Viscosity solutions of path-dependent pdes with randomized time

Description

We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. With the new definition, we prove the two important results, until now missing in the literature, namely, a general stability result and a comparison result for semicontinuous sub-/supersolutions. As an application, we prove the existence of viscosity solutions using the Perron method. Moreover, we connect viscosity solutions of path-dependent PDEs with viscosity solutions of partial differential equations on Hilbert spaces.

Additional details