The Annals of Probability

On viscosity solutions of path dependent PDEs

Ibrahim Ekren, Christian Keller, Nizar Touzi, and Jianfeng Zhang

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Abstract

In this paper we propose a notion of viscosity solutions for path dependent semi-linear parabolic PDEs. This can also be viewed as viscosity solutions of non-Markovian backward SDEs, and thus extends the well-known nonlinear Feynman–Kac formula to non-Markovian case. We shall prove the existence, uniqueness, stability and comparison principle for the viscosity solutions. The key ingredient of our approach is a functional Itô calculus recently introduced by Dupire [Functional Itô calculus (2009) Preprint].

Article information

Source
Ann. Probab., Volume 42, Number 1 (2014), 204-236.

Dates
First available in Project Euclid: 9 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1389278524

Digital Object Identifier
doi:10.1214/12-AOP788

Mathematical Reviews number (MathSciNet)
MR3161485

Zentralblatt MATH identifier
1320.35154

Subjects
Primary: 35D40: Viscosity solutions 35K10: Second-order parabolic equations 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Path dependent PDEs backward SDEs functional Itô formula viscosity solutions comparison principle

Citation

Ekren, Ibrahim; Keller, Christian; Touzi, Nizar; Zhang, Jianfeng. On viscosity solutions of path dependent PDEs. Ann. Probab. 42 (2014), no. 1, 204--236. doi:10.1214/12-AOP788. https://projecteuclid.org/euclid.aop/1389278524


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References

  • [1] Bank, P. and Baum, D. (2004). Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14 1–18.
  • [2] Bayraktar, E. and Sîrbu, M. (2012). Stochastic Perron’s method and verification without smoothness using viscosity comparison: The linear case. Proc. Amer. Math. Soc. 140 3645–3654.
  • [3] Cheridito, P., Soner, H. M., Touzi, N. and Victoir, N. (2007). Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60 1081–1110.
  • [4] Cont, R. and Fournié, D.-A. (2013). Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41 109–133.
  • [5] Crandall, M. G., Ishii, H. and Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1–67.
  • [6] Dupire, B. (2009). Functional Itô calculus. Preprint. Available at papers.ssrn.com.
  • [7] Ekren, I., Touzi, N. and Zhang, J. Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. Available at arXiv:1210.0006.
  • [8] Ekren, I., Touzi, N. and Zhang, J. Optimal stopping under nonlinear expectation. Available at arXiv:1209.6601.
  • [9] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1997). Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 702–737.
  • [10] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • [11] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25. Springer, New York.
  • [12] Ishii, H. (1987). Perron’s method for Hamilton–Jacobi equations. Duke Math. J. 55 369–384.
  • [13] Lions, P. L. (1988). Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. I. The case of bounded stochastic evolutions. Acta Math. 161 243–278.
  • [14] Lions, P. L. (1989). Viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions. II. Optimal control of Zakai’s equation. In Stochastic Partial Differential Equations and Applications, II (Trento, 1988). Lecture Notes in Math. 1390 147–170. Springer, Berlin.
  • [15] Lions, P. L. (1989). Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. III. Uniqueness of viscosity solutions for general second-order equations. J. Funct. Anal. 86 1–18.
  • [16] Lukoyanov, N. (2007). On viscosity solution of functional Hamilton–Jacobi type equations for hereditary systems. In Proceedings of the Steklov Institute of Mathematics, Suppl. 2, 190–200. Pleiades Publishing. Original Russian Text: N. Yu. Lukoyanov, Trudy Instituta Matematiki i Mekhaniki UrO RAN 13 (2007).
  • [17] Ma, J., Protter, P. and Yong, J. M. (1994). Solving forward–backward stochastic differential equations explicitly—a four step scheme. Probab. Theory Related Fields 98 339–359.
  • [18] Ma, J., Zhang, J. and Zheng, Z. (2008). Weak solutions for forward–backward SDEs—a martingale problem approach. Ann. Probab. 36 2092–2125.
  • [19] Pardoux, É. and Peng, S. (1992). Backward stochastic differential equations and quasilinear parabolic partial differential equations. In Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991). Lecture Notes in Control and Information Sciences 176 200–217. Springer, Berlin.
  • [20] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • [21] Pardoux, E. and Tang, S. (1999). Forward–backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Related Fields 114 123–150.
  • [22] Peng, S. (1997). Backward SDE and related $g$-expectation. In Backward Stochastic Differential Equations (Paris, 19951996). Pitman Research Notes in Mathematics Series 364 141–159. Longman, Harlow.
  • [23] Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probab. Theory Related Fields 113 473–499.
  • [24] Peng, S. (2007). $G$-Brownian motion and dynamic risk measure under volatility uncertainty. Available at arXiv:0711.2834.
  • [25] Peng, S. (2010). Backward stochastic differential equation, nonlinear expectation and their applications. In Proceedings of the International Congress of Mathematicians. Volume I 393–432. Hindustan Book Agency, New Delhi.
  • [26] Peng, S. (2011). Note on viscosity solution of path-dependent PDE and G-martingales. Available at arXiv:1106.1144.
  • [27] Peng, S. and Wang, F. (2011). BSDE, path-dependent PDE and nonlinear Feynman–Kac formula. Available at arXiv:1108.4317.
  • [28] Soner, H. M., Touzi, N. and Zhang, J. (2013). Dual formulation of second order target problems. Ann. Appl. Probab. 23 308–347.
  • [29] Soner, H. M., Touzi, N. and Zhang, J. (2012). Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153 149–190.
  • [30] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin.