Journal of the Mathematical Society of Japan

$G$-expectation weighted Sobolev spaces, backward SDE and path dependent PDE

Abstract

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's process. Consequently, a new path-dependent Itô's 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 BSDE is identified with the solution of the corresponding quasi-linear path-dependent PDE (PPDE). Rich and important results of 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 $P$-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 to a type of backward SDEs driven by $G$-Brownian motion.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1725-1757.

Dates
First available in Project Euclid: 27 October 2015

https://projecteuclid.org/euclid.jmsj/1445951164

Digital Object Identifier
doi:10.2969/jmsj/06741725

Mathematical Reviews number (MathSciNet)
MR3417511

Zentralblatt MATH identifier
1335.60098

Citation

PENG, Shige; SONG, Yongsheng. $G$-expectation weighted Sobolev spaces, backward SDE and path dependent PDE. J. Math. Soc. Japan 67 (2015), no. 4, 1725--1757. doi:10.2969/jmsj/06741725. https://projecteuclid.org/euclid.jmsj/1445951164

References

• J. M. Bismut, Conjugate Convex Functions in Optimal Stochastic Control, J. Math. Anal. Apl., 44 (1973), 384–404.
• R. Cont and D. Fournie, Functional Itô calculus and stochastic integral representation of martingales, Ann. Prob., 41 (2013), 109–133.
• 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.
• 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.
• 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.
• 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.
• B. Dupire, Functional Itô calculus, papers.ssrn.com., 2009.
• I. Ekren, Ch. Keller, N. Touzi and J. Zhang, On Viscosity Solutions of Path Dependent PDEs, Ann. Probab., 42 (2014), 204–236.
• H. Föllmer and A. Schied, Statistic Finance, Walter de Gruyter, 2004.
• I. Ekren, N. Touzi and J. Zhang, Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I–II, 2012, arXiv:1210.0007v1.
• N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1–71.
• 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.
• 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.
• N. V. Krylov, Nonlinear Parabolic and Elliptic Equations of the Second Order, Reidel Publishing Company, (Original Russian Version by Nauka, Moscow, 1985), 1987.
• E. Pardoux and S. Peng, Adapted Solutions of Backward Stochastic Equations, Systerm and Control Letters, 14 (1990), 55–61.
• 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.
• S. Peng, Probabilistic Interpretation for Systems of Quasilinear Parabolic Partial Differential Equations, Stochastics, 37 (1991), 61–74.
• 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.
• 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.
• S. Peng, Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 26B (2005), 159–184.
• 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.
• S. Peng, $G$-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty, 2007, arXiv:0711.2834v1.
• S. Peng, Multi-Dimensional $G$-Brownian Motion and Related Stochastic Calculus under $G$-Expectation, Stochastic Process. Appl., 118 (2008), 2223–2253.
• S. Peng, Nonlinear Expectations and Stochastic Calculus under Uncertainty, 2010, arXiv:1002.4546v1.
• S. Peng, Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications, In: Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010.
• 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.
• S. Peng and F. Wang, BSDE, Path-dependent PDE and Nonlinear Feynman–Kac Formula, 2011, arXiv:1108.4317v1.
• Z. Ren, N. Touzi and J. Zhang, Comparison of viscosity solutions of semilinear path-dependent partial differential equations, 2014, arXiv:1410.7291.
• M. Soner, N. Touzi and J. Zhang, Martingale Representation Theorem under $G$-expectation, Stochastic Processes and their Applications, 121 (2011), 265–287.
• M. Soner, N. Touzi and J. Zhang, Well-posedness of Second Order Backward SDEs, Probability Theory and Related Fields, 153 (2012), 149–190.
• Y. Song, Some properties on $G$-evaluation and its applications to $G$-martingale decomposition, Science China Mathematics, 54 (2011), 287–300.
• Y. Song, Properties of hitting times for $G$-martingales and their applications, Stochastic Processes and their Applications, 121 (2011), 1770–1784.
• Y. Song, Uniqueness of the representation for $G$-martingales with finite variation, Electron. J. Probab., 17 (2012), 1–15.
• Y. Song, Characterizations of processes with stationary and independent increments under $G$-expectation, Annales de l'Institut Henri Poincare, 49 (2013), 252–269.