformable fractional-order nonlinear systems depending on a parameter is presented and described. Furthermore, The design of a feedback controller for the same class of conformable fractional-order systems is introduced. Illustrative examples are given at the end of the paper to show the effectiveness of the proposed results.

We study a nonlocal problem for ordinary differential equations of 2⁢n{2n}-order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

We continue our investigation on general large deviation principles (LDPs) for longest runs. Previously, a general LDP for the longest success run in a sequence of independent Bernoulli trails was derived in [Z. Liu and X. Yang, A general large deviation principle for longest runs, Statist. Probab. Lett. 110 2016, 128–132]. In the present note, we establish a general LDP for the longest success run in a two-state ( success or failure ) Markov chain which recovers the previous result in the aforementioned paper. The main new ingredient is to implement suitable estimates of the distribution function of the longest success run recently established in [Z. Liu and X. Yang, On the longest runs in Markov chains, Probab. Math. Statist. 38 2018, 2, 407–428].

In this paper, a class of time fractional partial integro-differential equations (FPIDEs) with initial conditions is studied. Some operational matrices are used to reduce a FPIDE problem to a system of algebraic equations with special properties. The resulted system is solved to give an approximate solution to the problem. Error estimation is also discussed for the approximate solution. Finally, some numerical examples are given to show the accuracy of the proposed method.