Lipschitz Regularity of Viscosity Solutions to the Infinity Laplace Equation

In this paper, we study the viscosity solutions of the Neumann problem in a bounded C2 domain Ω, where ΔN∞ is called the normalized infinity Laplacian. The normalized infinity Laplacian was first studied by Peres, Shramm, Sheffield and Wilson from the point of randomized theory named tug-of-war, which has wide applications in optimal mass transportation, financial option price problems, digital image processing, physical engineering, etc. We give the Lipschitz regularity of the viscosity solutions of the Neumann problem. The method we adopt is to choose suitable auxiliary functions as barrier functions and combine the perturbation method and viscosity solutions theory. .

Uniqueness of the Viscosity Solutions for the Initial Value Problem of One Type of Second Order Partial Differential Equation

Aim To prove the uniqueness of the viscosity solutions for the initial value problems of one type of second order parabolic partial differential equations: Methods Using comparison theorem. Results and Conclusion If u0 is uniform continuousfunction in RN , F is continuous function in RNx(N) and F is degenerate elliptic, then thisequation has the sole viscosity solution.

VISCOSITY SOLUTIONS OF HJB EQUATIONS ARISING FROM THE VALUATION OF EUROPEAN PASSPORT OPTIONS

The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the issuer is liable for the net loss. In this article, the mathematical foundation for pricing the European passport option is established. The pricing equation which is a fully nonlinear equation is derived using the dynamic programming principle. The comparison principle, uniqueness and convexity preserving of the viscosity solutions of related H J13 equation are proved. A relationship between the passport and lookback options is discussed.

A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR,UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we de- rive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense, where detachment can occur. Another consequence is this： if interior gra- dient blow up occurs, Perron-type solutions can in general become discontinuous, so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.

Stochastic Viscosity Solutions for SPDEs with Discontinuous Coefficients

In this paper, a class of nonlinear stochastic partial differential equations with discontinuous coefficients is investigated. This study is motivated by some research on stochastic viscosity solutions under non-Lipschitz conditions recently. By studying the solutions of backward doubly stochastic differential equations with discontinuous coefficients and constructing a new approximation function fn to the coefficient f, we get the existence of stochastic viscosity sub-solutions (or super-solutions).The results of this paper can be seen as the extension and application of related articles.

Convergence of the viscosity solution of non-autonomous Hamilton-Jacobi equations

In this paper,we investigate the non-autonomous Hamilton-Jacobi equation{ə_(t)u+H(t,x,ə_(x)=0,u(x,t)_(0))=φ(x),x∈M where H is 1-periodic with respect to t and M is a compact Riemannian manifold without boundary.We obtain the viscosity solution denoted by T_(t_(0))^(t)φ(x)and show T_(t_(0))^(t)φ(x)converges uniformly to a time-periodic viscosity solution u^(*)(x,t)ofə_(t)u+H(t,x,ə_(x)u,u)=0.

Existence of viscosity solutions for ~Iessian equations in exterior domains

The Perron method is used to establish the existence of viscosity solutions to the exterior Dirichlet problems for a class of Hessian type equations with prescribed behavior at infinity.

Viscosity Solutions to a Parabolic Inhomogeneous Equation Associated with Infinity Laplacian

We obtain the existence and uniqueness results of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate and singular parabolic inhomogeneous equation of the form ut-△ ∞N u=f, where An denotes the so-called normalized infinity Laplacian given by △∞ Nu=1/｜Du｜2〈D2uDu,Du〉.

Regularity of viscosity solutions near KAM torus

In this paper, we give weak regularity theorems on P of u ?(x, P), where u ?(x, P) is the viscosity solution of the cell problem H ?(P + D x u ?, x) = $\overline H _\varepsilon $ (P).

VISCOSITY SOLUTIONS DEFINED BY RIGHT DIFFERENTIALS

Abstract. It is proved that for some partial differential equations, the classical notion ofviscosity solution can be defined via right-subdifferentials and superdifferentials of contin-uous functions.

Existence and Uniqueness of Viscosity Solutions for Nonlinear Variational Inequalities Associated with Mixed Control

The author investigates the nonlinear parabolic variational inequality derived from the mixed stochastic control problem on finite horizon.Supposing that some suffi-ciently smooth conditions hold,by the dynamic programming principle,the author builds the Hamilton-Jacobi-Bellman(HJB for short)variational inequality for the value function.The author also proves that the value function is the unique viscosity solution of the HJB variational inequality and gives an application to the quasi-variational inequality.

Viscosity Solutions of Monotonic Functional Parabolic PDE

In this paper,by the technique of coupled solutions,the notion of viscosity solution is ex- tended 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.

Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation

In this paper,we are interested in the regularity estimates of the nonnegative viscosity super solution of theβ−biased infinity Laplacian equationΔ^(β)_(∞)u=0,whereβ∈R is a fixed constant andΔ^(β)_(∞)u:=Δ^(N)_(∞)u+β|D u|,which arises from the random game named biased tug-of-war.By studying directly theβ−biased infinity Laplacian equation,we construct the appropriate exponential cones as barrier functions to establish a key estimate.Based on this estimate,we obtain the Harnack inequality,Hopf boundary point lemma,Lipschitz estimate and the Liouville property etc.

Exponential Convergence to Time-Periodic Viscosity Solutions in Time-Periodic Hamilton-Jacobi Equations

Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold,where the Hamiltonian satisfies the condition:The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit.It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity.

Wavelet Collocation Methods for Viscosity Solutions to Swing Options in Natural Gas Storage

This paper presents the wavelet collocation methods for the numerical ap- proximation of swing options for natural gas storage in a mean reverting market. The model is characterized by the Hamilton-Jacobi-Bellman （HJB） equations which only have the viscosity solution due to the irregularity of the swing option. The differential operator is formulated exactly and efficiently in the second generation interpolating wavelet setting. The convergence and stability of the numerical scheme are studied in the framework of viscosity solution theory. Numerical experiments demonstrate the accuracy and computational efficiency of the methods.

Fully nonlinear stochastic and rough PDEs:Classical and viscosity solutions

We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions.Our notion of viscosity solutions is equivalent to the alternative using semi-jets.Next,we prove basic properties such as consistency,stability,and a partial comparison principle in the general setting.If the diffusion coefficient is semilinear(i.e,linear in the gradient of the solution and nonlinear in the solution;the drift can still be fully nonlinear),we establish a complete theory,including global existence and a comparison principle.

Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

In this paper,we propose a new type of viscosity solutions for fully nonlinear path-dependent PDEs.By restricting the solution to a pseudo-Markovian structure defined below,we remove the uniform non-degeneracy condition needed in our earlier works(Ekren,I,Touzi,N,Zhang,J,Ann Probab,44:1212–1253,2016a;Ekren,I,Touzi,N,Zhang,J,Ann Probab,44:2507–2553,2016b)to establish the uniqueness result.We establish the comparison principle under natural and mild conditions.Moreover,we apply our results to two important classes of PPDEs:the stochastic HJB equations and the path-dependent Isaacs equations,induced from the stochastic optimization with random coefficients and the path-dependent zero-sum game problem,respectively.

THE BERNSTEIN ESTIMATES OF VISCOSITY SOLUTIONS OF LINEAR PARABOLIC EQUATIONS

This paper is concerned with the Bernstein estimates of viscosity solutions of the Cauchy problems for linear parabolic equations. The techniques of viscosity solution method given by H.Ishii and P.L. Lions in [1] allow us to deduce the estimates without differentiating the equation,which is in a way completely different from the classical one. We mainly get the estimate of under the corresponding assumptions on the smoothness of solutions and the known functions in the equation.

VISCOSITY SOLUTION THEORY OF A CLASS OF NONLINEAR DEGENERATE PARABOLIC EQUATIONS Ⅱ. LIPSCHITZ CONTINUITY OF FREE BOUNDARY

In [1] we construct a unique bounded H■lder continuous viscosity solution for the nonlinear PDEs with the evolution p-Laplacian equation and its anisotropic version as typical examples. In this part, we investigate the Lipschitz continuity of the free boundary of viscosity solution and its asymptotic spherical symmetricity, however,this result does not include the anisotropic case.

VISCOSITY SOLUTION THEORY OF A CLASS OF NONLINEAR DEGENERATE PARABOLIC EQUATIONS Ⅰ. UNIQUENESS AND EXISTENCE OF VISCOSITY SOLUTIONS

In this part we construct a unique bounded Holder continuous viscosity solution for the nonlinear PDEs with the evolution p-Laplacian equation and its anisotropic version as typical examples. The existence and properties of free boundaries will be discussed in part Ⅱ.