TY - JOUR

T1 - Comparison of viscosity solutions of semilinear path-dependent PDEs

AU - Ren, Zhenjie

AU - Touzi, Nizar

AU - Zhang, Jianfeng

N1 - Funding Information:
∗Received by the editors January 18, 2019; accepted for publication (in revised form) October 24, 2019; published electronically January 21, 2020. https://doi.org/10.1137/19M1239404 Funding: The second author was financially supported by the ERC Advanced Grant 321111, and the Chairs Financial Risk and Finance and Sustainable Development of the Louis Bachelier Institute. The research of the third author was supported in part by NSF grant DMS-1413717. †UniversitéParis-Dauphine, PSL Research University, CNRS, UMR [7534], Ceremade, 75016 Paris, France (ren@ceremade.dauphine.fr). ‡CMAP, Ecole Polytechnique Paris, Palaiseau, 91128, France (nizar.touzi@polytechinque.edu). §University of Southern California, Department of Mathematics, Los Angeles, CA, 90089-2532 (jianfenz@usc.edu).
Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Optimal stopping

KW - Path-dependent PDEs

KW - Stochastic control

KW - Viscosity solutions

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U2 - 10.1137/19M1239404

DO - 10.1137/19M1239404

M3 - Article

AN - SCOPUS:85079757824

SN - 0363-0129

VL - 58

SP - 277

EP - 302

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

IS - 1

ER -