2012/10/31 by Ibrahim Ekren, Nizar Touzi, Jianfeng Zhang · 1 citation
Mathematics · #math.PR
paper · pdf · doi:10.1214/15-aop1027
published as Annals of Probability 2016, Vol. 44, No. 4, 2507-2553 · Published at http://dx.doi.org/10.1214/15-AOP1027 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
arxiv created 2016/09/27 · arxiv updated 2016/09/28
In our previous paper [Ekren, Touzi and Zhang (2015)], we introduced a notion of viscosity solutions for fully nonlinear path-dependent PDEs, extending the semilinear case of Ekren et al. [Ann. Probab. 42 (2014) 204-236], which satisfies a partial comparison result under standard Lipshitz-type assumptions. The main result of this paper provides a full, well-posedness result under an additional assumption, formulated on some partial differential equation, defined locally by freezing the path. Namely, assuming further that such path-frozen standard PDEs satisfy the comparison principle and the Perron approach for existence, we prove that the nonlinear path-dependent PDE has a unique viscosity solution. Uniqueness is implied by a comparison result.