The Annals of Probability

A weak version of path-dependent functional Itô calculus

Dorival Leão, Alberto Ohashi, and Alexandre B. Simas

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We introduce a variational theory for processes adapted to the multidimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the “sensitivities” of processes, namely derivatives of martingale components and a weak notion of infinitesimal generator, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t. Brownian motion driving noise.

Article information

Ann. Probab., Volume 46, Number 6 (2018), 3399-3441.

Received: November 2015
Revised: November 2017
First available in Project Euclid: 25 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60H25: Random operators and equations [See also 47B80]

Stochastic calculus of variations functional Itô calculus


Leão, Dorival; Ohashi, Alberto; Simas, Alexandre B. A weak version of path-dependent functional Itô calculus. Ann. Probab. 46 (2018), no. 6, 3399--3441. doi:10.1214/17-AOP1250.

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  • [1] Bertoin, J. (1986). Les processus de Dirichlet en tant qu’espace de Banach. Stochastics 18 155–168.
  • [2] Bertoin, J. (1989). Sur une intégrale pour les processus à $\alpha$-variation bornée. Ann. Probab. 17 1521–1535.
  • [3] Bezerra, S. C., Ohashi, A. and Russo, F. (2018). Discrete-type approximations for non-Markovian optimal stopping problems: Part II. Available at arXiv:1707.05250.
  • [4] Buckdahn, R., Ma, J. and Zhang, J. (2015). Pathwise Taylor expansions for random fields on multiple dimensional paths. Stochastic Process. Appl. 125 2820–2855.
  • [5] Burq, Z. A. and Jones, O. D. (2008). Simulation of Brownian motion at first-passage times. Math. Comput. Simulation 77 64–71.
  • [6] Cohen, S. N. and Elliott, R. J. (2015). Stochastic Calculus and Applications, 2nd ed. Springer, Cham.
  • [7] Cont, R. and Fournié, D.-A. (2010). Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal. 259 1043–1072.
  • [8] Cont, R. and Fournié, D.-A. (2013). Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41 109–133.
  • [9] Cont, R. and Fournié, D. A. (2016). Functional Itô calculus and functional Kolmogorov equations. In Stochastic Integration by Parts and Functional Itô Calculus 115–207. Birkhäuser/Springer, Cham.
  • [10] Coquet, F., Mackevičius, V. and Mémin, J. (1998). Stability in $\mathbf{D}$ of martingales and backward equations under discretization of filtration. Stochastic Process. Appl. 75 235–248.
  • [11] Coquet, F., Mémin, J. and Słominski, L. (2001). On weak convergence of filtrations. In Séminaire de Probabilités, XXXV. Lecture Notes in Math. 1755 306–328. Springer, Berlin.
  • [12] Coquet, F. and Słomiński, L. (1999). On the convergence of Dirichlet processes. Bernoulli 5 615–639.
  • [13] Cosso, A. and Russo, F. (2015). Strong-viscosity solutions: Semilinear parabolic PDEs and path-dependent PDEs. Available at arXiv:1505.02927.
  • [14] Cosso, A. and Russo, F. (2016). Functional Itô versus Banach space stochastic calculus and strict solutions of semilinear path-dependent equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19 Art. ID 1650024.
  • [15] Cosso, A. and Russo, F. (2016). Functional and Banach space stochastic calculi: Path-dependent Kolmogorov equations associated with the frame of a Brownian motion. In Stochastics of Environmental and Financial Economics—Centre of Advanced Study, Oslo, Norway, 20142015 (F. E. Benth and G. Di Nunno, eds.). Springer Proc. Math. Stat. 138 27–80. Springer, Cham.
  • [16] Dellacherie, C. and Meyer, P.-A. (1975). Probabilités et Potentiel. Publications de l’Institut de Mathématique de l’Université de Strasbourg XV. Actualités Scientifiques et Industrielles 1372. Hermann, Paris. Chapitres I à IV, Édition entièrement refondue.
  • [17] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential B: Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam. Translated from the French by J. P. Wilson.
  • [18] Diestel, J., Ruess, W. M. and Schachermayer, W. (1993). On weak compactness in $L^{1}(\mu,X)$. Proc. Amer. Math. Soc. 118 447–453.
  • [19] Dupire, B. Functional Itô calculus. Portfolio Research Paper 2009-04, Bloomberg.
  • [20] Ekren, I., Keller, C., Touzi, N. and Zhang, J. (2014). On viscosity solutions of path dependent PDEs. Ann. Probab. 42 204–236.
  • [21] Ekren, I., Touzi, N. and Zhang, J. (2016). Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. Ann. Probab. 44 1212–1253.
  • [22] Ekren, I., Touzi, N. and Zhang, J. (2016). Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. Ann. Probab. 44 2507–2553.
  • [23] Ekren, I. and Zhang, J. (2016). Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs. Probab. Uncertain. Quant. Risk 1 Paper No. 6.
  • [24] Flandoli, F. and Zanco, G. (2016). An infinite-dimensional approach to path-dependent Kolmogorov equations. Ann. Probab. 44 2643–2693.
  • [25] He, S. W., Wang, J. G. and Yan, J. A. (1992). Semimartingale Theory and Stochastic Calculus. Kexue Chubanshe (Science Press), Beijing; CRC Press, Boca Raton, FL.
  • [26] Jacod, J. and Skorohod, A. V. (1994). Jumping filtrations and martingales with finite variation. In Séminaire de Probabilités, XXVIII. Lecture Notes in Math. 1583 21–35. Springer, Berlin.
  • [27] Keller, C. and Zhang, J. (2016). Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients. Stochastic Process. Appl. 126 735–766.
  • [28] Khoshnevisan, D. and Lewis, T. M. (1999). Stochastic calculus for Brownian motion on a Brownian fracture. Ann. Appl. Probab. 9 629–667.
  • [29] Le Jan, Y. (1978). Temps d’arret stricts et martingales de sauts. Z. Wahrsch. Verw. Gebiete 44 213–225.
  • [30] Leão, D. and Ohashi, A. (2013). Weak approximations for Wiener functionals. Ann. Appl. Probab. 23 1660–1691.
  • [31] Leão, D. and Ohashi, A. (2017). Corrigendum: Weak approximations for Wiener functionals [Ann. Appl. Probab. (2013) 23 1660–1691] [MR3098445]. Ann. Appl. Probab. 27 1294–1295.
  • [32] Leão, D., Ohashi, A. and Russo, F. (2018). Discrete-type approximations for non-Markovian optimal stopping problems: Part I. Available at arXiv:1707.05234.
  • [33] Leão, D., Ohashi, A. and Simas, A. B. (2018). Supplement to “A weak version of path-dependent functional Itô calculus.” DOI:10.1214/17-AOP1250SUPP.
  • [34] Leão, D., Ohashi, A. and Simas, A. B. (2018). Weak differentiability of Wiener functionals and occupation times. Bull. Sci. Math. To appear. Available at arXiv:1711.10895.
  • [35] Leão, D., Ohashi, A. and Souza, F. (2018). Stochastic near-optimal controls for path-dependent systems. Available at arXiv:1707.04976.
  • [36] Mémin, J. (2003). Stability of Doob–Meyer decomposition under extended convergence. Acta Math. Appl. Sin. Engl. Ser. 19 177–190.
  • [37] Oberhauser, H. (2016). The functional Itô formula under the family of continuous semimartingale measures. Stoch. Dyn. 16 Art. ID 1650010.
  • [38] Ohashi, A., Shamarova, E. and Shamarov, N. N. (2016). Path-dependent Itô formulas under $(p,q)$-variations. ALEA Lat. Am. J. Probab. Math. Stat. 13 1–31.
  • [39] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • [40] Peng, S. (2010). Backward stochastic differential equation, nonlinear expectation and their applications. In Proceedings of the International Congress of Mathematicians, Vol. I 393–432. Hindustan Book Agency, New Delhi.
  • [41] Peng, S. and Song, Y. (2015). $G$-expectation weighted Sobolev spaces, backward SDE and path dependent PDE. J. Math. Soc. Japan 67 1725–1757.
  • [42] Peng, S. and Wang, F. (2016). BSDE, path-dependent PDE and nonlinear Feynman–Kac formula. Sci. China Math. 59 19–36.

Supplemental materials

  • Supplement to “A weak version of path-dependent functional Itô calculus”. The proofs of Lemmas 2.1, 2.3, 3.2, 4.4 and Theorem 4.4 are provided in the online supplement [33].