Degenerate Variational Inequalities Research Articles

Asymptotic behavior of optimal exercise strategy for a small number of executive stock options

Abstract We investigate the asymptotic behavior, as the number of options approaches zero, of the optimal exercise strategy of continuous-exercise perpetual executive stock options. Here the optimal strategy is described in terms of the free boundary of a degenerate variational inequality. To our knowledge, our method based on the convexity of the variational inequality at the initial time is new. Also, the unique formal asymptotic expansions of the free boundary and the solution of the variational inequality are derived.

H-розв’язність задачі оптимального керування для виродженої еліптичної варіаційної нерівності

<i>H</i>-розв’язність задачі оптимального керування для виродженої еліптичної варіаційної нерівності

The existence of a solution to a class of degenerate parabolic variational inequalities

In this paper, we study the degenerate parabolic variational inequality problem in a bounded domain. First, the weak solutions of the variational inequality are defined. Second, the existence of the solutions in the weak sense are proved by using the penalty method and the reduction method.

Optimal Stopping of a Hilbert Space Valued Diffusion: An Infinite Dimensional Variational Inequality

A finite horizon optimal stopping problem for an infinite dimensional diffusion $X$ is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space $\mathcal{H}$ with a non-linear diffusion coefficient $\sigma(X)$ and a generic unbounded operator $A$ in the drift term. When the gain function $\Theta$ is time-dependent and fulfils mild regularity assumptions, the value function $\mathcal{U}$ of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient $\sigma(X)$ is specified, the solution of the variational problem is found in a suitable Banach space $\mathcal{V}$ fully characterized in terms of a Gaussian measure $\mu$. This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions \cite{Ben-Lio82}, of well-known results on optimal stopping theory and variational inequalities in $\mathbb{R}^n$. These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model.

Convergence of a finite element approximation to a degenerate parabolic variational inequality with non-smooth data arising from American option valuation

This paper deals with the approximation of solutions to a degenerate variational inequality of parabolic type with a non-smooth final condition arising from American option pricing by the piecewise linear finite element method in space and an implicit time-stepping scheme. We show that the error of the approximation in a weighted Sobolev norm is of order 𝒪(h 2/3+Δ t 1/3) under some realistic regularity assumptions on the exact solution, where h and Δ t denote the mesh parameters in space and time, respectively. Numerical examples are presented to confirm our theoretical results.

A Singular Control Problem with Discretionary Stopping for Geometric Brownian Motions

In this paper, we discuss a combined singular control problem with discretionary stopping for geometric Brownian motions. By the method of penalization, we solve the degenerate variational inequality associated with this problem. Its solution v coincides with the value function and, by the concavity of v, an optimal control is shown to exist.

Local Indices for Degenerate Variational Inequalities

We provide an index formula for solutions of variational inequality problems defined by a continuously differentiable function F over a convex set M represented by a finite number of inequality constraints. Our index formula can be applied when the solutions are nonsingular and possibly degenerate, as long as they also satisfy the injective normal map (INM) property, which is implied by strong stability. We show that when the INM property holds, the degeneracy in a solution can be removed by perturbing the function F slightly, i.e., the index of a degenerate solution is equal to the index of a nondegenerate solution of a slightly perturbed variational inequality problem. We further show that our definition of the index is equivalent to the topological index of the normal map at the zero corresponding to the solution. As an application of our index formula, we provide a global index theorem for variational inequalities which holds even when the solutions are degenerate.

Existence for a degenerate diffusion problem with a nonlinear operator

We study a degenerate nonlinear variational inequality which can be reduced to a multivalued inclusion by an appropriate change of the unknown function. We establish existence, uniqueness and regularity results. An application arising in the theory of water diffusion in porous media is discussed as an example.

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.

A dual approach to regularity in thin film micromagnetics

We prove a regularity result in the two-dimensional theory of soft ferromagnetic films. The associated Euler–Lagrange equation is given by a nonlocal degenerate variational inequality involving fractional derivatives. A difference quotient type argument based on a dual formulation in terms of magnetostatic potentials yields a Holder estimate for the uniquely determined gradient projection of the magnetization field.

Application of Relaxation Scheme to Degenerate Variational Inequalities

In this paper we are concerned with the solution of degenerate variational inequalities. To solve this problem numerically, we propose a numerical scheme which is based on the relaxation scheme using non-standard time discretization. The approximate solution on each time level is obtained in the iterative way by solving the corresponding elliptic variational inequalities. The convergence of the method is proved.

On the Porosity of Free Boundaries in Degenerate Variational Inequalities

In this note we consider a certain degenerate variational problem with constraint identically zero. The exact growth of the solution near the free boundary is established. A consequence of this is that the free boundary is porous and therefore its Hausdorff dimension is less than N and hence it is of Lebesgue measure zero.

A DEGENERATE PARABOLIC VARIATIONAL INEQUALITY FOR THE AMERICAN OPTION PRICING PROBLEM

A formulation in terms of degenerate parabolic variational inequalities for the price of an American put option is stated. The usual uniform ellipticity condition is weakened by the presence of a weight function. Then the appropriate framework is that of some suitable "weighted" Sobolev spaces. One applies a result previously stated by the authors in Ref. 8.

Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear parabolic equations and variational inequalities

In this paper we study the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in Rd with d=l or 2. Given T>0, f and uO; find u∊K, where K is a closed convex subset of the So bo lev space , such that for x∊Ω and for any v∊K for a.e. t∊(0,T], where k∊C(0,∞) is a given nonnegative function with k(s)s strictly increasing for s≥O, but possibly degenerate, and p∊(1,∞) depends on k. For such a general problem we establish error bounds in energy type norms for a fully discrete approximation based on the backward Euler time discretisation. We show that these error bounds converge at the optimal rate with respect to the space discretisation, provided p≤2 and the solution u is sufficiently regular.

Hölder continuity of the gradient for degenerate variational inequalities

Hölder continuity of the gradient for degenerate variational inequalities

Degenerate variational inequalities of evolution

On etudie des inegalites variationnelles de la forme Re ≥0 pour tout v∈K, By(0)=Bu 0 , ou B est lineaire mais peut s'annuler, A est non lineaire et K est un ensemble convexe ferme

On a degenerate variational inequality with Neumann boundary conditions

A stopping time problem for degenerate reflected diffusions is studied in this paper. We give a characterization of the optimal cost as the maximum solution of a degenerate elliptic variational inequality with Neumann boundary conditions.

Gradient bounds for solutions of degenerate variational inequalities

We note here that a technique of Oleinik for obtaining derivative estimates for degenerate second-order elliptic equations applies even to certain degenerate variational inequalities.

On the degenerate variational inequality associated to the stopping time problem

On the degenerate variational inequality associated to the stopping time problem