Parabolic Variational Inequalities Research Articles

Exact penalty operators and regularization of parabolic variational inequalities with an obstacle inside a domain

A class of parabolic variational inequalities with zero obstacle inside the domain is considered. For this class, exact penalty operators are defined, which are then used to construct and study regularization methods. The following estimates are obtained for the closeness of the original and regularized problem: $$ u - \varepsilon _1 \leqslant u_\varepsilon \leqslant u + \varepsilon _1 and \left\| {u - u_\varepsilon } \right\|_{L_2 (0,T;V^0 )} = O(\varepsilon ^{3/4} ). $$

Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain

This paper is aimed at a homogenization problem for a parabolic variational inequality with unilateral constraints. The constraints on solutions are imposed on disk-shaped subsets belonging to the boundary of the domain and forming a periodic structure, so that one has a problem with rapidly oscillating boundary conditions on a part of the boundary. Under certain conditions on the relation between the period of the structure and the radius of the disks, the homogenized problem is obtained. With the help of special auxiliary functions, the solutions of the original variational inequalities are shown to converge to the solution of the homogenized problem in Sobolev space as the period of the structure tends to zero.

A Variational Inequality Arising from European Installment Call Options Pricing

In this paper we consider a parabolic variational inequality arising from European continuous installment call options pricing and prove the existence and uniqueness of the solution to the problem. Moreover, we obtain $C^\infty$ regularity and the bounds of the free boundary, as well as the limit of the free boundary as $\tau=T-t\rightarrow+\infty$. Eventually we show its numerical result by the binomial method.

A class of Crouzeix-Raviart type nonconforming finite element methods for parabolic variational inequality problem with moving grid on anisotropic meshes

A class of Crouzeix-Raviart type nonconforming finite element methods are proposed for the parabolic variational inequality problem with moving grid on anisotropic meshes. By using some novel approaches and techniques, the same optimal error estimates are obtained as the traditional ones. It is shown that the classical regularity condition or quasi-uniform assumption on meshes is not necessary for the finite element analysis.

Variational Inequalities in Hilbert Spaces with Measures and Optimal Stopping Problems

We study the existence theory for parabolic variational inequalities in weighted L 2 spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted L 2 setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.

An inverse coefficient problem for a nonlinear parabolic variational inequality

This study is related to inverse coefficient problems for a nonlinear parabolic variational inequality with an unknown leading coefficient in the equation for the gradient of the solution. An inverse method, involving minimization of a least-squares cost functional, is developed to identify the unknown coefficient. It is proved that the solution of the corresponding direct problem depends continuously on the coefficient. On the basis of this, the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.

On nonlinear parabolic variational inequalities containing nonlocal terms

We investigate nonlinear parabolic variational inequalities which contain functional dependence on the unknown function. Such parabolic functional differential equations were studied e.g. by L. Simon in [8] (which was motivated by the work of M. Chipot and L. Molinet in [4]), where the following equation was considered: $$ \begin{array}{*{20}c} {D_t u(t,x) - \sum\limits_{i = 1}^n {D_i \left[ {a_i (t,x,u(t,x),Du(t,x);u)} \right]} } \\ { + a_0 (t,x,u(t,x),Du(t,x);u) = f(t,x)} \\ {(t,x) \in Q_T = (0,T) \times \Omega ,a_i :Q_T \times R^{n + 1} \times L^p (0,T;V) \to R,} \\ \end{array} $$ (1) where V denotes a closed linear subspace of the Sobolev-space W1,p(Ω) (2 ≦ p < ∞). In the above mentioned paper existence of weak solutions of the above equation is shown. These results were extended to systems of functional differential equations in [2]. In the following, we extend these existence results to variational inequalities by using the (less known) results of [6]. Finally, we show some examples.

Analysis of a Degenerate Parabolic Variational Inequality with Gradient Constraints from Optimal Investment and Consumption with Transaction Costs

This paper is concerned with a degenerate parabolic variational inequality with gradient constraints arising from an optimal investment and consumption problem with proportional transaction costs. We are most interested in the resulting free boundaries that correspond to the optimal trading strategies. Under a reasonable technical condition, we present a complete analysis of the smoothness of solution and the behaviors of free boundaries.

Bilateral Obstacle Control Problem of Parabolic Variational Inequalities

In this paper, the optimality system as well as the existence theorem for an obstacle optimal control problem are established, in which the governing system is a parabolic bilateral variational inequality and the input control is the pair of upper and lower obstacles.

Finite Horizon Optimal Investment and Consumption with Transaction Costs

This paper concerns continuous-time optimal investment and consumption decision of a CRRA investor who faces proportional transaction costs and finite time horizon. In the no consumption case, it has been studied by Liu and Loewenstein (2002) and Dai and Yi (2006). Mathematically, it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraints. The problem gives rise to two free boundaries which stand for the optimal buying and selling strategies, respectively. We present an analytical approach to analyze the behaviors of free boundaries. The regularity of the value function is studied as well. Our approach is essentially based on the connection between singular control and optimal stopping, which is first revealed to the present problem.

Well-posedness for an extended Penrose–Fife phase-field model with energy balance supplied by Dirichlet boundary conditions

In this study we consider the non-isothermal phase-field model proposed by Penrose and Fife [Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 (1990) 44–62]. The system consists of the energy balance law (a nonlinear heat equation) and an equation that describes space-time changes in the order parameter (the Ginzburg–Landau equation). For the energy balance law, we consider the general nonlinear heat flux arising in non-equilibrium thermodynamics and impose the Dirichlet boundary condition. For the order parameter, we impose a constraint and thus consider a parabolic variational inequality. We prove the well-posedness of the problem: the system yields a unique solution that depends continuously upon given data.

On Stability and Convergence of a Finite Difference Approximation to a Parabolic Variational Inequality Arising From American Option Valuation

In this article we study the numerical solution of a degenerate variational inequality of parabolic type arising from the valuation of American options by a finite difference method. Stability and convergence of the finite difference method are established. Numerical results are also presented to show the effectiveness and usefulness of the discretization scheme.

An adaptive method for the Stefan problem and its application to endoglacial conduits

This paper concerns an adaptive finite element method for the Stefan one-phase problem. We derive a parabolic variational inequality using the Duvaut transformation. In each time-step we consider an adaptive algorithm based on a combination of the Uzawa method associated with the corresponding multivalued operator and a convergent adaptive method for the linear problem. We justify the convergence of the method. As an application we model an endoglacial conduit in which a phase change phenomenon takes place.

A parabolic variational inequality arising from the valuation of strike reset options

A strike reset option is an option that allows its holder to reset the strike price to the prevailing underlying asset price at a moment chosen by the holder. The pricing model of the option can be formulated as a one-dimensional parabolic variational inequality, or equivalently, a free boundary problem, where the free boundary just corresponds to the optimal reset strategy adopted by the holder of the option. This paper is concerned with the theoretical analysis of the model. The existence and uniqueness of the solution are established. Furthermore, we study properties of the free boundary. The monotonicity and C ∞ smoothness of the free boundary are proven in some situations.

Optimal Control Problems for Elliptic–Parabolic Variational Inequalities with Time-Dependent Constraints

In this paper, we study optimal control problems for quasi-linear elliptic–parabolic variational inequalities with time-dependent constraints. We prove the existence of an optimal control that minimizes the nonlinear cost functional. Moreover, we apply our general results to some model problems. In particular, we show the necessary condition of optimal pair for a problem of partial differential equation (PDE) with a non-homogeneous Dirichlet boundary condition.

Domain decomposition and heterogeneous modeling used in solving a moving boundary problem

PurposeTo provide an excellent numerical method to simulate an important industrial process – wet chemical etching problem.Design/methodology/approachA mathematical model describing the wet chemical etching process is formulated in terms of a parabolic variational inequality and a non‐overlapping domain decomposition (DD) method is proposed in the etching region, where a partial differential equation is treated in one sub‐domain, while a variational inequality is considered in the second sub‐domain. A Robin boundary condition with a parameter whose optimal value is to be found is enforced on the common boundary between these two sub‐domains. Finite difference technique with projection and the Crank‐Nicholson scheme for the time discretization are the major numerical tools utilized in this paper.FindingsThe proposed numerical method has achieved the best numerical performance for the famous wet chemical etching process among all the numerical schemes applied to this problem. It also shows the great computational power of domain splitting technique.Research limitations/implicationsLack of parallel supercomputer system limits the authors to perform further numerical tests with extremely large data. This will be done in the future.Practical implicationsThis is a very useful paper for the academic researchers and industry engineers who wish to develop more advanced numerical methods to simulate various kinds of industry processes along the research direction from this paper.Originality/valueThis paper provides a new and promising version of DD method in the field of moving boundary problems and offers a practical way to simulate wet chemical etching process for the engineers and scientists in the related field.

Finite volume methods for the valuation of American options

We consider the use of finite volume methods for the approximation of a parabolic variational inequality arising in financial mathematics. We show, under some regularity conditions, the convergence of the upwind implicit finite volume scheme to a weak solution of the variational inequality in a bounded domain. Some results, obtained in comparison with other methods on two dimensional cases, show that finite volume schemes can be accurate and efficient.

Optimal Control of Variational Inequalities with Delays in the Highest Order Spatial Derivatives

In this paper, an optimal control problem for parabolic variational inequalities with delays in the highest order spatial derivatives is investigated. The well–posedness of such kinds of variational inequalities is established. The existence of optimal controls under a Cesari–type condition is proved, and the necessary conditions of Pontryagin type for optimal controls is derived.

Evolution variational inequalities and projected dynamical systems with application to human migration

In this paper, we explore the relationship between projected dynamical systems and evolution variational inequalities (also sometimes referred to as parabolic variational inequalities). The methodology of evolution variational inequalities is then utilized for the first time to model the dynamic adjustment of a socio-economic process in the context of human migration. The questions of dynamics and convergence of algorithms in this framework are addressed and answered. In particular, we provide existence and uniqueness results for the solution path without assuming Lipschitz continuity and propose a finite-difference scheme for the solution of the human migration problem. The algorithm, an ordinary implicit scheme, is a discrete-time version of the model. Its convergence estimate is also established.

Parabolic variational inequalities: The Lagrange multiplier approach

Parabolic variational inequalities are discussed and existence and uniqueness of strong as well as weak solutions are established. Our approach is based on a Lagrange multiplier treatment. Existence is obtained as the unique asymptotic limit of solutions to a family of appropriately regularized nonlinear parabolic equations. Two regularization techniques are presented resulting in feasible and unfeasible approximations respectively. Monotonicity results of the regularized solutions and convergence rate estimate are established. The results are applied to the Black–Scholes model for American options. The case of the bilateral constraints is also treated. Numerical results for the Black–Scholes model are presented and prove the practical efficiency of our results.