Framework Of Viscosity Solutions Research Articles

Pathwise Stochastic Control and a Class of Stochastic Partial Differential Equations

In this article, we study a stochastic optimal control problem in the pathwise sense, as initially proposed by Lions and Souganidis in [C. R. Acad. Sci. Paris Ser. I Math., 327 (1998), pp. 735-741]. The corresponding Hamilton-Jacobi-Bellman (HJB) equation, which turns out to be a non-adapted stochastic partial differential equation, is analyzed. Making use of the viscosity solution framework, we show that the value function of the optimal control problem is the unique solution of the HJB equation. When the optimal drift is defined, we provide its characterization. Finally, we describe the associated conserved quantities, namely the space-time transformations leaving our pathwise action invariant.

The evolution problem associated with the fractional first eigenvalue

In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as t→∞ with a bound that is given by the first eigenvalue for this problem that we also study.

On a new singular and degenerate extension of the [formula omitted]-Laplace operator

We study a novel degenerate and singular elliptic operator Δ˜(τ,χ) defined by Δ˜(τ,χ)u=τ(x,Du)(|Du|Δ1u+χ(x,Du)Δ∞u), where the singular weights τ(x,s)>0 and χ(x,s)≥0 are continuous functions on Ω×Rn∖{0}. The operator Δ˜(τ,χ) is an extension of Δ(p,q)u=|Du|qΔ1u+(p−1)|Du|p−2Δ∞u,p≥1,q≥0, introduced by the authors in Baravdishet al. (2020), which in turn is an extension of the p-Laplace operator Δp. We establish the well-posedness of the Neumann boundary value problem for the parabolic equation ut=Δ˜(τ,χ)u in the framework of viscosity solutions. For the solution u, the weight χ controls the evolution along the tangential and the normal directions, respectively, on the level surface of u. The weight τ controls the total speed of the evolution of u. We also prove the consistency and the convergence of the numerical scheme for the finite differences method of the parabolic equation above. Numerical simulations show that our novel nonlinear operator Δ˜(τ,χ) gives better results than both the Perona–Malik (Perona and Malik, 1990) and total variation (TV) methods (Chan and Shen, 2005) when applied to image enhancement.

Fractional convexity

We introduce a notion of fractional convexity that extends naturally the usual notion of convexity in the Euclidean space to a fractional setting. With this notion of fractional convexity, we study the fractional convex envelope inside a domain of an exterior datum (the largest possible fractional convex function inside the domain that is below the datum outside) and show that the fractional convex envelope is characterized as a viscosity solution to a non-local equation that is given by the infimum among all possible directions of the 1-dimensional fractional laplacian. For this equation we prove existence, uniqueness and a comparison principle (in the framework of viscosity solutions). In addition, we find that solutions to the equation for the convex envelope are related to solutions to the fractional Monge–Ampere equation.

Stochastic recursive optimal control problem with mixed delay under viscosity solution's framework

SummaryThis article is concerned with the stochastic recursive optimal control problem with mixed delay. The connection between Pontryagin's maximum principle and Bellman's dynamic programming principle is discussed. Without containing any derivatives of the value function, relations among the adjoint processes and the value function are investigated by employing the notions of super‐ and sub‐jets introduced in defining the viscosity solutions. Stochastic verification theorem is also given to verify whether a given admissible control is really optimal.

Two-species competing population dynamics with the population-dependent environmental capacities under random disturbance.

We propose and analyze a stochastic competing two-species population dynamics model subject to jump and continuous correlated noises. Competing benthic algae population dynamics in river environment, which is an important engineering problem, motivates this new model. The model is a system of stochastic differential equations having a characteristic that the two populations are competing with each other through the environmental capacities; an increase in one population decreases the other's environmental capacity. Unique existence of the uniformly bounded strong solution is proven, and attractors of the solutions are identified depending on the parameter values. The Kolmogorov's backward equation associated with the population dynamics is formulated and its unique solvability in a Banach space with a weighted norm is discussed. A novel uncertain correlation case is also analyzed in the framework of viscosity solutions. Numerical computation results using a finite difference scheme and a Monte-Carlo method are presented to deeper analyze the model. Our analysis results can be utilized for establishment of a foundation for modeling, analysis and control of the competing population dynamics.

Stochastic control of single-species population dynamics model subject to jump ambiguity

A logistic type stochastic control model for cost-effective single-species population management subject to an ambiguous jump intensity is presented based on the modern multiplier-robust formulation. The Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation for finding the optimal control is then derived. Mathematical analysis of the HJBI equation from the viewpoint of viscosity solutions is carried out with an emphasis on the non-linear and non-local term, which is a key term arising due to the jump ambiguity. We show that this term can be efficiently handled in the framework of viscosity solutions by utilizing its monotonicity property. A numerical scheme to discretize the HJBI equation is presented as well. Our model is finally applied to management of algae population in river environment. Optimal management policies ranging from the short-term to long-term viewpoints are numerically investigated.

A unique value function for an optimal control problem of irrigation water intake from a reservoir harvesting flash floods

Operation of reservoirs is a fundamental issue in water resource management. We herein investigate well-posedness of an optimal control problem for irrigation water intake from a reservoir in an irrigation scheme, the water dynamics of which is modeled with stochastic differential equations. A prototype irrigation scheme is being developed in an arid region to harvest flash floods as a source of water. The Hamilton–Jacobi–Bellman (HJB) equation governing the value function is analyzed in the framework of viscosity solutions. The uniqueness of the value function, which is a viscosity solution to the HJB equation, is demonstrated with a mathematical proof of a comparison theorem. It is also shown that there exists such a viscosity solution. Then, an approximate value function is obtained as a numerical solution to the HJB equation. The optimal control strategy derived from the approximate value function is summarized in terms of rule curves to be presented to the operator of the irrigation scheme.

Numerical investigation of the free boundary regularity for a degenerate advection-diffusion problem

We study the free boundary regularity of the traveling wave solutions to a degenerate advection-diffusion problem of Porous Medium type, whose existence was proved in [24]. We set up a finite difference scheme allowing to compute approximate solutions and capture the free boundaries, and we carry out a numerical investigation of their regularity. Based on some nondegeneracy assumptions supported by solid numerical evidence, we prove the Lipschitz regularity of the free boundaries. Our simulations indicate that this regularity is optimal, and the free boundaries seem to develop Lipschitz corners at least for some values of the nonlinear diffusion exponent. We discuss analytically the existence of corners in the framework of viscosity solutions to certain periodic Hamilton–Jacobi equations, whose validity is again supported by numerical evidence.

Existence and uniqueness for parabolic problems with Caputo time derivative

In this paper we are interested in the well-posedness of fully nonlinear Cauchy problems in which the time derivative is of Caputo type. We address this question in the framework of viscosity solutions, obtaining the existence via Perron's method, and comparison for bounded sub and supersolutions by a suitable regularization through inf and sup convolution in time. As an application, we prove the steady-state large time behavior in the case of proper nonlinearities and provide a rate of convergence by using the Mittag–Leffler operator.

Connection between MP and DPP for Stochastic Recursive Optimal Control Problems: Viscosity Solution Framework in the General Case

This paper deals with a stochastic recursive optimal control problem, where the diffusion coefficient depends on the control variable and the control domain is not necessarily convex. We focus on the connection between the general maximum principle and the dynamic programming principle for such a control problem without the assumption that the value is smooth enough; the set inclusions among the sub- and super-jets of the value function and the first-order and second-order adjoint processes as well as the generalized Hamiltonian function are established. Moreover, by comparing these results with the classical ones in Yong and Zhou [Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999], it is natural to obtain the first- and second-order adjoint equations of Hu [Probability, Uncertainty and Quantitative Risk, 2 (2017), 1].

Convergence of meshfree collocation methods for fully nonlinear parabolic equations

We prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions, provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step. A numerical experiment with a radial basis function demonstrates the convergence property.

Analysis and Approximation of Some Shape-from-Shading Models for Non-Lambertian Surfaces

The reconstruction of a 3D object or a scene is a classical inverse problem in Computer Vision. In the case of a single image this is called the Shape-from-Shading (SfS) problem and it is known to be ill-posed even in a simplified version like the vertical light source case. A huge number of works deals with the orthographic SfS problem based on the Lambertian reflectance model, the most common and simplest model which leads to an eikonal-type equation when the light source is on the vertical axis. In this paper, we want to study non-Lambertian models since they are more realistic and suitable whenever one has to deal with different kind of surfaces, rough or specular. We will present a unified mathematical formulation of some popular orthographic non-Lambertian models, considering vertical and oblique light directions as well as different viewer positions. These models lead to more complex stationary non-linear partial differential equations of Hamilton---Jacobi type which can be regarded as the generalization of the classical eikonal equation corresponding to the Lambertian case. However, all the equations corresponding to the models considered here (Oren---Nayar and Phong) have a similar structure so we can look for weak solutions to this class in the viscosity solution framework. Via this unified approach, we are able to develop a semi-Lagrangian approximation scheme for the Oren---Nayar and the Phong model and to prove a general convergence result. Numerical simulations on synthetic and real images will illustrate the effectiveness of this approach and the main features of the scheme, also comparing the results with previous results in the literature.

Iterative scheme for an elliptic non-local free boundary problem

In this paper we treat a non-local free boundary problem arising in financial bubbles, where the model is set in the framework of viscosity solutions. We suggest an iterative scheme which consists of a sequence of obstacle problems at each step to be solved, that in turn gives the next obstacle function in the iteration scheme. The suggested approach gives a solution to the theoretical problem and paves the way for the numerical implementation, as done in the text below.

A non-local free boundary problem arising in a theory of financial bubbles

We consider an evolution non-local free boundary problem that arises in the modelling of speculative bubbles. The solution of the model is the speculative component in the price of an asset. In the framework of viscosity solutions, we show the existence and uniqueness of the solution. We also show that the solution is convex in space, and establish several monotonicity properties of the solution and of the free boundary with respect to parameters of the problem. To study the free boundary, we use, in particular, the fact that the odd part of the solution solves a more standard obstacle problem. We show that the free boundary is [Formula: see text] and describe the asymptotics of the free boundary as c, the cost of transacting the asset, goes to zero.

Existence of viscosity solution for a singular Hamilton–Jacobi equation

In this paper we study the existence of a singular Hamilton–Jacobi equation under the framework of viscosity solutions. The analysis is inspired by the arguments of [8] where a study of a model on dislocation dynamics was considered.

Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies

Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is the selection of tumor cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the expression level of a resistance gene (or genes, yielding a phenotype) influencing in healthy and tumor cells birth/death rates, effects of chemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work by demonstrating how qualitatively different actions of chemotherapeutic and cytostatic treatments may induce different levels of resistance. The mathematical interest of our study is in the formalism of constrained Hamilton–Jacobi equations in the framework of viscosity solutions. We derive the long-term temporal dynamics of the fittest traits in the regime of small mutations. In the context of adaptive cancer management, we also analyse whether an optimal drug level is better than the maximal tolerated dose.

A Non Local Free Boundary Problem Arising in a Theory of Financial Bubbles

In this paper, we consider an evolution non local free boundary problem that arises in the modeling of speculative bubbles. The solution of the model is the speculative component in the price of an asset. In the framework of viscosity solutions, we show the existence and uniqueness of the solution. We also show the convexity in space of the solution, and various monotonicity properties of the solution and of the free boundary with respect to parameters of the problem.In order to study the free boundary, we use in particular the fact that the odd part of the solution solves a more standard obstacle problem. We show moreover that the free boundary is Lipschitz continuous, and describe the asymptotics of the free boundary as c, the cost of transacting the asset, goes to zero.

A semi-Lagrangian scheme for the game p-Laplacian via p-averaging

We present and analyze an approximation scheme for the two-dimensional game p-Laplacian in the framework of viscosity solutions. The approximation is based on a semi-Lagrangian scheme which exploits the idea of p-averages. We study the properties of the scheme and prove that it converges, in particular cases, to the viscosity solution of the game p-Laplacian. We also present a numerical implementation of the scheme for different values of p; the numerical tests show that the scheme is accurate.

Stochastic verification theorem of forward–backward controlled systems for viscosity solutions

In this paper, we investigate the controlled systems described by forward–backward stochastic differential equations with the control contained in drift, diffusion and generator of BSDEs. A new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions. It is worth pointing out that this theorem has wider applicability than the restrictive classical verification theorems. As a relevant problem, the optimal stochastic feedback controls for forward–backward systems are discussed as well.