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Register a new userViscosity solutions of obstacle problems for Fully nonlinear path-dependent PDEs
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Authors
Ibrahim Ekren
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In this article, we adapt the definition of viscosity solutions to the obstacle problem for fully nonlinear path-dependent PDEs with data uniformly continuous in $(t,omega)$, and generator Lipschitz continuous in $(y,z,gamma)$. We prove that our definition of viscosity solutions is consistent with the classical solutions, and satisfy a stability result. We show that the value functional defined via the second order reflected backward stochastic differential equation is the unique viscosity solution of the variational inequalities.
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This paper introduces a convenient solution space for the uniformly elliptic fully nonlinear path dependent PDEs. It provides a wellposedness result under standard Lipschitz-type assumptions on the nonlinearity and an additional assumption formulated on some partial differential equation defined locally by freezing the path.
We propose a definition of viscosity solutions to fully nonlinear PDEs driven by a rough path via appropriate notions of test functions and rough jets. These objects will be defined as controlled processes with respect to the driving rough path. We show that this notion is compatible with the seminal results of Lions and Souganidis and with the recent results of Friz and coauthors on fully non-linear SPDEs with rough drivers.
We introduce a notion of approximate viscosity solution for a class of nonlinear path-dependent PDEs (PPDEs), including the Hamilton-Jacobi-Bellman type equations. Existence, comparaison and stability results are established under fairly general conditions. It is also consistent with smooth solutions when the dimension is less or equal to two, or the non-linearity is concave in the second order space derivative. We finally investigate the regularity (in the sense of Dupire) of the solution to the PPDE.
We address our interest to the development of a theory of viscosity solutions {`a} la Crandall-Lions for path-dependent partial differential equations (PDEs), namely PDEs in the space of continuous paths C([0, T ]; R^d). Path-dependent PDEs can play a central role in the study of certain classes of optimal control problems, as for instance optimal control problems with delay. Typically, they do not admit a smooth solution satisfying the corresponding HJB equation in a classical sense, it is therefore natural to search for a weaker notion of solution. While other notions of generalized solution have been proposed in the literature, the extension of the Crandall-Lions framework to the path-dependent setting is still an open problem. The question of uniqueness of the solutions, which is the more delicate issue, will be based on early ideas from the theory of viscosity solutions and a suitable variant of Ekelands variational principle. This latter is based on the construction of a smooth gauge-type function, where smooth is meant in the horizontal/vertical (rather than Fr{e}chet) sense. In order to make the presentation more readable, we address the path-dependent heat equation, which in particular simplifies the smoothing of its natural candidate solution. Finally, concerning the existence part, we provide a new proof of the functional It{^o} formula under general assumptions, extending earlier results in the literature.
We prove existence and uniqueness of Crandall-Lions viscosity solutions of Hamilton-Jacobi-Bellman equations in the space of continuous paths, associated to the optimal control of path-dependent SDEs. This seems the first uniqueness result in such a context. More precisely, similarly to the seminal paper of P.L. Lions, the proof of our core result, that is the comparison theorem, is based on the fact that the value function is bigger than any viscosity subsolution and smaller than any viscosity supersolution. Such a result, coupled with the proof that the value function is a viscosity solution (based on the dynamic programming principle, which we prove), implies that the value function is the unique viscosity solution to the Hamilton-Jacobi-Bellman equation. The proof of the comparison theorem in P.L. Lions paper, relies on regularity results which are missing in the present infinite-dimensional context, as well as on the local compactness of the finite-dimensional underlying space. We overcome such non-trivial technical difficulties introducing a suitable approximating procedure and a smooth gauge-type function, which allows to generate maxima and minima through an appropriate version of the Borwein-Preiss generalization of Ekelands variational principle on the space of continuous paths.
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