Notion Of Viscosity Solutions Research Articles

Optimal Control of Infinite-Dimensional Differential Systems with Randomness and Path-Dependence and Stochastic Path-Dependent Hamilton–Jacobi Equations

This paper is devoted to the stochastic optimal control problem of infinite-dimensional differential systems allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases studied by Bayraktar and Keller [J. Funct. Anal. 275 (2018) 2096-2161], the value function turns out to be a random field on the path space and it is characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.

Controlled ordinary differential equations with random path-dependent coefficients and stochastic path-dependent Hamilton–Jacobi equations

This paper is devoted to the stochastic optimal control problem of ordinary differential equations allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases, the value function turns out to be a random field on the path space and it is characterized by a stochastic path-dependent Hamilton–Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations

In this paper, we consider the two-player state and control path-dependent stochastic zero-sum differential game. In our problem setup, the state process, which is controlled by the players, is dependent on (current and past) paths of state and control processes of the players. Furthermore, the running cost of the objective functional depends on both state and control paths of the players. We use the notion of non-anticipative strategies to define lower and upper value functionals of the game, where unlike the existing literature, these value functions are dependent on the initial states and control paths of the players. In the first main result of this paper, we prove that the (lower and upper) value functionals satisfy the dynamic programming principle (DPP), for which unlike the existing literature, the Skorohod metric is necessary to maintain the separability of càdlàg (state and control) spaces. We introduce the lower and upper Hamilton–Jacobi–Isaacs (HJI) equations from the DPP, which correspond to the state and control path-dependent nonlinear second-order partial differential equations. In the second main result of this paper, we show that by using the functional Itô calculus, the lower and upper value functionals are viscosity solutions of (lower and upper) state and control path-dependent HJI equations, where the notion of viscosity solutions is defined on a compact κ-Hölder space to use several important estimates and to guarantee the existence of minimum and maximum points between the (lower and upper) value functionals and the test functions. Based on these two main results, we also show that the Isaacs condition and the uniqueness of viscosity solutions imply the existence of the game value. Finally, we prove the uniqueness of classical solutions for the (state path-dependent) HJI equations in the state path-dependent case, where its proof requires establishing an equivalent classical solution structure as well as an appropriate contradiction argument.

Viscosity solutions to first order path-dependent Hamilton–Jacobi–Bellman equations in Hilbert space

In this article, a notion of viscosity solutions is introduced for first order path-dependent Hamilton–Jacobi–Bellman (PHJB) equations associated with optimal control problems for path-dependent evolution equations in Hilbert space. We identify the value functional of optimal control problems as unique viscosity solution to the associated PHJB equations without the assumption (A.4) on page 231 of Li and Yong (1995). We also show that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property.

Stochastic optimal control with random coefficients and associated stochastic Hamilton\u2013Jacobi\u2013Bellman equations

We consider the optimal control problem for stochastic differential equations (SDEs) with random coefficients under the recursive-type objective functional captured by the backward SDE (BSDE). Due to the random coefficients, the associated Hamilton–Jacobi–Bellman (HJB) equation is a class of second-order stochastic PDEs (SPDEs) driven by Brownian motion, which we call the stochastic HJB (SHJB) equation. In addition, as we adopt the recursive-type objective functional, the drift term of the SHJB equation depends on the second component of its solution. These two generalizations cause several technical intricacies, which do not appear in the existing literature. We prove the dynamic programming principle (DPP) for the value function, for which unlike the existing literature we have to use the backward semigroup associated with the recursive-type objective functional. By the DPP, we are able to show the continuity of the value function. Using the Itô–Kunita’s formula, we prove the verification theorem, which constitutes a sufficient condition for optimality and characterizes the value function, provided that the smooth (classical) solution of the SHJB equation exists. In general, the smooth solution of the SHJB equation may not exist. Hence, we study the existence and uniqueness of the solution to the SHJB equation under two different weak solution concepts. First, we show, under appropriate assumptions, the existence and uniqueness of the weak solution via the Sobolev space technique, which requires converting the SHJB equation to a class of backward stochastic evolution equations. The second result is obtained under the notion of viscosity solutions, which is an extension of the classical one to the case for SPDEs. Using the DPP and the estimates of BSDEs, we prove that the value function is the viscosity solution to the SHJB equation. For applications, we consider the linear-quadratic problem, the utility maximization problem, and the European option pricing problem. Specifically, different from the existing literature, each problem is formulated by the generalized recursive-type objective functional and is subject to random coefficients. By applying the theoretical results of this paper, we obtain the explicit optimal solution for each problem in terms of the solution of the corresponding SHJB equation.

A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution

Determining water surface profiles of steady open channel flows in a one-dimensional bounded domain is one of the well-trodden topics in conventional hydraulic engineering. However, it involves Dirichlet problems of scalar first-order quasilinear ordinary differential equations, which are of mathematical interest. We show that the notion of viscosity solution is useful in thoroughly describing the characteristics of possibly non-smooth and discontinuous solutions to such problems, achieving the conservation of momentum and the entropy condition. Those viscosity solutions are the generalized solutions in the space of bounded measurable functions. Generalized solutions to some Dirichlet problems are not always unique, and a necessary condition for the non-uniqueness is derived. A concrete example illustrates the non-uniqueness of discontinuous viscosity solutions in a channel of a particular cross-sectional shape.

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.

Cauchy noise removal by nonlinear diffusion equations

This paper focuses on the problem of image restoration under Cauchy noise. The variational method, which constructs the data fidelity term involving the Cauchy distribution by MAP estimator, has been proven to be a successful approach. In this paper, a nonlinear diffusion equation is proposed to deal with it. The main ingredients of the proposed equation are a gray level based diffusivity that estimates the amplitude of the noise and a classical gradient based diffusivity that controls the anisotropic diffusion according to the image’s local structure. The proposed equation has the nondivergence form, and its properties, including the existence, uniqueness, and stability of solutions, are established by the notion of viscosity solution. Experimental results show the superiority of the proposed equation over variational methods in restoring small details of images.

Viscosity Solutions to HJB Equations for Boundary-Noise and Boundary-Control Problems

We consider a boundary-noise and boundary-control problem for an equation of parabolic type in a bounded domain in $\mathbb{R}^d$ with a Neumann boundary condition. We study the Hamilton--Jacobi--Bellman equation associated to this problem and introduce an appropriate notion of viscosity solution. We prove the comparison principle for viscosity subsolutions and viscosity supersolutions of the Hamilton--Jacobi--Bellman equation. We also prove various continuity properties of the value function, show that it satisfies the dynamic programming principle, and prove that the value function is the unique viscosity solution of the Hamilton--Jacobi--Bellman equation.

Delay optimal control and viscosity solutions to associated Hamilton–Jacobi–Bellman equations

ABSTRACTIn this article, optimal control problems of differential equations with delays are investigated for which the associated Hamilton–Jacobi–Bellman (HJB) equations are nonlinear partial differential equations with delays. This type of HJB equation has not been previously studied and is difficult to solve because the state equations do not possess smoothing properties. We introduce a new notion of viscosity solutions and identify the value functional of the optimal control problems as the unique solution to the associated HJB equations. An analytical example is given as application.

Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations

In this paper we study the fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and we prove that the value function of the optimal stochastic control problem is the maximal viscosity solution of the associated stochastic HJB equation. For the superparabolic cases when the diffusion coefficients are deterministic functions of time, states and controls, the uniqueness is addressed as well.

Existence and Uniqueness for Integro-Differential Equations with Dominating Drift Terms

In this paper we are interested on the well-posedness of Dirichlet problems associated to integro-differential elliptic operators of order α <1 in a bounded smooth domain Ω. The main difficulty arises because of losses of the boundary condition for sub and supersolutions due to the lower diffusive effect of the elliptic operator compared with the drift term. We consider the notion of viscosity solution with generalized boundary conditions, concluding strong comparison principles in under rather general assumptions over the drift term. As a consequence, existence and uniqueness of solutions in is obtained via Perron's method.

An Approximation Scheme for an Eikonal Equation with Discontinuous Coefficient

We consider the stationary Hamilton--Jacobi equation $\sum_{i,j=1}^N b_{ij}(x)u_{x_i}u_{x_j}= \left[f(x)\right]^2$, in $\Omega$, where $\Omega$ is an open set of $\mathbb{R}^n$, $b$ can vanish at some points, and the right-hand-side $f$ is strictly positive and is allowed to be discontinuous. More precisely, we consider a special class of discontinuities for which the notion of viscosity solution is well-suited. We propose a semi-Lagrangian scheme for the numerical approximation of the viscosity solution in the sense of Ishii and we study its properties. We also prove an a priori error estimate for the scheme in $L^1$. The last section contains some applications to control and image processing problems.

An approximation scheme for a Hamilton–Jacobi equation defined on a network

In this paper we study approximation of Hamilton–Jacobi equations defined on a network. We introduce an appropriate notion of viscosity solution on networks which satisfies existence, uniqueness and stability properties. Then we define an approximation scheme of semi-Lagrangian type by discretizing in time the representation formula for the solution of Hamilton–Jacobi equations and we prove that the discrete problem admits a unique solution. Moreover we prove that the solution of the approximation scheme converges to the solution of the continuous problem uniformly on the network.In the second part of the paper we study a fully discrete scheme obtained via a finite elements discretization of the semi-discrete problem. Also for fully discrete scheme we prove the well posedness and the convergence to the viscosity solution of the Hamilton–Jacobi equation. We also discuss some issues concerning the implementation of the algorithm and we present some numerical examples.

Hamilton-Jacobi Equations with Discontinuous Source Terms

We study the initial-value problem for a Hamilton-Jacobi equation whose Hamiltonian is discontinuous with respect to state variables. Our motivation comes from a model describing the two dimensional nucleation in crystal growth phenomena. A typical equation has a semicontinuous source term. We introduce a new notion of viscosity solutions and prove among other results that the initial-value problem admits a unique global-in-time uniformly continuous solution for any bounded uniformly continuous initial data. We also give a representation formula of the solution as a value function by the optimal control theory with a semicontinuous running cost function.

Uniqueness for Integro-PDE in Hilbert Spaces

The paper deals with fully nonlinear integro-PDE of Hamilton-Jacobi-Bellman type with unbounded terms in infinite dimensional Hilbert spaces. A notion of viscosity solution is introduced for such integro-PDE and a general comparison theorem is proved.

Solvability via viscosity solutions for a model of phase transitions driven by configurational forces

This article is concerned with an initial–boundary value problem for an elliptic–parabolic coupled system arising in martensitic phase transition theory of elastically deformable solid materials, e.g., steel. This model was proposed in Alber and Zhu (2007) [4], and investigated in Alber and Zhu (2006) [3] the existence of weak solutions which are defined in a standard way, however the key technique used in Alber and Zhu (2006) [3] is not applicable to multi-dimensional problem. One of the motivations of this study is to solve this multi-dimensional problem, and another is to investigate the sharp interface limits. Thus we define weak solutions in a way, which is different from Alber and Zhu (2006) [3], by using the notion of viscosity solution. We do prove successfully the existence of weak solutions in this sense for one-dimensional problem, yet the multi-dimensional problem is still open.

Asymptotic behavior of the generalized Korteweg–de Vries–Burgers equation

Cauchy’s problem for a generalization of the KdV–Burgers equation is considered in Sobolev spaces \({H^1(\mathbb{R})}\) and \({H^2(\mathbb{R})}\). We study its local and global solvability and the asymptotic behavior of solutions (in terms of the global attractors). The parabolic regularization technique is used in this paper which allows us to extend the strong regularity properties and estimates of solutions of the fourth order parabolic approximations onto their third order limit—the generalized Korteweg–de Vries–Burgers (KdVB) equation. For initial data in \({H^2(\mathbb{R})}\) we study the notion of viscosity solutions to KdVB, while for the larger \({H^1(\mathbb{R})}\) phase space we introduce weak solutions to that problem. Finally, thanks to our general assumptions on the nonlinear term f guaranteeing that the global attractor is usually nontrivial (i.e., not reduced to a single stationary solution), we study an upper semicontinuity property of the family of global attractors corresponding to parabolic regularizations when the regularization parameter \({\epsilon}\) tends to 0+ (which corresponds the passage to the KdVB equation).

Analytical binomial lookback options with double-exponential jumps

We study the problem of the convergence of the adjusted binomial lookback option in double-exponential jump diffusion models. By using the results of [Dai, M., (2000). A modified binomial tree method for currency lookback options. Acta Mathematica Sinica, 16, 445–454; Kou, S., & Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Management Science, 50, 1178–1192] and [Park, H.S., Kim, K.I., & Qian, X. (2009). A Mathematical modeling for the Lookback option with jump diffusion using Binomial tree method. Journal of Computational and Applied Mathematics, preprint], we show the equivalence between the adjusted binomial tree method and the explicit difference scheme. The convergence is also theoretically proved through the notion of viscosity solution. Numerical results coincide with the theoretical results.

Finite Difference Approximation for Pricing the American Lookback Option

In this paper we are concerned with the pricing of lookback options with American type constrains. Based on the difierential linear complementary formula associated with the pricing problem, an implicit difierence scheme is constructed and analyzed. We show that there exists a unique difierence solution which is unconditionally stable. Using the notion of viscosity solutions, we also prove that the flnite difierence solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the O(¢t+¢x 2 )-order error estimate is derived in the discrete L2-norm provided that the continuous problem is su‐ciently regular. In addition, a numerical example is provided to illustrate the theoretical results.