Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness.Access to the published version may require subscription. N.B. When citing this work, cite the original published paper. Abstract. In this paper we study viscosity solutions to the systemwherewhere L is a linear, possibly degenerate, parabolic operator of second order and I is a non-local integro-partial differential operator. A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems when the dynamics of the underlying state variables is described by an N -dimensional Levy process. We establish a general comparison principle for viscosity sub-and supersolutions to the system under mild regularity, growth and structural assumptions on the data, i.e., on the operator H and on continuous functions ψ i , c i,j , and g i . Using the comparison principle we prove the existence of a unique viscosity solution (u 1 , . . . , u d ) to the system by Perron's method. Our main contribution is that we establish existence and uniqueness of viscosity solutions, in the setting of Levy processes and non-local operators, with no sign assumption on the switching costs {c i,j } and allowing c i,j to depend on x as well as t.2000 Mathematics Subject Classification.
Abstract. In this paper we formulate and study an optimal switching problem under partial information. In our model the agent/manager/investor attempts to maximize the expected reward by switching between different states/investments. However, he is not fully aware of his environment and only an observation process, which contains partial information about the environment/underlying, is accessible. It is based on the partial information carried by this observation process that all decisions must be made. We propose a probabilistic numerical algorithm based on dynamic programming, regression Monte Carlo methods, and stochastic filtering theory to compute the value function. In this paper, the approximation of the value function and the corresponding convergence result are obtained when the underlying and observation processes satisfy the linear Kalman-Bucy setting. A numerical example is included to show some specific features of partial information.2000 Mathematics Subject Classification: 60C05, 60F25, 60G35, 60G40, 60H35, 62J02.
We study viscosity solutions to a system of nonlinear degenerate parabolic partial integrodifferential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variable is described by an n-dimensional Lévy process. We first establish a continuous dependence estimate for viscosity sub-and supersolutions to the system under mild regularity, growth and structural assumptions on the partial integro-differential operator and on the obstacles and terminal conditions. Using the continuous dependence estimate, we obtain the comparison principle and uniqueness of viscosity solutions as well as Lipschitz regularity in the spatial variables. Our main contribution is construction of suitable families of viscosity sub-and supersolutions which we use as "barrier functions" to prove Hölder continuity in the time variable, and, through Perron's method, existence of a unique viscosity solution. This paper generalizes parts of the results of Biswas, Jakobsen and Karlsen (2010) [BJK10] and of Lundström, Nyström and Olofsson (2014) [LNO14, LNO14b] to hold for more general systems of equations.2000 Mathematics Subject Classification: 35R09, 49L25, 45G15.
We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear elliptic PDEs on the form F (x, u, Du, D 2 u) = 0 under suitable structure conditions on the equation allowing for non-Lipschitz growth in the gradient terms. In case of smooth boundaries, we also prove the Hopf lemma, the boundary Harnack inequality and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.