On viscosity solutions of path dependent PDEs

By Ibrahim Ekren, Christian Keller, Nizar Touzi, Jianfeng Zhang. 2011

211 citations recorded by the source corpus.

In this paper we propose a notion of viscosity solutions for path dependent semilinear parabolic PDEs. This can also be viewed as viscosity solutions of non-Markovian Backward SDEs, and thus extends the well known nonlinear Feynman-Kac formula to non-Markovian case. We shall prove the existence, uniqueness, stability, and comparison principle for the viscosity solutions. The key ingredient of our approach is a functional It^ o’s calculus recently introduced by Dupire [6].

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7b5e7e80e68e86d0e87c2b3503ec9f42c928e7bb

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Citations in the corpus

Optimal Stopping under Nonlinear Expectation (2012, methodology): “Our previous paper [5] was restricted to the context of semilinear path-dependent partial differential equations.”
Backward stochastic differential equations driven by G-Brownian motion (2012, background): “[5,6] through which a new result of the existence and uniqueness theorem was obtained.”
Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012, methodology): “One particular application is the existence of the game value for a path dependent zero sum stochastic differential game, due to our wellposedness result of the path dependent Bellman-Isaacs equation, see Pham and Zhang [17].”
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 .”
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.”
Path-Dependent Optimal Stochastic Control and Viscosity Solution of Associated Bellman Equations (2012, background): “[15] studied in a quite different way viscosity solution to second order path-dependent PDE.”
Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data (2019, methodology): “We shall adopt the viscosity solution approach for PPDEs developed by Ekren-Keller-TouziZhang [34], Ekren-Touzi-Zhang [35, 36], and Ren-Touzi-Zhang [63].”