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Decoupled Mild Solutions of Path-Dependent PDEs and Integro PDEs Represented by BSDEs Driven by Cadlag Martingales

  • Adrien Barrasso
  • Francesco RussoEmail author
Article
  • 22 Downloads

Abstract

We focus on a class of path-dependent problems which include path-dependent PDEs and Integro PDEs (in short IPDEs), and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of decoupled mild solution for which, under general assumptions, we study existence and uniqueness and its representation via the aforementioned BSDEs. This concept generalizes a similar notion introduced by the authors in recent papers in the framework of classical PDEs and IPDEs. For every initial condition (s, η), where s is an initial time and η an initial path, the solution of such BSDE produces a couple of processes (Ys, η, Zs, η). In the classical (Markovian or not) literature the function \(u(s,\eta ):= Y^{s,\eta }_{s}\) constitutes a viscosity type solution of an associated PDE (resp. IPDE); our approach allows not only to identify u as the unique decoupled mild solution, but also to solve quite generally the so called identification problem, i.e. to also characterize the (Zs, η)s, η processes in term of a deterministic function v associated to the (above decoupled mild) solution u.

Keywords

Decoupled mild solutions Martingale problem Cadlag martingale Path-dependent PDEs Backward stochastic differential equation Identification problem 

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Notes

Acknowledgments

The authors are grateful to the anonymous Referee and Associated Editor for their stimulating comments on the first version of the paper. The research of the first named author was provided by a PhD fellowship (AMX) of the Ecole Polytechnique. The contribution of the second named author was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

References

  1. 1.
    Aliprantis, C.D., Border, K.C.: Infinite-Dimensional Analysis, 2nd edn. Springer, Berlin (1999). A hitchhiker’s guideCrossRefGoogle Scholar
  2. 2.
    Barrasso, A., Russo, F.: Backward stochastic differential equations with no driving martingale, Markov processes and associated pseudo partial differential equations. Preprint, hal-01431559, v2 (2017)Google Scholar
  3. 3.
    Barrasso, A., Russo, F.: Backward stochastic differential equations with no driving martingale, Markov processes and associated pseudo partial differential equations. part II: Decoupled mild solutions and examples. Preprint, hal-01505974 (2017)Google Scholar
  4. 4.
    Barrasso, A., Russo, F.: Martingale driven BSDEs, PDEs and other related deterministic problems. Preprint, hal-01566883 (2017)Google Scholar
  5. 5.
    Barrasso, A., Russo, F.: Gâteaux type path-dependent PDEs and BSDEs with Gaussian forward processes. In preparation (2019)Google Scholar
  6. 6.
    Barrasso, A., Russo, F.: Path-dependent martingale problems and additive functionals. Stochastics and Dynamics, 19 no 1, Preprint, hal-01775200 (2019)Google Scholar
  7. 7.
    Bion-Nadal, J.: Dynamic risk reasures and path-dependent second order PDEs. Stoch. Environ. Financ. Econ. 138, 147–178 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bismut, J.M.: Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Carbone, R., Ferrario, B., Santacroce, M.: Backward stochastic differential equations driven by càdlàg martingales. Veroyatn. Primen. 52(2), 375–385 (2007)zbMATHCrossRefGoogle Scholar
  10. 10.
    Cont, R., Fournié, D.-A.: Change of variable formulas for non-anticipative functionals on path space. J. Funct Anal. 259(4), 1043–1072 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Cont, R., Fournié, D.-A.: Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41(1), 109–133 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Cosso, A., Russo, F.: Strong-viscosity solutions: semilinear parabolic PDEs and path-dependent PDEs. To appear: Osaka Journal of Mathematics Preprint HAL-01145301 (2015)Google Scholar
  13. 13.
    Cosso, A., Russo, F.: Functional Itô versus Banach space stochastic calculus and strict solutions of semilinear path-dependent equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19(4), 1650024, 44 (2016)zbMATHCrossRefGoogle Scholar
  14. 14.
    Cruzeiro, A.B., Qian, Z.M.: Backward stochastic differential equations associated with the vorticity equations. J. Funct. Anal. 267(3), 660–677 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Di Girolami, C., Russo, F.: Infinite dimensional stochastic calculus via regularization and applications. Preprint HAL-INRIA, inria-00473947 version 1 (Unpublished) (2010)Google Scholar
  16. 16.
    Dupire, B.: Functional Itô calculus. Portfolio Research Paper, Bloomberg (2009)Google Scholar
  17. 17.
    Ekren, I., Keller, C., Touzi, N., Zhang, J.: On viscosity solutions of path dependent PDEs. Ann. Probab. 42(1), 204–236 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ekren, I., Touzi, N., Zhang, J.: Viscosity solutions of fully nonlinear parabolic path dependent PDEs. Part I. To appear in Annals of Probability (2013)Google Scholar
  19. 19.
    El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Financ. 7(1), 1–71 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ethier, S.N., Kurtz, T.G.: Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986). Characterization and convergenceGoogle Scholar
  21. 21.
    Flandoli, F., Zanco, G.: An infinite-dimensional approach to path-dependent Kolmogorov equations. Ann. Probab. 44(4), 2643–2693 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Fuhrman, M., Masiero, F., Tessitore, G.: Stochastic equations with delay: optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 48(7), 4624–4651 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Fuhrman, M., Tessitore, G.: Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control. Ann. Probab. 30(3), 1397–1465 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Jacod, J.: Calcul Stochastique et problèmes de Martingales, Volume 714 of Lecture Notes in Mathematics. Springer, Berlin (1979)CrossRefGoogle Scholar
  25. 25.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, Volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer, Berlin (2003)Google Scholar
  26. 26.
    Leão, D., Ohashi, A., Simas, A.B.: A weak version of path-dependent functional Itô calculus. Ann Probab. 46(6), 3399–3441 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Liang, G., Lyons, T., Qian, Zh.: Backward stochastic dynamics on a filtered probability space. Ann Probab. 39(4), 1422–1448 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Pardoux, É., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Pardoux, É. , Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991), Volume 176 of Lecture Notes in Control and Inform. Sci., pp 200–217. Springer, Berlin (1992)Google Scholar
  30. 30.
    Peng, S., Wang, F.: BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci. Chin/ Math. 59(1), 19–36 (2016)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.ENSTA Paristech, Institut Polytechnique de ParisUnité de Mathématiques AppliquéesPalaiseauFrance
  2. 2.Ecole PolytechniquePalaiseauFrance

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