TY - JOUR
T1 - Comparison of viscosity solutions of fully nonlinear degenerate parabolic path-dependent PDEs
AU - Ren, Zhenjie
AU - Touzi, Nizar
AU - Zhang, Jianfeng
N1 - Funding Information:
∗Received by the editors August 19, 2016; accepted for publication (in revised form) May 3, 2017; published electronically October 24, 2017. http://www.siam.org/journals/sima/49-5/M109033.html Funding: The second author was supported by ANR, the Chair Financial Risks of the Risk Foundation sponsored by SociétéGénérale, and the Chair Finance and Sustainable Development sponsored by EDF and Calyon. The third author was supported in part by NSF grant DMS 1413717. †UniversitéParis–Dauphine, PSL Research University, CNRS, UMR [7534], Ceremade, 75016 Paris, France ([email protected]). ‡CMAP, Ecole Polytechnique, Paris, France ([email protected]). §Department of Mathematics, University of Southern California, Los Angeles, CA 90089 ([email protected]).
Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.
PY - 2017
Y1 - 2017
N2 - We prove a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs. In contrast with the previous result in Ekren, Touzi, and Zhang [Ann. Probab., 44 (2016), pp. 2507{2553], our conditions are easier to check and allow for the degenerate case, thus including first order path-dependent PDEs. Our argument follows the reg- ularization method as introduced by Jensen, Lions, and Souganidis [Proc. Amer. Math. Soc., 102, (1988)] in the corresponding finite-dimensional PDE setting. The present argument significantly sim-plifies the comparison proof of Ekren, Touzi, and Zhang but requires an Lp-type of continuity (with respect to the path) for the viscosity semisolutions and for the nonlinearity defining the equation.
AB - We prove a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs. In contrast with the previous result in Ekren, Touzi, and Zhang [Ann. Probab., 44 (2016), pp. 2507{2553], our conditions are easier to check and allow for the degenerate case, thus including first order path-dependent PDEs. Our argument follows the reg- ularization method as introduced by Jensen, Lions, and Souganidis [Proc. Amer. Math. Soc., 102, (1988)] in the corresponding finite-dimensional PDE setting. The present argument significantly sim-plifies the comparison proof of Ekren, Touzi, and Zhang but requires an Lp-type of continuity (with respect to the path) for the viscosity semisolutions and for the nonlinearity defining the equation.
KW - Comparison principle
KW - Path-dependent PDEs
KW - Regularization
KW - Viscosity solutions
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U2 - 10.1137/16M1090338
DO - 10.1137/16M1090338
M3 - Article
AN - SCOPUS:85034024831
SN - 0036-1410
VL - 49
SP - 4093
EP - 4116
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
IS - 5
ER -