Double Obstacle Problem Research Articles

Nonlocal Tug‐of‐War and the Infinity Fractional Laplacian

AbstractMotivated by the “tug‐of‐war” game studied by Peres et al. in 2009, we consider a nonlocal version of the game that goes as follows: at every step two players pick, respectively, a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount ϵ > 0 (as is done in the classical case), it is an s‐stable Levy process that chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically derive a deterministic nonlocal integrodifferential equation that we call the “infinity fractional Laplacian.” We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double‐obstacle problem, both problems having a natural interpretation as tug‐of‐war games. © 2011 Wiley Periodicals, Inc.

Parabolic variational inequality with parameter and gradient constraints

This paper concerns a singular control problem whose value function is governed by a time-dependent HJB equation with gradient constraints. The method is to transform a two-dimensional parabolic variational inequality with gradient constraints into a double obstacle problem with parameter involving two free boundaries that correspond to the investment and disinvestment policies. Moreover we analyze the behaviors of the free boundary surfaces. The main difficulties are to show the free boundary surfaces to be smooth with respect to time and to find the properties of free boundaries with respect to parameter.

Regularity of the Very Weak Solutions for Double Obstacle Problems

The apply of harmonic eauation is know to all. Our interesting is to get the regularity os their solution, recently for obstacle problems. Many interesting results have been obtained for the solutions of harmonic equation ande their obstacle problems, however the double obstacle problems about the definition and regularity results for non-homogeneous elliptic equation. In this paper , the basic tool for the Young inequality,Hölder inequality, Minkowski inequality, Poincaré inequality and a basic inequality. The definition of very weak solutions for double obstacle problems associated with non-homogeneous elliptic equation is given, and the local integrability result is obtained by using the technique of Hodge decomposition.

Continuous dependence on obstacles for the double obstacle problem on metric spaces

Let X be a complete metric space equipped with a doubling Borel measure supporting a p -Poincaré inequality. We obtain various convergence results for solutions of double obstacle problems on open subsets of X . In particular, we consider a sequence of double obstacle problems with converging obstacles and show that the corresponding solutions converge as well. We use the convergence properties to define and study a generalized solution of the double obstacle problem.

Convergence results for obstacle problems on metric spaces

Let X be a complete metric space equipped with a doubling Borel measure supporting a p-Poincaré inequality. We obtain various convergence results for the single and double obstacle problems on open subsets of X. In particular, we consider single and double obstacle problems with fixed obstacles and boundary data on an increasing sequence of open sets.

Free boundaries in optimal transport and Monge-Ampère obstacle problems

Given compactly supported 0 f,g 2 L 1 (R n ), the problem of trans- porting a fraction m min{kfkL1,kgkL1} of the mass of f onto g as cheaply as possible is considered, where cost per unit mass transported is given by a cost function c, typically quadratic c(x,y) = |x y| 2 /2. This question is shown to be equivalent to a double obstacle problem for the Monge-Ampere equation, for which sucient conditions are given to guarantee uniqueness of the solution, such as f vanishing on sptg in the quadratic case. The part of f to be transported increases monotonically with m, and if sptf and sptg are separated by a hyperplane H, then this part will separated from the balance of f by a semiconcave Lipschitz graph over the hyperplane. If f = f and g = g are bounded away from zero and infinity on separated strictly convex domains , R n , for the quadratic cost this graph is shown to be a C 1, loc hypersurface in whose normal coincides with the direction transported; the optimal map between f and g is shown to be Holder continuous up to this free bound- ary, and to those parts of the fixed boundary @ which map to locally convex parts of the path-connected target region.

Continuous-Time Markowitz's Model with Transaction Costs

A continuous-time Markowitz's mean-variance portfolio selection problem is studied in a market with one stock, one bond, and proportional transaction costs. This is a singular stochastic control problem,inherently in a finite time horizon. With a series of transformations, the problem is turned into a so-called double obstacle problem, a well studied problem in physics and partial differential equation literature, featuring two time-varying free boundaries. The two boundaries, which define the buy, sell, and no-trade regions, are proved to be smooth in time. This in turn characterizes the optimal strategy, via a Skorokhod problem, as one that tries to keep a certain adjusted bond-stock position within the no-trade region. Several features of the optimal strategy are revealed that are remarkably different from its no-transaction-cost counterpart. It is shown that there exists a critical length in time, which is dependent on the stock excess return as well as the transaction fees but independent of the investment target and the stock volatility, so that an expected terminal return may not be achievable if the planning horizon is shorter than that critical length (while in the absence of transaction costs any expected return can be reached in an arbitrary period of time). It is further demonstrated that anyone following the optimal strategy should not buy the stock beyond the point when the time to maturity is shorter than the aforementioned critical length. Moreover, the investor would be less likely to buy the stock and more likely to sell the stock when the maturity date is getting closer. These features, while consistent with the widely accepted investment wisdom, suggest that the planning horizon is an integral part of the investment opportunities.

A double obstacle problem arising in differential game theory

A Hamilton–Jacobi equation involving a double obstacle problem is investigated. The link between this equation and the notion of dual solutions—introduced in [S. As Soulaimani, Infinite horizon differential games with asymmetric information, PhD thesis; P. Cardaliaguet, Differential games with asymmetric information, SIAM J. Control Optim. 46 (3) (2007) 816–838; P. Cardaliaguet, C. Rainer, Stochastic differential games with asymmetric information, Appl. Math. Optim. 59 (1) (2009) 1–36] in the framework of differential games with lack of information—is established. As an application we characterize the convex hull of a function in the simplex as the unique solution of some nonlinear obstacle problem.

Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem

This paper concerns optimal investment problem of a CRRA investor who faces proportional transaction costs and finite time horizon. From the angle of stochastic control, it is a singular control problem, whose value function is governed by a time-dependent HJB equation with gradient constraints. We reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. This enables us to make use of the well-developed theory of obstacle problem to attack the problem. The C 2 , 1 regularity of the value function is proven and the behaviors of the free boundaries are completely characterized.

An algorithm for solving the double obstacle problems

In this paper, we extend the algorithm in Xue and Cheng [L. Xue, X.L. Cheng, An algorithm for solving the obstacle problems, Comput. Math. Appl. 48 (2004) 1651–1657] for solving the double obstacle problem. We try to find the approximated region of the contact in the double obstacle problem with less computation time by iteration. Numerical example is given for the double obstacle problem for the elastic–plastic torsion problem to support the algorithm.

Some Convergence Results for Howard's Algorithm

This paper deals with convergence results of Howard's algorithm for the resolution of $\min_{a\in\mathcal{A}} (B^a x - c^a) = 0$, where $B^a$ is a matrix, $c^a$ is a vector, and $\mathcal{A}$ is a compact set. We show a global superlinear convergence result, under a monotonicity assumption on the matrices $B^a$. Extensions of Howard's algorithm for a max-min problem of the form $\max_{b\in\mathcal{B}} \min_{a\in\mathcal{A}} (B^{a,b}x - c^{a,b}) = 0$ are also proposed. In the particular case of an obstacle problem of the form $\min (Ax - b,$ $x - g) = 0$, where A is an $N \times N$ matrix satisfying a monotonicity assumption, we show the convergence of Howard's algorithm in no more than N iterations, instead of the usual $2^N$ bound. Still in the case of an obstacle problem, we establish the equivalence between Howard's algorithm and a primal-dual active set algorithm [M. Hintermüller, K. Ito, and K. Kunisch, SIAM J. Optim., 13 (2002), pp. 865–888]. The algorithms are illustrated on the discretization of nonlinear PDEs arising in the context of mathematical finance (American option and Merton's portfolio problem), of front propagation problems, and for the double-obstacle problem.

Finite-Horizon Optimal Investment with Transaction Costs: A Parabolic Double Obstacle Problem

This paper concerns optimal investment problem of a CRRA investor who faces proportional transaction costs and finite time horizon. Using a partial differential equation approach, we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. This enables us to make use of the well developed theory of variational inequality to study the problem. The $C^{2,1}$ regularity of the value function is proven and the optimal investment policies are completely characterized. Relying on the double obstacle problem, we extend the binomial method widely used in option pricing to determine the optimal investment policies. Numerical examples are presented as well.

Lipschitz Character of Solutions to the Double Inner Obstacle Problems

In our paper we consider the inner problem with l 2 N impediments from below, the inner problem with m 2 N impediments from above and the double inner problem with l + m impediments. Assuming the Lipschitz character of the obstacles we show that the corresponding solutions are also Lipschitz. We extend here the result given in (SV), where the author considered the inner obstacle problem with a single impediment from below. Our work is based on the ideas introduced by J. Jordanov from 1982 who investigated H 1;p () regularity of solutions to inner obstacle problems. In the 1970's there was considerable interest in the analysis of obstacle problems. This was connected with the development of research on variational inequalities and has been studied by many authors (see (BC), (BS), (T) and references therein). The majority of results concentrated on, natural from a mathematical point of view, problems of existence and uniqueness of the solutions. However, in case of variational inequalities corresponding to obstacle problems additional questions regarding, e.g., the coincidence set (cf. (DS1), (DS2)) or regularity of the solutions (cf. (BS)) can be posed. These problems seem to be interesting due to possible applications.

On the minimization of some nonconvex double obstacle problems

We consider a nonconvex variational problem for which the set of admissible functions consists of all Lipschitz functions located between two fixed obstacles. It turns out that the value of the minimization problem at hand is equal to zero when the obstacles do not touch each other; otherwise, it might be positive. The results are obtained through the construction of suitable minimizing sequences. Interpolating these minimizing sequences in some discrete space, a numerical analysis is then carried out.

A Posteriori Error Estimators for Regularized Total Variation of Characteristic Functions

We consider a nonuniformly elliptic double obstacle problem arising from "convexifying" and regularizing a minimization of a functional with total variation in the set of characteristic functions. We derive a posteriori estimators for the discretization error with linear finite elements, which are uniform in the regularization, incorporate computable and local information on the conditioning, vanish in the intersection of discrete and exact contact sets, and are not affected by possible nonuniqueness. Moreover, we integrate these estimators in an adaptive algorithm and illustrate their properties by various numerical experiments.

Stability and higher integrability of derivatives of solutions in double obstacle problems

Higher integrability of the derivatives of solutions to double obstacle problems associated with the second-order quasilinear elliptic differential equation ∇· A( x,∇ u)=0 is obtained under natural assumptions on obstacles. This result is used to prove a stability result for solutions to double obstacle problems for varying equations.

On Singular Limits of Mean-Field Equations

Mean-field equations arise as steady state versions of convection-diffusion systems where the convective field is determined by solution of a Poisson equation whose right-hand side is affine in the solutions of the convection-diffusion equations. In this paper we consider the repulsive coupling case for a system of two convection-diffusion equations. For general diffusivities we prove the existence of a unique solution of the mean-field equation by a variational analysis of a saddle point problem (usually without coercivity). Also we analyze the small-Debye-length limit and prove convergence to either the so-called charge-neutral case or to a double obstacle problem for the limiting potential depending on the data.

Residual type a posteriori error estimates for elliptic obstacle problems

A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed.

The obstacle problem for functions of least gradient

For a given domain $\Omega \subset \Bbb{R}^n$, we consider the variational problem of minimizing the $L^1$-norm of the gradient on $\Omega$ of a function $u$ with prescribed continuous boundary values and satisfying a continuous lower obstacle condition $u\ge \Psi$ inside $\Omega$. Under the assumption of strictly positive mean curvature of the boundary $\partial\Omega$, we show existence of a continuous solution, with H\older exponent half of that of data and obstacle. This generalizes previous results obtained for the unconstrained and double-obstacle problems. The main new feature in the present analysis is the need to extend various maximum principles from the case of two area-minimizing sets to the case of one sub- and one superminimizing set. This we accomplish subject to a weak regularity assumption on one of the sets, sufficient to carry out the analysis. Interesting open questions include the uniqueness of solutions and a complete analysis of the regularity properties of area superminimizing sets. We provide some preliminary results in the latter direction, namely a new monotonicity principle for superminimizing sets, and the existence of ``foamy'' superminimizers in two dimensions.

Travelling waves for a nonlocal double-obstacle problem

Existence, uniqueness and regularity properties are established for monotone travelling waves of a convolution double-obstacle problemut =J*u−u−f (u),the solution u(x, t) being restricted to taking values in the interval [−1, 1]. When u=±1, the equation becomes an inequality. Here the kernel J of the convolution is nonnegative with unit integral and f satisfies f(−1)>0>f(1). This is an extension of the theory in Bates et al. (1997), which deals with this same equation, without the constraint, when f is bistable. Among many other things, it is found that the travelling wave profile u(x−ct) is always ±1 for sufficiently large positive or negative values of its argument, and a necessary and sufficient condition is given for it to be piecewise constant, jumping from −1 to 1 at a single point.