Strong-viscosity solutions: semilinear parabolic PDEs and path-dependent PDEs

april, 2019
Publication type:
Paper in peer-reviewed journals
Journal:
Osaka Journal of Mathematics., vol. 56, No.2
Keywords :
strong-viscosity solutions; viscosity solutions; backward stochastic differential equations; path-dependent partial differential equations.
Abstract:
The aim of the present work is the introduction of a viscosity type solution, called \emph{strong-viscosity solution} to distinguish it from the classical one, with the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. First, we introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semilinear parabolic partial differential equations; we compare it with the standard definition of viscosity solution. Afterwards, we extend the concept of strong-viscosity solution to the case of semilinear parabolic path-dependent partial differential equations, providing an existence and uniqueness result.
BibTeX:
@article{Cos-Rus-2019,
    author={Andrea Cosso and Francesco Russo },
    title={Strong-viscosity solutions: semilinear parabolic PDEs and 
           path-dependent PDEs },
    journal={Osaka Journal of Mathematics. },
    year={2019 },
    month={4},
    volume={56, No.2 },
}