Nonlocal Integral Operator Research Articles

An Efficient Numerical Scheme for a Time-Fractional Black–Scholes Partial Differential Equation Derived from the Fractal Market Hypothesis

Since the early 1970s, the study of Black–Scholes (BS) partial differential equations (PDEs) under the Efficient Market Hypothesis (EMH) has been a subject of active research in financial engineering. It has now become obvious, even to casual observers, that the classical BS models derived under the EMH framework fail to account for a number of realistic price evolutions in real-time market data. An alternative approach to the EMH framework is the Fractal Market Hypothesis (FMH), which proposes better and clearer explanations of market behaviours during unfavourable market conditions. The FMH involves non-local derivatives and integral operators, as well as fractional stochastic processes, which provide better tools for explaining the dynamics of evolving market anomalies, something that classical BS models may fail to explain. In this work, using the FMH, we derive a time-fractional Black–Scholes partial differential equation (tfBS-PDE) and then transform it into a heat equation, which allows for ease of implementing a high-order numerical scheme for solving it. Furthermore, the stability and convergence properties of the numerical scheme are discussed, and overall techniques are applied to pricing European put option problems.

On a nonlocal integral operator commuting with the Laplacian and the Sturm–Liouville problem: Low rank perturbations of the operator

We reformulate all general real coupled self-adjoint boundary value problems as integral operators and show that they are all finite rank perturbations of the free space Green’s function on the real line. This free space Green’s function corresponds to the nonlocal boundary value problem proposed earlier by Saito [Appl. Comput. Harmon. Anal. 25, 68–97 (2008)]. We prove these perturbations to be polynomials of rank up to 4. They encapsulate in a fundamental way the corresponding boundary conditions.

Wave propagation through a stationary field of clouds: A homogenisation approach

AbstractThe effect of a subgrid‐scale cloud field on the propagation of long atmospheric waves is investigated using a new scale‐consistent formulation based on the asymptotic theory of homogenisation. A key aim is to quantify potential model errors in wave propagation speeds, introduced by using averaged fields in place of the fully resolved circulation, in the setting of a simple stratified Boussinesq midlatitude ‐channel model. The effect of the cloud field, represented here by a random array of strongly nonlinear axisymmetric circulations, is found to appear in the large‐scale governing equations through new terms which redistribute the large‐scale buoyancy and horizontal momentum fields in the vertical. These new terms, which have the form of nonlocal integral operators, are linear in the cloud number density and are fully determined by the solution of a linear elliptic equation known as a cell problem. The cell problem in turn depends on the details of the nonlinear cloud circulations. The integral operators are calculated explicitly for example cloud fields and then dispersion relations are compared for different waves in the presence of clouds at realistic densities. The main finding is that baroclinic Rossby waves are significantly slowed and damped by the clouds, whilst inertia–gravity waves are affected almost exclusively by damping, most strongly at the lowest frequencies. In contrast, all waves with a barotropic structure are found to be almost unaffected by the presence of clouds, even at the highest realistic cloud densities. An important consequence of this study is a new approach to the closure of subgrid‐scale cloud fields in the parameterisation of convection in large‐scale atmospheric models.

The energy–momentum complex in non-local gravity

In General Relativity, the issue of defining the gravitational energy contained in a given spatial region is still unresolved, except for particular cases of localized objects where the asymptotic flatness holds for a given spacetime. In principle, a theory of gravity is not self-consistent, if the whole energy content is not uniquely defined in a specific volume. Here, we generalize the Einstein gravitational energy–momentum pseudotensor to non-local theories of gravity where analytic functions of the non-local integral operator [Formula: see text] are taken into account. We apply the Noether theorem to a gravitational Lagrangian, supposed invariant under the one-parameter group of diffeomorphisms, that is, the infinitesimal rigid translations. The invariance of non-local gravitational action under global translations leads to a locally conserved Noether current, and thus, to the definition of a gravitational energy–momentum pseudotensor, which is an affine object transforming like a tensor under affine transformations. Furthermore, the energy–momentum complex remains locally conserved, thanks to the non-local contracted Bianchi identities. The continuity equations for the gravitational pseudotensor and the energy–momentum complex, taking into account both gravitational and matter components, can be derived. Finally, the weak field limit of pseudotensor is performed to lowest order in metric perturbation in view of astrophysical applications.

On a class of nonlocal obstacle type problems related to the distributional Riesz fractional derivative

In this work, we consider the nonlocal obstacle problem with a given obstacle \psi in a bounded Lipschitz domain \Omega in \mathbb{R}^{d} , such that \mathbb{K}_\psi^s=\{v\in H^s_0(\Omega):v\geq\psi \text{ a.e. in }\Omega\}\neq\emptyset , given by u\in\mathbb{K}_\psi^s:\quad\langle\mathcal{L}_au,v-u\rangle\geq\langle F,v-u\rangle\quad\forall v\in\mathbb{K}^s_\psi, for F in H^{-s}(\Omega) , the dual space of the fractional Sobolev space H^s_0(\Omega) , 0<s<1 . The nonlocal operator \mathcal{L}_a:H^s_0(\Omega)\to H^{-s}(\Omega) is defined with a measurable, bounded, strictly positive singular kernel a(x,y):\mathbb{R}^d\times\mathbb{R}^d\to[0,\infty) , by the bilinear form \langle\mathcal{L}_au,v\rangle=\mathrm{P.V.}\int_{\mathbb{R}^d}\int_{\mathbb{R}^d} \tilde{v}(x)(\tilde{u}(x)-\tilde{u}(y))a(x,y) \,dy\,dx=\mathcal{E}_a(u,v), which is a (not necessarily symmetric) Dirichlet form, where \tilde{u},\tilde{v} are the zero extensions of u and v outside \Omega respectively. Furthermore, we show that the fractional operator \tilde{\mathcal{L}}_A=-D^s\cdot AD^s:H^s_0(\Omega)\to H^{-s}(\Omega) defined with the distributional Riesz fractional D^s and with a measurable, bounded matrix A(x) corresponds to a nonlocal integral operator \mathcal{L}_{k_A} with a well-defined integral singular kernel a=k_A . The corresponding s -fractional obstacle problem for \tilde{\mathcal{L}}_A is shown to converge as s\nearrow1 to the obstacle problem in H^1_0(\Omega) with the operator -D\cdot AD given with the classical gradient D . We mainly consider obstacle type problems involving the bilinear form \mathcal{E}_a with one or two obstacles, as well as the N -membranes problem, thereby deriving several results, such as the weak maximum principle, comparison properties, approximation by bounded penalization, and also the Lewy–Stampacchia inequalities. This provides regularity of the solutions, including a global estimate in L^\infty(\Omega) , local Hölder regularity of the solutions when a is symmetric, and local regularity in fractional Sobolev spaces W^{2s,p}_{\mathrm{loc}}(\Omega) and in C^1(\Omega) when \mathcal{L}_a=(-\Delta)^s corresponds to fractional s -Laplacian obstacle type problems u\in\mathbb{K}^s_\psi(\Omega):\quad\int_{\mathbb{R}^d}(D^su-{f})\cdot D^s(v-u)\,dx\geq0\quad\forall v\in\mathbb{K}^s_\psi\text{ for }{f}\in [L^2(\mathbb{R}^d)]^d. These novel results are complemented with the extension of the Lewy–Stampacchia inequalities to the order dual of H^s_0(\Omega) and some remarks on the associated s -capacity and the s -nonlocal obstacle problem for a general \mathcal{L}_a .

Eliminating Artificial Boundary Conditions in Time-Dependent Density Functional Theory Using Fourier Contour Deformation.

We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain, FCD yields a high-order accurate numerical solution of the time-dependent Schrödinger equation directly in free space, without the need for artificial boundary conditions. Of the many existing artificial boundary condition schemes, FCD is most similar to an exact nonlocal transparent boundary condition, but it works directly on Cartesian grids in any dimension, and runs on top of the fast Fourier transform rather than fast algorithms for the application of nonlocal history integral operators. We adapt FCD to time-dependent density functional theory (TDDFT), and describe a simple algorithm to smoothly and automatically truncate long-range Coulomb-like potentials to a time-dependent constant outside of a bounded domain of interest, so that FCD can be used. This approach eliminates errors originating from the use of artificial boundary conditions, leaving only the error of the potential truncation, which is controlled and can be systematically reduced. The method enables accurate simulations of ultrastrong nonlinear electronic processes in molecular complexes in which the interference between bound and continuum states is of paramount importance. We demonstrate results for many-electron TDDFT calculations of absorption and strong field photoelectron spectra for one and two-dimensional models, and observe a significant reduction in the size of the computational domain required to achieve high quality results, as compared with the popular method of complex absorbing potentials.

A sufficient condition for optimal control problem of fully coupled forward‐backward stochastic systems with jumps: A state‐constrained control approach

AbstractWe study the stochastic optimal control problem for fully coupled forward‐backward stochastic differential equations (FBSDEs) with jump diffusions. A major technical challenge of such problems arises from the dependence of the (forward) diffusion term on the backward SDE and the presence of jump diffusions. Previously, this class of problems has been solved via only the stochastic maximum principle, which guarantees only the necessary condition of optimality and requires identifying unknown parameters in the corresponding variational inequality. Our paper provides an alternative approach, which constitutes the sufficient condition for optimality. Specifically, the original fully coupled FBSDE control problem (referred to as (P)) is converted into the terminal state‐constrained forward stochastic control problem (referred to as ) that includes additional (possibly unbounded) control variables. Then is solved via the backward reachability analysis, by which the value function of is expressed as the zero‐level set of the value function for the auxiliary unconstrained (forward) control problem (referred to as ). Unlike ), is an unconstrained problem, which includes additional control variables as a consequence of the martingale representation theorem. We show that the value function for is the unique viscosity solution to the associated integro‐type Hamilton‐Jacobi‐Bellman (HJB) equation. The viscosity solution analysis presented in our paper requires a new technique due to additional control variables in the Hamiltonian maximization and the presence of the nonlocal integral operator in terms of the (singular) Lévy measure. To solve the original problem (P), we reverse our approach. Specifically, we first solve to obtain the value function using the verification theorem and the viscosity solution of the HJB equation. Then is solved by characterizing the zero‐level set of the value function of , from which the optimal solution of (P) can be constructed. To illustrate the theoretical results of this paper, applications to the linear‐quadratic problem for fully coupled FBSDEs with jumps are also presented.

State and Control Path-Dependent Stochastic Optimal Control With Jumps

We consider the state and control path-dependent stochastic optimal control problem for jump-diffusion models, where the dynamics and the objective functional are dependent on (current and past) paths of state and control processes. We prove the dynamic programming principle of the value functional, for which, unlike the existing literature, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Skorohod metric</i> is necessary to maintain the separability of càdlàg (state and control) spaces. We introduce the state and control path-dependent integro-type Hamilton–Jacobi–Bellman (PIHJB) equation, which includes the Lévy measure in the corresponding nonlocal path-dependent integral operator. Then, by using the functional Itô calculus of a càdlàg path, we show the verification theorem, which constitutes the sufficient condition for optimality in terms of the solution to the PIHJB equation. We finally apply our verification theorem to the linear-quadratic optimal control problem of jump-diffusion models with delay and the control path-dependent problem, for which the explicit optimal solutions are obtained by solving the corresponding PIHJB equation.

Computation of Eigenvalues for Nonlocal Models by Spectral Methods

The purpose of this work is to study spectral methods to approximate the eigenvalues of nonlocal integral operators. Indeed, even if the spatial domain is an interval, it is very challenging to obtain closed analytical expressions for the eigenpairs of peridynamic operators. Our approach is based on the weak formulation of eigenvalue problem and in order to compute the eigenvalues we consider an orthogonal basis consisting of a set of Fourier trigonometric or Chebyshev polynomials. We show the order of convergence for eigenvalues and eigenfunctions in $$L^2$$ -norm, and finally, we perform some numerical simulations to compare the two proposed methods.

Convergence of Nonlocal Integral Operator in Peridynamics

Convergence of Nonlocal Integral Operator in Peridynamics

New Aspects of Bloch Model Associated with Fractal Fractional Derivatives

Abstract To model complex real world problems, the novel concept of non-local fractal-fractional differential and integral operators with two orders (fractional order and fractal dimension) have been used as mathematical tools in contrast to classical derivatives and integrals. In this paper, we consider Bloch equations with fractal-fractional derivatives. We find the general solutions for components of magnetization ℳ = (Mu , Mv , Mw ) by using descritization and Lagrange's two step polynomial interpolation. We analyze the model with three different kernels namely power function, exponential decay function and Mittag-Leffler type function. We provide graphical behaviour of magnetization components ℳ = (Mu , Mv , Mw ) on different orders. The examination of Bloch equations with fractal-fractional derivatives show new aspects of Bloch equations.

Fractional B-spline collection method for solving fractal-differential equations

This study used the fractional B-spline collocation technique to obtain the numerical solution of fractal-fractional differential equations. The technique was considered to solve the fractal-fractional differential equations (FFDEs) with ($0 <gamma_i < 1,, i = 1, 2, cdots, N$). In this suggested technique, the B-spline of fractional order was utilised in the collocation technique. The scheme was easily attained, efficient, and relatively precise with reduced computational work numerical findings. Via the proposed technique, FFDEs can be reduced for solving a system of linear algebraic equations using an appropriate numerical approach. The verified numerical illustrative experiments were presented will show the effectiveness of the technique proposed in this study in solving FFDEs in three cases of nonlocal integral and differential operators namely power law kernel, when the kernels are exponential and the generalization of Mittag-Leffler kernel. The approximate solution is very good and accurate to the exact solution.

Bivariate Hermite interpolation by a limiting case of the cross approximation algorithm

Computing low-rank approximations of a given function is a key step for implementing efficiently numerous algorithms in various fields, including the discretisation of non-local integral operators and Isogeometric Analysis. The adaptive cross approximation (ACA) algorithm is an efficient method requiring few computational resources introduced by Bebendorf. We introduce in the present paper the new paradigm of approximating the given function by a piecewise low-rank function with C1-regularity. The proposed approximation is based on the ACA algorithm and our main contribution is the extension of the interpolation property characterising this algorithm to Hermite interpolation. Therefore, we introduce a new method for low-rank Hermite interpolation using a limiting case of the ACA algorithm. The proposed method has full approximation order. We then propose a piecewise low-rank approximation with adaptive refinement using either the ACA algorithm or our new method to compute each piece. We finally compare the results obtained for the two methods.

Analysis of fractal fractional differential equations

Nonlocal differential and integral operators with fractional order and fractal dimension have been recently introduced and appear to be powerful mathematical tools to model complex real world problems that could not be modeled with classical and nonlocal differential and integral operators with single order. To stress further possible application of such operators, we consider in this work an advection-dispersion model, where the velocity is considered to be 1. We consider three cases of the models, when the kernels are power law, exponential decay law and the generalized Mittag-Leffler kernel. For each case, we present a detailed analysis including, numerical solution, stability analysis and error analysis. We present some numerical simulation.

Branching Random Walk Solutions to the Wigner Equation

The stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator $\Theta_V$ with an anti-symmetric kernel as {the generator of two branches of jump processes}, are analyzed. All existing branching random walk solutions are formulated based on the Hahn-Jordan decomposition $\Theta_V=\Theta^+_V-\Theta^-_V$, i.e., treating $\Theta_V$ as the difference of two positive operators $\Theta^\pm_V$, each of which characterizes the transition of states for one branch of particles. Despite the fact that the first moments of such models solve the Wigner equation, we prove that the bounds of corresponding variances grow exponentially in time with the rate depending on the upper bound of $\Theta^\pm_V$, instead of $\Theta_V$. In other words, the decay of high-frequency components is totally ignored, resulting in a severe {numerical sign problem}. {To fully utilize such decay property}, we have recourse to the stationary phase approximation for $\Theta_V$, which captures essential contributions from the stationary phase points as well as the near-cancelation of positive and negative weights. The resulting branching random walk solutions are then proved to asymptotically solve the Wigner equation, but {gain} a substantial reduction in variances, thereby ameliorating the sign problem. Numerical experiments in 4-D phase space validate our theoretical findings.

Affine Approximation of Parametrized Kernels and Model Order Reduction for Nonlocal and Fractional Laplace Models

We consider parametrized problems driven by spatially nonlocal integral operators with parameter-dependent kernels. In particular, kernels with varying nonlocal interaction radius $\delta > 0$ and ...

Non-local fractional calculus from different viewpoint generated by truncated [formula omitted]-derivative

The core of this paper is to propose more favored novel non-local fractional integral and derivative operators with three parameters by using the iteration manner on truncated M-derivative involving truncated Mittag-Leffler function. This is related closely to the fact that, in a certain sense, to iterate a convenient local derivative for real phenomena under investigation is one of the best ways for obtaining more comprehensive fractional integration and differentiation. In order to gain a deeper grip, we address some significant theorems, reflected accomplishedly the features of the standard fractional calculus, about these aidful operators encompassing some other types of fractional operators. The presented new operators in Riemann–Liouville and Caputo sense are utilized to solve Sturm–Liouville problem subjected to the initial conditions. As a useful method, Laplace transform of the aforementioned fractional integral and derivative is introduced to solve anomalous dynamics of various complex systems. Moreover, we carry out comparison analysis by plotting governing Sturm–Liouville equation in order to grasp the advantages of these new fractional operators performing much better on account of the additional parameter γ.

Regularity analyses and approximation of nonlocal variational equality and inequality problems

We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These types of operators are used to model anomalous diffusion and, for a special choice of the integral kernels, reduce to the fractional Laplace operator on a bounded domain. By means of a nonlocal vector calculus we recast the problems in a weak form, leading to corresponding nonlocal variational equality and inequality problems. We prove optimal regularity results for both problems, including a higher regularity of the solution and the Lagrange multiplier. Based on the regularity results, we analyze the convergence of finite element approximations for a linear problem and illustrate the theoretical findings by numerical results.

Charged particle nonlinear resonance with localized electrostatic wave-packets

A resonant wave-particle interaction, in particular a nonlinear resonance characterized by particle phase trapping, is an important process determining charged particle energization in many space and laboratory plasma systems. Although an individual charged particle motion in the nonlinear resonance is well described theoretically, the kinetic equation modeling the long-term evolution of the resonant particle ensemble has been developed only recently. This study is devoted to generalization of this equation for systems with localized wave packets propagating with the wave group velocity different from the wave phase velocity. We limit our consideration to the Landau resonance of electrons and waves propagating in an inhomogeneous magnetic field. Electrons resonate with the wave field-aligned electric fields associated with gradients of wave electrostatic potential or variations of the field-aligned component of the wave vector potential. We demonstrate how wave-packet properties determine the efficiency of resonant particle acceleration and derive the nonlocal integral operator acting on the resonant particle distribution. This operator describes particle distribution variations due to interaction with one wave-packet. We solve kinetic equation with this operator for many wave-packets and show that solutions coincide with the results of the numerical integration of test particle trajectories. To demonstrate the range of possible applications of the proposed approach, we consider the electron evolution induced by the Landau resonances with packets of kinetic Alfven waves, electron acoustic waves, and very oblique whistler waves in the near-Earth space plasma.

Modelling of activator-inhibitor dynamics via nonlocal integral operators

This paper proposes application of nonlocal operators to represent the biological pattern formation mechanism of self-activation and lateral inhibition. The blue-green algae Anabaena is discussed as a model example. The patterns are determined by the kernels of the integrals representing the nonlocal operators. The emergence of patters when varying the size of the support of the kernels is numerically investigated.