Abstract
We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables (t, x) ∈ [0, T] × Rd. This new type of PDEs are formulated through a classical BSDE in which the terminal values and the generators are allowed to be general function of Brownian motion paths. In this way, we establish the nonlinear Feynman-Kac formula for a general non-Markovian BSDE. Some main properties of solutions of this new PDEs are also obtained.
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Bismut J M. Conjugate convex functions in optimal stochastic control. J Math Anal Appl, 1973, 44: 384–404
Bouchard B, Touzi N. Discrete time approximation and Monte Carlo simulation of backward stochastic differential equations. Stochastic Process Appl, 2004, 111: 175–206
Briand P, Delyon B, Hu Y, et al. L p-solutions of backward stochastic differential equations. Stochastic Process Appl, 2003, 108: 109–129
Cont R, Fournié D-A. Change of variable formulas for non-anticipative functionals on path space. J Funct Anal, 2010, 259: 1043–1072
Cont R, Fournié D-A. A functional extension of the Itô formula. C R Math Acad Sci Paris, 2010, 348: 57–61
Cont R, Fournié D-A. Functional Itô calculus and stochastic integral representation of martingales. Ann Probab, 2013, 41: 109–133
Douglas J, Ma J, Protter P. Numerical methods for forward-backward stochastic differential equations. Ann Appl Probab, 1996, 6: 940–968
Dupire B. Functional Itô calculus. Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS. Http://ssrn.com/abstract=1435551
Ekren I, Keller C, Touzi N, et al. On viscosity solutions of path dependent PDEs. Ann Probab, 2014, 42: 204–236
Ekren I, Touzi N, Zhang J F. Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. ArXiv:1210.0006, 2012
Ekren I, Touzi N, Zhang J F. Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. ArXiv:1210.0007, 2012
El Karoui N, Peng S G, Quenez M C. Backward stochastic differential equation in finance. Math Finance, 1997, 7: 1–71
Gobet E, Lemor J P, Warin X. A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann Appl Probab, 2005, 15: 2172–2202
Hu Y, Ma J. Nonlinear Feynman-Kac formula and discrete-functional-type BSDEs with continuous coefficients. Stochastic Process Appl, 2004, 112: 23–51
Ma J, Zhang J F. Representation theorems for backward stochastic differential equations. Ann Appl Probab, 2002, 12: 1390–1418
Ma J, Zhang J F. Path regularity for solutions of backward stochastic differential equations. Probab Theory Related Fields, 2002, 122: 163–190
Pardoux E, Peng S G. Adapted solutions of backward stochastic equations. Systems Control Lett, 1990, 14: 55–61
Pardoux E, Peng S G. Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Rozuvskii B L, Sowers R B, eds. Stochastic Partial Differential Equations and Their Applications. Lect Notes Control Inf Sci, vol. 176. Berlin-Heidelberg-New York: Springer, 1992, 200–217
Peng S G. Probabilistic interpretation for systems of quasilinear parabolic partial differential equation. Stochastics Stochastics Rep, 1991, 37: 61–74
Peng S G. A nonlinear Feynman-Kac formula and applications. In: Control Theory, Stochastic Analysis and Applications. River Edge: World Sci Publ, 1991, 173–184
Peng S G. Nonlinear expectations and nonlinear Markov chain. Chin Ann Math B, 2005, 26: 159–184
Peng S G. G-expectation, G-Brownian Motion and Related Stochastic Calculus of Itô type. In: Stochastic Analysis and Applications. Abel Symp, vol. 2. Berlin: Springer, 2007, 541–567
Peng S G. Backward stochastic differential equation, nonlinear expectation and their applications. In: Proceedings of the International Congress of Mathematicians, vol. 1. New Delhi: Hindustan Book Agency, 2011, 393–432
Peng S G. Note on viscosity solution of path-dependent PDE and G-martingales lecture notes. ArXiv:1106.1144v2, 2011
Peng S G, Song Y S. G-expectation weighted Sobolev spaces, backward SDE and path dependent PDE. ArXiv:1305.4722, 2013
Peng S G, Xu M Y. Numerical algorithms for backward stochastic differential equations with 1-dimenisonal Brownian motion: Convergence and simulations. ArXiv:0611864, 2006
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Citation: Peng S G, Wang F L. BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci China Math, 2016, 59, doi: 10.1007/s11425-015-5086-1
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Peng, S., Wang, F. BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci. China Math. 59, 19–36 (2016). https://doi.org/10.1007/s11425-015-5086-1
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DOI: https://doi.org/10.1007/s11425-015-5086-1