Viscosity Solutions of Path-Dependent PDEs with Randomized Time

Abstract

We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [8]. With the new definition, we prove the two important results till now missing in the literature, namely, a general stability result and a comparison result for semicontinuous sub-/super-solutions. 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. . Research supported by ERC 321111Rofirm. The authors are sincerely grateful to Nizar Touzi for valuable discussions.It is not difficult to show that H δ is continuous. Let θ ∈ Θ, with t < T. For u : Θ → R upper semicontinuous, locally bounded from above, the subjet of u in θ is defined byIn a symmetric way, for a lower semicontinuous function u : Θ → R, locally bounded from below, the superjet of u in θ is defined by J L u(θ) := (α, β, γ) ∈ R × R m × S m : u(θ) = E L (u θ − ϕ α,β,γ )(T ∧ H δ , B) for some δ ∈ (0, T − t] .We denote R := R ∪ {−∞, +∞}.

Path-dependent Hamilton-Jacobi equations with super-quadratic growth in the gradient and the vanishing viscosity method

“…as well as bounded and T 0 ℓ(t, a N (t)) dt < ∞, which follows from T 0 ℓ(t, a(t)) dt < ∞, (H2), and the convexity of ℓ(t, •) (cf. the first displayed equation after (38) in [2]). Finally, letting N → ∞ concludes the proof as in [2].…”

Viscosity Solutions to Second Order Elliptic Hamilton-Jacobi-Bellman Equation with infinite delay

“…We also mention that a new concept of viscosity solutions for semi-linear path-dependent partial differential equations was introduced by Ekren, Keller, Touzi and Zhang [9] in terms of a nonlinear expectation, and further extended to fully nonlinear parabolic equations by Ekren, Touzi, and Zhang [10,11], elliptic equations by Ren [30], obstacle problems by Ekren [8] when the Hamilton function H is uniformly nondegenerate, and degenerate second-order equations by Ren, Touzi, and Zhang [31] and Ren and Rosestolato [32] when the nonlinearity H is d p -uniformly continuous in the path function. However, none of the results we know are directly applicable to our situation as in our case the Hamilton function H may be degenerate and is only required to have continuity…”

Partial smoothing of delay transition semigroups acting on special functions

“…In this framework a more general structure than (1.7) can be considered, see e.g. [9] and [24], see also [3] where infinite dimensional path dependent PDEs are considered.…”