The Annals of Probability

Density analysis of BSDEs

Thibaut Mastrolia, Dylan Possamaï, and Anthony Réveillac

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In this paper, we study the existence of densities (with respect to the Lebesgue measure) for marginal laws of the solution $(Y,Z)$ to a quadratic growth BSDE. Using the (by now) well-established connection between these equations and their associated semi-linear PDEs, together with the Nourdin–Viens formula, we provide estimates on these densities.

Article information

Ann. Probab., Volume 44, Number 4 (2016), 2817-2857.

Received: February 2014
Revised: January 2015
First available in Project Euclid: 2 August 2016

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

BSDEs Malliavin calculus density analysis Nourdin–Viens’ formula PDEs


Mastrolia, Thibaut; Possamaï, Dylan; Réveillac, Anthony. Density analysis of BSDEs. Ann. Probab. 44 (2016), no. 4, 2817--2857. doi:10.1214/15-AOP1035.

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  • [1] Aboura, O. and Bourguin, S. (2013). Density estimates for solutions to one dimensional backward SDE’s. Potential Anal. 38 573–587.
  • [2] Ankirchner, S., Imkeller, P. and Dos Reis, G. (2007). Classical and variational differentiability of BSDEs with quadratic growth. Electron. J. Probab. 12 1418–1453 (electronic).
  • [3] Antonelli, F. and Kohatsu-Higa, A. (2005). Densities of one-dimensional backward SDEs. Potential Anal. 22 263–287.
  • [4] Barrieu, P. and El Karoui, N. (2007). Pricing, hedging and optimally designing derivatives via minimization of risk measures. Available at arXiv:0708.0948 [math.PR].
  • [5] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [6] Chassagneux, J.-F. and Richou, A. (2016). Numerical simulation of quadratic BSDEs. Ann. Appl. Probab. 26 262–304.
  • [7] dos Reis, G. and dos Reis, R. J. N. (2013). A note on comonotonicity and positivity of the control components of decoupled quadratic FBSDE. Stoch. Dyn. 13 1350005, 11.
  • [8] Dos Reis, G. (2010). On some properties of solutions of quadratic growth BSDE and applications in finance and insurance. Ph.D. thesis, Humboldt Univ., Berlin. Available at
  • [9] El Karoui, N., Hamadene, S. and Matoussi, A. (2008). Backward stochastic differential equations and applications. In Indifference Pricing: Theory and Applications (R. Carmona, ed.) 267–320. Springer, Berlin.
  • [10] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • [11] Fournier, N. and Printems, J. (2010). Absolute continuity for some one-dimensional processes. Bernoulli 16 343–360.
  • [12] Hu, Y., Imkeller, P. and Müller, M. (2005). Utility maximization in incomplete markets. Ann. Appl. Probab. 15 1691–1712.
  • [13] Imkeller, P. and Dos Reis, G. (2010). Path regularity and explicit convergence rate for BSDE with truncated quadratic growth. Stochastic Process. Appl. 120 348–379.
  • [14] Imkeller, P., Réveillac, A. and Richter, A. (2012). Differentiability of quadratic BSDEs generated by continuous martingales. Ann. Appl. Probab. 22 285–336.
  • [15] Jacobson, D. H. (1973). Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. IEEE Trans. Automat. Control AC-18 124–131.
  • [16] Kazamaki, N. (1994). Continuous Exponential Martingales and BMO. Lecture Notes in Math. 1579. Springer, Berlin.
  • [17] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558–602.
  • [18] Kohatsu-Higa, A. (2003). Lower bounds for densities of uniformly elliptic non-homogeneous diffusions. In Stochastic Inequalities and Applications. Progress in Probability 56 323–338. Birkhäuser, Basel.
  • [19] Kohatsu-Higa, A. (2003). Lower bounds for densities of uniformly elliptic random variables on Wiener space. Probab. Theory Related Fields 126 421–457.
  • [20] Ma, J. and Zhang, J. (2002). Path regularity for solutions of backward stochastic differential equations. Probab. Theory Related Fields 122 163–190.
  • [21] Ma, J. and Zhang, J. (2002). Representation theorems for backward stochastic differential equations. Ann. Appl. Probab. 12 1390–1418.
  • [22] Nourdin, I. and Viens, F. G. (2009). Density formula and concentration inequalities with Malliavin calculus. Electron. J. Probab. 14 2287–2309.
  • [23] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [24] Nualart, E. and Quer-Sardanyons, L. (2012). Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension. Stochastic Process. Appl. 122 418–447.
  • [25] 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 Inform. Sci. 176 200–217. Springer, Berlin.
  • [26] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • [27] Rouge, R. and El Karoui, N. (2000). Pricing via utility maximization and entropy. Math. Finance 10 259–276.
  • [28] Seneta, E. (1976). Regularly Varying Functions. Lecture Notes in Mathematics 508. Springer, Berlin.