\(G\)-expectation weighted Sobolev spaces, backward SDE and path dependent PDE. (English) Zbl 1335.60098

Peng, Shige; Song, Yongsheng

J. Math. Soc. Japan 67, No. 4, 1725-1757 (2015).

Summary: Beginning from a space of smooth, cylindrical and non-anticipative processes defined on a Wiener probability space \((\Omega, \mathcal{F}, P)\), we introduce a \(P\)-weighted Sobolev space, or “\(P\)-Sobolev space”, of non-anticipative path-dependent processes \(u=u(t,\omega)\) such that the corresponding Sobolev derivatives \(\mathcal{D}_{t}+(1/2)\Delta_x\) and \(\mathcal{D}_{x}u\) of Dupire’s type are well defined on this space. We identify each element of this Sobolev space with the one in the space of classical \(L_P^p\) integrable Itô processes. Consequently, a new path-dependent Itô formula is applied to all such Itô processes. {
} It follows that the path-dependent nonlinear Feynman-Kac formula is satisfied for most \(L^p_P\)-solutions of backward SDEs: each solution of such a BSDE is identified with the solution of the corresponding quasi-linear path-dependent PDE (PPDE). Rich and important results on existence, uniqueness, monotonicity and regularity of BSDEs obtained in the past decades can be directly applied to obtain their corresponding properties in the new fields of PPDEs. {
} In the above framework of a \(P\)-Sobolev space based on the Wiener probability measure \(P\), only the derivatives \(\mathcal{D}_{t}+(1/2)\Delta_x\) and \(\mathcal{D}_{x}u\) are well-defined and well-integrated. This prevents us from formulating and solving a fully nonlinear PPDE. We then replace the linear Wiener expectation \(E_P\) by a sublinear \(G\)-expectation \(\mathbb{E}^G\) and thus introduce the corresponding \(G\)-expectation weighted Sobolev space, or “\(G\)-Sobolev space”, in which the derivatives \(\mathcal{D}_{t}u\), \(\mathcal{D}_xu\) and \(\mathcal{D}^2_{x}u\) are all well-defined separately. We then formulate a type of fully nonlinear PPDEs in the \(G\)-Sobolev space and then identify them with a type of backward SDEs driven by \(G\)-Brownian motion.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30	Applications of stochastic analysis (to PDEs, etc.)
60E05	Probability distributions: general theory
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J15	Second-order elliptic equations
35K10	Second-order parabolic equations

Keywords:

backward stochastic differential equations; \(P\)-weighted Sobolev spaces; \(G\)-expectation weighted Soboloev spaces; path dependent partial differential equations; \(G\)-Brownian motion; \(G\)-martingale

References:

[1]	J. M. Bismut, Conjugate Convex Functions in Optimal Stochastic Control, J. Math. Anal. Apl., 44 (1973), 384-404. · Zbl 0276.93060
[2]	R. Cont and D. Fournie, Functional Itô calculus and stochastic integral representation of martingales, Ann. Prob., 41 (2013), 109-133. · Zbl 1272.60031
[3]	F. Coquet, Y. Hu, J. Memin and S. Peng, Filtration Consistent Nonlinear Expectations and Related g-Expectations, Probab. Theory Relat. Fields, 123 (2002), 1-27. · Zbl 1007.60057
[4]	M. Crandall, H. Ishii and P.-L. Lions, User’s Guide to Viscosity Solutions of Second Order Partial Differential Equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. · Zbl 0755.35015
[5]	F. Delbaen, Coherent Risk Measures (Lectures given at the Cattedra Galileiana at the Scuola Normale di Pisa, March 2000), the Scuola Normale di Pisa, 2002.
[6]	L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a sublinear expectation: application to \(G\)-Brownian motion pathes, Potential Anal., 34 (2011), 139-161. · Zbl 1225.60057
[7]	B. Dupire, Functional Itô calculus, papers.ssrn.com., 2009.
[8]	I. Ekren, Ch. Keller, N. Touzi and J. Zhang, On Viscosity Solutions of Path Dependent PDEs, Ann. Probab., 42 (2014), 204-236. · Zbl 1320.35154
[9]	H. Föllmer and A. Schied, Statistic Finance, Walter de Gruyter, 2004. · Zbl 1126.91028
[10]	I. Ekren, N. Touzi and J. Zhang, Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I-II, 2012, arXiv: arXiv: arXiv:1210.0007v1 · Zbl 1375.35250
[11]	N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71. · Zbl 0884.90035
[12]	M. Hu, S. Ji, S. Peng and Y. Song, Backward Stochastic Differential Equations Driven by \(G\)-Brownian Motion, Stochastic Processes and their Applications, 124 (2014), 759-784. · Zbl 1300.60074
[13]	M. Hu and S. Peng, On Representation Theorem of \(G\)-Expectations and Paths of \(G\)-Brownian Motion, Acta Math. Appl. Sin. Engl. Ser., 25 (2009), 539-546. · Zbl 1190.60043
[14]	N. V. Krylov, Nonlinear Parabolic and Elliptic Equations of the Second Order, Reidel Publishing Company, (Original Russian Version by Nauka, Moscow, 1985), 1987. · Zbl 0586.35002
[15]	E. Pardoux and S. Peng, Adapted Solutions of Backward Stochastic Equations, Systerm and Control Letters, 14 (1990), 55-61. · Zbl 0692.93064
[16]	E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Stochastic partial differential equations and their applications, Proc. IFIP, LNCIS, 176 (1992), 200-217. · Zbl 0766.60079
[17]	S. Peng, Probabilistic Interpretation for Systems of Quasilinear Parabolic Partial Differential Equations, Stochastics, 37 (1991), 61-74. · Zbl 0739.60060
[18]	S. Peng, BSDE and related g-expectation, In: Pitman Research Notes in Mathematics Series, 364 , Backward Stochastic Differential Equation, (eds. N. El Karoui and L. Mazliak), 1997, 141-159. · Zbl 0892.60066
[19]	S. Peng, Nonlinear Expectations, Nonlinear Evaluations and Risk Measures, Lectures Notes in CIME-EMS Summer School, 2003, Bressanone, Springer’s Lecture Notes in Math., 1856. · Zbl 1127.91032
[20]	S. Peng, Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 26B (2005), 159-184. · Zbl 1077.60045
[21]	S. Peng, \(G\)-expectation, \(G\)-Brownian Motion and Related Stochastic Calculus of Itô type, Stochastic analysis and applications, Abel Symp., 2 , Springer, Berlin, 2007, 541-567.
[22]	S. Peng, \(G\)-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty, 2007, arXiv: arXiv: arXiv:0711.2834v1
[23]	S. Peng, Multi-Dimensional \(G\)-Brownian Motion and Related Stochastic Calculus under \(G\)-Expectation, Stochastic Process. Appl., 118 (2008), 2223-2253. · Zbl 1158.60023
[24]	S. Peng, Nonlinear Expectations and Stochastic Calculus under Uncertainty, 2010, arXiv: arXiv: arXiv:1002.4546v1 · Zbl 1233.60031
[25]	S. Peng, Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications, In: Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010. · Zbl 1233.60031
[26]	S. Peng, Y. Song and J. Zhang, A Complete Representation Theorem for \(G\)-martingales, Stochastics: An International Journal of Probability and Stochastic Processes, 86 (2014), 609-631. · Zbl 1337.60130
[27]	S. Peng and F. Wang, BSDE, Path-dependent PDE and Nonlinear Feynman-Kac Formula, 2011, arXiv: arXiv: arXiv:1108.4317v1 · Zbl 1226.93127
[28]	Z. Ren, N. Touzi and J. Zhang, Comparison of viscosity solutions of semilinear path-dependent partial differential equations, 2014, arXiv: arXiv: arXiv:1410.7291
[29]	M. Soner, N. Touzi and J. Zhang, Martingale Representation Theorem under \(G\)-expectation, Stochastic Processes and their Applications, 121 (2011), 265-287. · Zbl 1228.60070
[30]	M. Soner, N. Touzi and J. Zhang, Well-posedness of Second Order Backward SDEs, Probability Theory and Related Fields, 153 (2012), 149-190. · Zbl 1252.60056
[31]	Y. Song, Some properties on \(G\)-evaluation and its applications to \(G\)-martingale decomposition, Science China Mathematics, 54 (2011), 287-300. · Zbl 1225.60058
[32]	Y. Song, Properties of hitting times for \(G\)-martingales and their applications, Stochastic Processes and their Applications, 121 (2011), 1770-1784. · Zbl 1231.60054
[33]	Y. Song, Uniqueness of the representation for \(G\)-martingales with finite variation, Electron. J. Probab., 17 (2012), 1-15. · Zbl 1244.60046
[34]	Y. Song, Characterizations of processes with stationary and independent increments under \(G\)-expectation, Annales de l’Institut Henri Poincare, 49 (2013), 252-269. · Zbl 1282.60050