Notion Of Viscosity Solutions Research Articles

Convergence of the trinomial tree method for pricing European/American options

In this paper, we present a trinomial tree method for pricing options and show its equivalence to certain explicit difference scheme. Using the numerical analysis and the notion of viscosity solution, we prove the uniform convergence of the trinomial tree method for European/American options. The accuracy and efficiency are shown through numerical simulations.

A Unifying and Rigorous Shape from Shading Method Adapted to Realistic Data and Applications

We propose a new method for the Lambertian Shape From Shading (SFS) problem based on the notion of Crandall-Lions viscosity solution. This method has the advantage of requiring the knowledge of the solution (the surface to be reconstructed) only on some part of the boundary and/or of the singular set (the set of the points at maximal intensity). Moreover it unifies in an unique mathematical formulation the works of Rouy et al. [34, 50], Falcone et al. [21], Prados et al. [46, 48, 49], based on the notion of viscosity solutions and the work of Dupuis and Oliensis [17] dealing with classical solutions and value functions. Also, it allows to generalize their results to the perspective problem recently simultaneously introduced in [13,46,55]. While the theoretical part has been developed in [44], in this paper we give some stability results and we describe numerical schemes for the SFS based on this method. We construct provably convergent and robust algorithms. Finally, we apply our SFS method to real images and we suggest some real-life applications.

Convergence of the binomial tree method for Asian options in jump-diffusion models

The binomial tree methods (BTM), first proposed by Cox, Ross and Rubinstein [J. Cox, S. Ross, M. Rubinstein, Option pricing: A simplified approach, J. Finan. Econ. 7 (1979) 229–264] in diffusion models and extended by Amin [K.I. Amin, Jump diffusion option valuation in discrete time, J. Finance 48 (1993) 1833–1863] to jump-diffusion models, is one of the most popular approaches to pricing options. In this paper, we present a binomial tree method for Asian options in jump-diffusion models and show its equivalence to certain explicit difference scheme. Employing numerical analysis and the notion of viscosity solution, we prove the uniform convergence of the binomial tree method for European-style and American-style Asian options.

A viscosity solution method for Shape-From-Shading without image boundary data

In this paper we propose a solution of the Lambertian shape-from-shading (SFS) problem by designing a new mathematical framework based on the notion of viscosity solution. The power of our approach is twofolds: (1) it defines a notion of weak solutions (in the viscosity sense) which does not necessarily require boundary data. Moreover, it allows to characterize the viscosity solutions by their “minimums”; and (2) it unifies the works of [Rouy and Tourin, SIAM J. Numer. Anal. 29 (1992) 867–884], [Lions et al ., Numer. Math. 64 (1993) 323–353], [Falcone and Sagona, Lect. Notes Math. 1310 (1997) 596–603], [Prados et al. , Proc. 7th Eur. Conf. Computer Vision 2351 (2002) 790–804; Prados and Faugeras, IEEE Comput. Soc. Press 2 (2003) 826–831], based on the notion of viscosity solutions and the work of [Dupuis and Oliensis, Ann. Appl. Probab. 4 (1994) 287–346] dealing with classical solutions.

Viscosity solutions of fully nonlinear functional parabolic PDE

By the technique of coupled solutions, the notion of viscosity solutions is extended to fully nonlinear retarded parabolic equations. Such equations involve many models arising from optimal control theory, economy and finance, biology, and so forth. The comparison principle is shown. Then the existence and uniqueness are established by the fixed point theory.

Viscosity Solutions of Monotonic Functional Parabolic PDE

In this paper, by the technique of coupled solutions, the notion of viscosity solution is extended to quasi–monotonic fully nonlinear parabolic equations with delay, which involves many models arising from optimal control theory, economy and finance, biology etc. The comparison, existence and uniqueness are proved. And the results are applied to the retarded Bellman equations.

Convergence of Binomial Tree Methods for European/American Path-Dependent Options

The binomial tree method, first proposed by Cox, Ross, and Rubinstein (Journal of Financial Economics, 7 (1979), pp. 229-263), is one of the most popular approaches to pricing options. By introducing an additional path-dependent variable, such methods can be readily extended to the valuation of path-dependent options. In this paper, using numerical analysis and the notion of viscosity solutions, we present a unifying theoretical framework to show the uniform convergence of binomial tree methods for European/American path-dependent options, including arithmetic average options, geometric average options, and lookback options.

A Finite Difference Scheme for Option Pricing in Jump-diffusion and Exponential Levy Models

We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit time-stepping scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Numerical tests are performed for the Merton jump-diffusion model and for the Variance Gamma model with smooth and non-smooth payoff functions. Our scheme can be used for European and barrier options, applies in the case of pure-jump models or degenerate diffusion coefficients, and extends to time-dependent coefficients.

Hamilton-Jacobi equations with measurable dependence on the state variable

We study the Hamilton-Jacobi equation \[H(x,Du)=0 , \] where $H(x,p)$ is assumed to be measurable in $x$, quasiconvex and continuous in $p$. The notion of viscosity solution is adapted to the measurable setting making use of suitable measure--theoretic devices. We obtain integral representation formulae generalizing the ones valid for continuous equations, comparison principles and uniqueness results. We examine stability properties of the new definition and present two approximation procedures: the first one is based on a regularization of the Hamiltonian by mollification and in the second one the approximating sequence is made up by minimizers of certain variational integrals.

A Modified Binomial Tree Method for Currency Lookback Options

The binomial tree method is the most popular numerical approach to pricing options. However, for currency lookback options, this method is not consistent with the corresponding continuous models, which leads to slow speed of convergence. On the basis of the PDE approach, we develop a consistent numerical scheme called the modified binomial tree method. It possesses one order of accuracy and its efficiency is demonstrated by numerical experiments. The convergence proofs are also produced in terms of numerical analysis and the notion of viscosity solution.

Plastic Torsion and Related Problems

A simple algorithm to calculate the maximum torsional load for a cylindrical shaft is presented. The algorithm is based on the notion of viscosity solutions to the eikonal equation, and is not restricted to simply-connected cross-sections. Applications to other, related problems, such as ferromagnetic thin films, and elastic buckling of thin film blisters are also discussed.

On viscosity solutions of fully nonlinear equations with measurable ingredients

We study fully nonlinear, uniformly elliptic equations with measurable ingredients. Caffarelli's recent work on W2,p estimates for viscosity solutions has led to significant progress in this area. Here we present a unified treatment of this theory based on an appropriate notion of viscosity solution. For instance, it is shown that strong solutions are viscosity solutions, that viscosity solutions are twice differentiable a.e., and that the pointwise derivatives satisfy the equation a.e. An important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations. This extends results of this type due to Evans and Krylov. Our work is to some extent expository, the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients. © 1996 John Wiley & Sons, Inc.

Global behaviour of solutions to a parabolic mean curvature equation

Consider (1.1) for a domain $\Omega$ for which there is no classical nonparametric solution of the stationary problem. We study viscosity solutions of (1.1). In general they fail to satisfy Dirichlet data on the boundary and "detach." In fact the solution tends to infinity with finite speed. The velocity stabilizes as $t\to \infty$, and we give some results on asymptotic growth. These new effects can be reconciled with the notion of viscosity solutions. The free boundary data are shown to be Lipschitzian for special domains $\Omega$. Problem (1.1) is related to some isoperimetric geometric problem.

On Hamilton–Jacobi equations in bounded domains

SynopsisInitial-boundary value problems for nonlinear first order partial differential equations ∂tu + H(x, t, u, Dxu) = 0 and corresponding boundary value problems H(x, u, Dxu) = 0 are studied in bounded sets, using Crandal's and Lions' notion of viscosity solutions. We give pointwise conditions on the boundary data that guarantee the existence of such solutions and estimate their moduli of continuity in terms of continuity properties of the data. The results are applied to properties of the value function for certain differential games.

Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman’s and Isaacs’ Equations

Recent work by the authors and others has demonstrated the connections between the dynamic programming approach to optimal control theory and to two-person, zero-sum differential games problems and the new notion of “Viscosity” solutions of Hamilton–Jacobi PDE’s introduced by M. G. Crandall and P.-L. Lions. In particular, it has been proved that the dynamic programming principle implies that the value function is the viscosity solution of the associated Hamilton–Jacobi–Bellman and Isaacs equations. In the present work, it is shown that viscosity super- and subsolutions of these equations must satisfy some inequalities called super- and subdynamic programming principle respectively. This is then used to prove the equivalence between the notion of viscosity solutions and the conditions, introduced by A. Subbotin, concerning the sign of certain generalized directional derivatives.

Remarques sur des résultats d’existence pour les équations de Hamilton-Jacobi du premier ordre

We complement some existence results obtained previously by P. L. Lions for first-order Hamilton Jacobi Equations. These results rely on the notion of viscosity solution.

Existence results for first order Hamilton Jacobi equations

We prove existence results for the classical first-order Hamilton Jacobi Equations. These results rely on the notion of viscosity solution and they are obtained under the same assumptions as the uniqueness optimal results for bounded, uniformly continuous solutions.