Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II

Abstract

In our previous paper [Ekren, Touzi and Zhang (2015)], we introduced a notion of viscosity solutions for fully nonlinear pathdependent PDEs, extending the semilinear case of Ekren et al. [Ann. Probab. 42 (2014) 204-236], which satisfies a partial comparison result under standard Lipshitz-type assumptions. The main result of this paper provides a full, well-posedness result under an additional assumption, formulated on some partial differential equation, defined locally by freezing the path. Namely, assuming further that such path-frozen standard PDEs satisfy the comparison principle and the Perron approach for existence, we prove that the nonlinear pathdependent PDE has a unique viscosity solution. Uniqueness is implied by a comparison result.

A BSDE with Delayed Generator Approach to Pricing under Counterparty Risk and Collateralization

“…Analogously, in [33][34][35], the authors proposed an ad hoc notion of viscosity solutions to path-dependent PDE which, similar to the relation established by the FeynmanKac theorem between a stochastic differential equation (SDE) and its deterministic counterpart, relates a path-dependent SDE to a corresponding path-dependent PDE, by exploiting the theory of BSDE, hence by using the notion of nonlinear expectations; see, for example, [36].…”

“…Eventually, summing up all the aforementioned cash flows (35), (36), (38), (40), and (42), we have that the value of the portfolio is given by…”

Functional and Banach Space Stochastic Calculi: Path-Dependent Kolmogorov Equations Associated with the Frame of a Brownian Motion

“…The issue of providing a suitable definition of viscosity solutions for pathdependent PDEs has attracted a great interest, see Peng [33] and Tang and Zhang [42], Ekren et al [18][19][20], Ren et al [34]. In particular, the definition of viscosity solution provided by [18][19][20]34] is characterized by the fact that the classical minimum/maximum property, which appears in the standard definition of viscosity solution, is replaced with an optimal stopping problem under nonlinear expectation [21].…”

“…In particular, the definition of viscosity solution provided by [18][19][20]34] is characterized by the fact that the classical minimum/maximum property, which appears in the standard definition of viscosity solution, is replaced with an optimal stopping problem under nonlinear expectation [21]. Then, probability plays an essential role in this latter definition, which can not be considered as a purely analytic object as the classical definition of viscosity solution is; it is, more properly, a probabilistic version of the classical definition of viscosity solution.…”

A Dual Algorithm for Stochastic Control Problems: Applications to Uncertain Volatility Models and CVA

“…The arguments in this subsection are based on the "frozen-path" approach developed in Ekren, Touzi and Zhang [6]. In order to apply their approach, we have restricted the class of diffusions X α we consider, compared to the Markovian control problem.…”