We extend the new approach introduced in [24] and [25] for dealing with stochastic Volterra equations using ideas of Rough Path theory prove global existence uniqueness results. The main idea this is simple: Instead iterated integrals a path comprising data necessary to solve any equation driven by that path, now integral convolutions kernel comprise said data. This leads corresponding abstract...

An explicit method is derived for collocating either of the convolution integrals p(x) = fi f(x t)g(t)dt or q(x) = /*/(< x)g(t)dt, where x 6 (a, b), a subinterval of M . The collocation formulas take the form p = F(Am)% or q = F(Bm)g, where g is an w-vector of values of the function g evaluated at the "Sine points", Am and Bm are explicitly described square matrices of order m, and F(s) = ¡Qexp...

We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates: when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra. Therefore we can define a “rectangular free convolution”, linearized by cumulants and by an analytic integral...

In this paper, we introduce an efficient method based on Haar wavelet to approximate a solutionfor the two-dimensional linear stochastic Fredholm integral equation. We also give an example to demonstrate the accuracy of the method.  

In the paper we study stochastic convolution appearing in Volterra equation driven by so called Lévy process. By Lévy process we mean a process with homogeneous independent increments, continuous in probability and cadlag.

We develop a stochastic formulation of the optimally tuned range-separated hybrid density functional theory that enables significant reduction of the computational effort and scaling of the nonlocal exchange operator at the price of introducing a controllable statistical error. Our method is based on stochastic representations of the Coulomb convolution integral and of the generalized Kohn-Sham...

For a component or a system subject to stochastic degradation with sporadic jumps that occur at random times and have random sizes, we propose to model the cumulative degradation with random jumps using a single stochastic process based on the characteristics of Lévy subordinators, the class of non-decreasing Lévy processes. Based on an inverse Fourier transform, we derive a new closed-form rel...

Considering the network structure is one of the new approaches in studying stochastic PERT networks (SPN). In this paper, planar networks are studied as a special class of networks. Two structural reducible mechanisms titled arc contraction and deletion are developed to convert any planar network to a series-parallel network structure. In series-parallel SPN, the completion time distribution f...

In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. particular, prove $L^p$-extrapolation results under a H\"ormander condition on the Sparse domination and sharp weighted bounds are obtained Dini kernel, leading to version of solution $A_2$-conjecture. The applied obtain $p$-independence maximal $L^p$-regularity both in complex re...