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

By Ibrahim Ekren, Nizar Touzi, Jianfeng Zhang. 2012

182 citations recorded by the source corpus.

The main objective of this paper and the accompanying one [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint] is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work [Ann. Probab. (2014) 42 204–236], focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in [Stochastic Process. Appl. (2014) 124 3277–3311]. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the well-posedness results established in [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint]. We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path-dependent dynamic programming equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/dd26ad98b1d6db41ff2b37c0c53436fcc614e0ca

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In Applied Mathematics

1. Cong Wu
2. Frederi G. Viens
3. Chao Zhou
4. Jianfeng Zhang
5. Xiaolu Tan
6. Dylan Possamaï

In Computational Theory and Mathematics

1. Frederi G. Viens
2. Cong Wu
3. Chao Zhou
4. Jianfeng Zhang
5. Xiaolu Tan
6. Dylan Possamaï

Citations in the corpus

• Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls (2018, methodology): “By Wu & Zhang [36] Lemma 2, we see that DF (t, X̃) · x is a constant on Ei: by setting E = Ei, DF (t, X̃) · x = lim ε→0 f ( t, ∑ j 6=i pjδ{ω̂j} + piδ{ω̂i+εx1[t,T ]} ) − f ( t, ∑ j≥1 pjδ{ω̂j} ) εpi .”
• Stochastic control for a class of nonlinear kernels and applications (2015, background): “In the case of a fully nonlinear PPDE, such a result has been recently achieved by Ren, Touzi and Zhang [82].”
• A martingale approach for fractional Brownian motions and related path dependent PDEs (2017, methodology): “In addition, our framework also covers the fractional Stein-Stein model, where √ V is Gaussian and is the fractional Ornstein-Uhlenbeck process; see Comte & Renault [9], Chronopoulou & Viens [8], and Gulisashvili, Viens, & Zhang [26], and references therein.”