Comparison of viscosity solutions of semi-linear path-dependent PDEs
Ren, Zhenjie; Touzi, Nizar; Zhang, Jianfeng (2020), Comparison of viscosity solutions of semi-linear path-dependent PDEs, SIAM Journal on Control and Optimization, 58, 1, p. 277–302. 10.1137/19M1239404
TypeArticle accepté pour publication ou publié
Journal nameSIAM Journal on Control and Optimization
SIAM - Society for Industrial and Applied Mathematics
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Centre de Mathématiques Appliquées [CMAP]
Abstract (EN)This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204--236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645--3654; SIAM J. Control Optim., 51 (2013), pp. 4274--4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204--236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution.
Subjects / Keywordsviscosity solutions; optimal stopping; stochastic control; path-dependent PDEs
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