Discontinuous Operators Research Articles

Three new weak formulations of the problem of American call options valuation

Three new weak formulations of the problem of American call options valuation are given. The first of these is a parabolic obstacle problem in a finite domain. The second is a parabolic variational inequality with a convex and Lipschitz-continuous functional and the last one is a semilinear parabolic equation with a discontinuous spatial operator. All these problems are equivalent – they have the same unique solution. Different formulations can be used both for theoretical research and for constructing numerical methods.

Integral Representation for Solution of Discontinuous Diffusion Operator with Jump Conditions

In this study, the diffusion operator with discontinuity function and the jump conditions is considered. Under certain initial and discontinuity conditions, integral equations have been derived for the solutions. Integral representations, which is too useful for this type equation, have been presented.

PMLSM position control based on continuous projection adaptive sliding mode controller

ABSTRACTIn this paper, the design of projection-based Adaptive Sliding Mode Controller (ASMC) is presented for position control of Permanent Magnet Linear Synchronous Motor (PMLSM) with unknown mover mass. The PMLSM model was first established and a vector control based on field orientation is used to decouple the cross-coupling in motor model. ASMC has been adopted to deal with the unknown mover mass and to give robust operation against external load thrust. Based on the Lyapunov method, the stability of adaptive sliding mode-controlled PMLSM has been proven and the adaptive law has been developed. Additionally, a continuous projection operator is applied to adaptive law such as to enforce the estimated mover mass within a pre-specified bound. The performance of ASMC based on continuous projection operator is investigated via simulation results within MATLAB environment. Also, a comparison study in ASMC performance is made due to the inclusion continuous and discontinuous projection operators. The simulated results showed that ASMC based on continuous projection gives better performance than that based on discontinuous one.

A version of Krasnosel'skiĭ's compression-expansion fixed point theorem in cones for discontinuous operators with applications

We introduce a new fixed point theorem of Krasnosel’skiĭ type for discontinuous operators. As an application we use it to study the existence of positive solutions of a second-order differential problem with separated boundary conditions and discontinuous nonlinearities.

On the continuous and discontinuous maximal operators

In the first part of this paper we study the regularity properties of a wide class of maximal operators. These results are used to show that the spherical maximal operator is continuous W1,p(Rn)↦W1,p(Rn), when p>nn−1. Other given applications include fractional maximal operators and maximal singular integrals. On the other hand, we show that the restricted Hardy–Littlewood maximal operator Mλ, where the supremum is taken over the cubes with radii greater than λ>0, is bounded from Lp(Rn) to W1,p(Rn) but discontinuous.

Extremal solutions for second-order fully discontinuous problems with nonlinear functional boundary conditions

We provide new results concerning the existence of extremal solutions for a class of second-order problems with nonlinear functional boundary conditions where the nonlinearity considered may be discontinuous with respect to all of its variables. The main result relies on recent fixed point theorems for discontinuous operators and the lower and upper solution method.

Spectral Analysis for Dissipative Singular Sturm–Liouville Operators with Transmission Conditions

In the present paper, we study a class of discontinuous dissipative differential operators in Weyl’s limit circle case. Using the characteristic determinant, we prove the completeness of the system of eigenvectors and associated vectors of this dissipative operator.

Inverse Problems for Parabolic Equation with Discontinuous Coefficients

Abstract We consider the reaction-diffusion equation with discontinuities in the diffusion coefficient and the potential term. We start by deriving the Carleman estimate for the discontinuous reaction-diffusion operator which is deployed in the inverse problems of finding the stability result of the two discontinuous coefficients from the internal observations of the given parabolic equation.

Convergence and Data Dependency of Normal−S Iterative Method for Discontinuous Operators on Banach Space

ABSTRACTWe study convergence, rate of convergence and data dependency of normal−S iterative method for a fixed point of a discontinuous operator T on a Banach space. We also prove some Collage type theorems for T. The main aim here is to show that there is a close relationship between the concepts of data dependency of fixed points and the collage theorems and show that the latter provides better estimate. Numerical examples in support of the results obtained are also given.

Erratum to: A Comparison Principle for PDEs Arising in Approximate Hedging Problems: Application to Bermudan Options

In a Markovian framework, we consider the problem of finding the minimal initial value of a controlled process allowing to reach a stochastic target with a given level of expected loss. This question arises typically in approximate hedging problems. The solution to this problem has been characterised by Bouchard et al. (SIAM J Control Optim 48(5):3123–3150, 2009) and is known to solve an Hamilton–Jacobi–Bellman PDE with discontinuous operator. In this paper, we prove a comparison theorem for the corresponding PDE by showing first that it can be rewritten using a continuous operator, in some cases. As an application, we then study the quantile hedging price of Bermudan options in the non-linear case, pursuing the study initiated in Bouchard et al. (J Financial Math 7(1):215–235, 2016).

Coupled best approximation theorems for discontinuous operators in partially ordered Banach spaces

Coupled best approximation theorems for discontinuous operators in partially ordered Banach spaces

Complete Controllability of Fractional Impulsive Multivalued Stochastic Partial Integrodifferential Equations with State-Dependent Delay

Abstract In this article, we consider a class of fractional impulsive multivalued stochastic partial integrodifferential equations with state-dependent delay in a real separable Hilbert space. Sufficient conditions for the complete controllability of impulsive fractional stochastic evolution systems are established by means of the fixed-point theorem for discontinuous multivalued operators due to Dhage and properties of the α $\alpha$ -resolvent operator combined with approximation techniques. Two examples are also given to illustrate the obtained theorem.

A Comparison Principle for PDEs Arising in Approximate Hedging Problems: Application to Bermudan Options

In a Markovian framework, we consider the problem of finding the minimal initial value of a controlled process allowing to reach a stochastic target with a given level of expected loss. This question arises typically in approximate hedging problems. The solution to this problem has been characterised by Bouchard, Elie and Touzi in [1] and is known to solve an Hamilton-Jacobi-Bellman PDE with discontinuous operator. In this paper, we prove a comparison theorem for the corresponding PDE by showing first that it can be rewritten using a continuous operator, in some cases. As an application, we then study the quantile hedging price of Bermudan options in the non-linear case, pursuing the study initiated in [2]. [1] Bruno Bouchard, Romuald Elie, and Nizar Touzi. Stochastic target problems with controlled loss. SIAM Journal on Control and Optimization, 48(5):3123-3150,2009. [2] Bruno Bouchard, Romuald Elie, Antony R\'eveillac, et al. Bsdes with weak terminal condition. The Annals of Probability, 43(2):572-604,2015.

Best proximity points of discontinuous operator in partially ordered metric spaces

Best proximity points of discontinuous operator in partially ordered metric spaces

On the completeness of eigenfunctions of a discontinuous Dirac operator with an eigenparameter in the boundary condition

This paper deals with a discontinuous Dirac operator with an eigenparameter in the boundary condition. By using Lidskii?s theorem, we investigate the completeness of the system of eigenfunctions of such an operator.

Existence of two nontrivial solutions for sufficiently large values of the spectral parameter in eigenvalue problems for equations with discontinuous right-hand sides

The question on the existence of solutions to eigenvalue problems is treated for nonlinear equations with discontinuous operators in a real Hilbert space. Using a variational method, theorems on the existence of two nontrivial solutions for sufficiently large values of the spectral parameter are proved. As an application, eigenvalue problems for elliptic-type equations with nonlinear terms which are discontinuous in the phase variable are investigated. Bibliography: 22 titles.

О базисности собственных функций разрывного дифференциального оператора второго порядка

О базисности собственных функций разрывного дифференциального оператора второго порядка

On maximal parabolic regularity for non-autonomous parabolic operators

We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r≠2. This allows us to prove maximal parabolic Lr-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.

Complexity of linear ill-posed problems in Hilbert space

Information complexity of ill-posed problems may be seen as controversial. On the one hand side there were pessimistic results stating that the complexity is infinite, while on the other hand side the theory of ill-posed problems is well developed. In contrast to well-posed problems (continuous solution operators) the complexity analysis of ill-posed problems (discontinuous solution operators) is impossible without taking into account the impact of noise in the information. Commonly used models consider bounded deterministic noise and unbounded stochastic (Gaussian white) noise. It is common belief that white noise makes ill-posed problems more complex than problems under bounded noise. In this study we shed light on a rigorous complexity analysis of ill-posed problems providing (tight) lower and upper bounds for both noise models. It will be shown that in contrast to the deterministic case statistical ill-posed problems have finite complexity at every prescribed error level. Moreover, the ill-posedness of the problem raises the issue of adaptation to unknown solution smoothness, and we provide results in this direction.

On discontinuous Dirac operator with eigenparameter dependent boundary and two transmission conditions

In this paper, we consider a discontinuous Dirac operator with eigenparameter dependent both boundary and two transmission conditions. We introduce a suitable Hilbert space formulation and get some properties of eigenvalues and eigenfunctions. Then we investigate the Green’s function, the resolvent operator, and some uniqueness theorems by using the Weyl function and some spectral data.