In the present article we address modelling and numerical computation of total value adjustment for European options in a multi-currency setting when foreign exchange rates between different involved currencies are assumed to be stochastic. Thus, extend more realistic approach previous work where constant have been considered. New models formulated both terms linear nonlinear PDEs expectations,...

The paper presents an iterative numerical method for approximating solutions of two-dimensional Fredholm–Volterra integral equations the second kind. As these arise in many applications, there is a constant need accurate, but fast and simple to use approximations their solutions. proposed here uses successive Mann type suitable cubature formula. Mann’s procedure known converge faster than class...

In 1916, Tricomi 1 introduced originally the concept of quasi-nonexpansive for real functions. Subsequently, this concept has studied for mappings in Banach and metric spaces see, e.g., 2–7 . Recently, some generalized types of quasi-nonexpansive mappings in metric and Banach spaces have appeared. For example, see Ahmed and Zeyada 8 , Qihou 9–11 and others. Unless stated to the contrary, we ass...

This paper offers an effective technique for seeking numerical solution of the interior Riemann problem on a simply connected region with a finite number of corners by Picard iteration method. Previously we have obtained an integral equation associated with the problem and constructed an iterative formula that facilitated numerical integrations. Numerical examples presented in this paper reveal...

In this paper we propose to replace the standard Picard iteration in the Newton–Raphson method by Mann and Ishikawa iterations. This iteration’s replacement influence the solution finding process that can be visualized as polynomiographs for the square systems of equations. Polynomiographs presented in the paper, in some sense, are generalization of Kalantari’s polynomiography from a single pol...

We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in [Hairer, Weber, Probab. Theory Related Fields, 2013]. The key idea was to use the theory of controlled rough paths to give definitions of weak / mild solutions and to set up a Picard iteratio...

In Chapter I, Livre III of his monograph “Systèmes Orthogonaux” Darboux stated three integrability theorems. They provide local existence and uniqueness of solutions to systems of first order Partial Differential Equations of the type: ∂xiuα(x) = f α i (x, u(x)), i ∈ Iα ⊆ {1, . . . , n}. Here x = (x1, . . . , xn) , u = (u1, . . . , um), and f α i (x, u(x)) are given functions. For each dependen...

This paper presents some new compact approximation stencils based on integrated radial basis functions (IRBFs) for numerically solving second-order elliptic differential problems Cartesian grids. Higher-order IRBF schemes are employed to approximate the field/dependent variable. The approximations in each direction 3 points and constructed independently, where derivatives of second, third, four...

this research work is devoted to the development of a mathematical model for the simulation of the flow of polymeric melts through the die region of extruders. a set of governing equations are solved using finite element method. standard galerkin technique is used in conjunction with the mixed scheme to solve the flow equation. due to the non-linear nature of global equations, the picard iterat...

The notion of H-covariant strong Morita equivalence is introduced for ∗-algebras over C = R(i) with an ordered ring R which are equipped with a ∗-action of a Hopf ∗-algebra H . This defines a corresponding H-covariant strong Picard groupoid which encodes the entire Morita theory. Dropping the positivity conditions one obtains H-covariant ∗-Morita equivalence with its H-covariant ∗-Picard groupo...