Title: On the regularity of solutions of some linear path-dependent parabolic PDEs
Abstract: We consider a family of linear parabolic partial differential equations defined on the space of continuous path x in C([0,T]), in which coefficients at time t depend on x(t) and the integral of x with respect to A, for some continuous process A with bounded variations. When the equation is uniformly elliptic, we provide conditions on A and the Hölder regularity of the coefficients under which existence of a smooth solution holds, when appealing to the notion of Dupire’s derivatives. It provides a generalization to the existing literature considering situations in which A admits a density with respect to the Lebesgue’s measure, and complements the recent work of Bouchard, Loeper and Tan (2022) on the regularity of approximate viscosity solutions for path-dependent parabolic partial differential equations. We shall also review some recent results on the Dupire-Itô’s formula for path-dependent functionals that are only C^{0,1}, in the sense of Dupire.