## Electronic Journal of Probability

### Quadratic BSDEs with Random Terminal Time and Elliptic PDEs in Infinite Dimension

#### Abstract

In this paper we study one dimensional backward stochastic differential equations (BSDEs) with random terminal time not necessarily bounded or finite when the generator $F(t,Y,Z)$ has a quadratic growth in $Z$. We provide existence and uniqueness of a bounded solution of such BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained results are then applied to prove existence and uniqueness of a mild solution to elliptic partial differential equations in Hilbert spaces. Finally we show an application to a control problem.

#### Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 54, 1529-1561.

Dates
Accepted: 17 September 2008
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819127

Digital Object Identifier
doi:10.1214/EJP.v13-514

Mathematical Reviews number (MathSciNet)
MR2438815

Zentralblatt MATH identifier
1191.60071

Rights

#### Citation

Confortola, Fulvia; Briand, Philippe. Quadratic BSDEs with Random Terminal Time and Elliptic PDEs in Infinite Dimension. Electron. J. Probab. 13 (2008), paper no. 54, 1529--1561. doi:10.1214/EJP.v13-514. https://projecteuclid.org/euclid.ejp/1464819127

#### References

• A. Ambrosetti, G. Prodi. A primer of nonlinear analysis. Corrected reprint of the 1993 original. Cambridge Studies in Advanced Mathematics 34 (1995). Cambridge University Press.
• Ph. Briand, F. Confortola. BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces. Stochastic Process. Appl. 118 (2008), 818-838.
• Ph. Briand, B. Delyon, Y. Hu, E. Pardoux, L. Stoica. Lp solutions of backward stochastic differential equations. Stochastic Process. Appl. 108 (2003), 109-129.
• Ph. Briand, Y. Hu. Stability of BSDEs with random terminal time and homogenization of semilinear elliptic PDEs. J. Funct. Anal. 155 (1998), 455-494.
• R. Buckdahn, S. Peng. Stationary backward stochastic differential equations and associated partial differential equations. Probab. Theory Related Fields 115 (1999), 383-399.
• S. Cerrai. Second order PDE's in finite and infinite dimension. A probabilistic approach. Lecture Notes in Mathematics 1762 (2001) Springer-Verlag.
• R.W.R. Darling, E. Pardoux. Backwards SDE with random terminal time and applications to semilinear elliptic PDE. Ann. Probab. 25 (1997), 1135-1159.
• N. El Karoui. Backward stochastic differential equation a general introduction. Backward stochastic differential equations (Paris, 1995–1996), 7–26, Pitman Res. Notes Math. Ser. 364 (1997) Longman, Harlow.
• G. Da Prato, J. Zabczyk. Ergodicity for infinite-dimensional systems. London Mathematical Society Lecture Note Series 229 (1996) Cambridge University Press
• G. Da Prato, J. Zabczyk. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications 44 (1992) Cambridge University Press
• G. Da Prato, J. Zabczyk. Second order partial differential equations in Hilbert spaces. London Mathematical Society Lecture Note Series 293 (2002) Cambridge University Press
• M. Fuhrman, Y. Hu, G. Tessitore. On a class of stochastic optimal control problems related to BSDEs with quadratic growth. SIAM J. Control Optim. 45 (2006), 1279-1296 (electronic)
• M. Fuhrman, G. Tessitore. Nonlinear Kolmogorov equations in infinite dimensional space the backward stochastic differential equations approach and applications to optimal control. Ann. Probab. 30 (2002), 1397-1465.
• M. Fuhrman, G. Tessitore. Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces. Ann. Probab. 32 (2004), 607-660.
• F. Gozzi, E. Rouy. Regular solutions of second-order stationary Hamilton-Jacobi equations. J. Differential Equations 130 (1996), 201-234.
• Y. Hu, P. Imkeller, M. Müller. Utility maximization in incomplete markets. Ann. Appl. Probab. 15 (2005), 1691-1712.
• Y. Hu, G. Tessitore. BSDE on an infinite horizon and elliptic PDEs in infinite dimension. NoDEA Nonlinear Differential Equations Appl. 14 (2007), 825-846.
• N. Kazamaki. Continuous exponential martingales and BMO. Lecture Notes in Mathematics 1579 (1994) Springer-Verlag
• M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 (2000), 558-602.
• F. Masiero. Infinite horizon stochastic optimal control problems with degenerate noise and elliptic equations in Hilbert spaces. Appl. Math. Optim. 55 (2007), 285-326.
• J.-P. Lepeltier, J. San Martín. Existence for BSDE with superlinear-quadratic coefficient. Stochastics Stochastics Rep. 63 (1998), 227-240.
• &Eacute.; Pardoux. Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. Stochastic analysis and related topics, VI (Geilo, 1996), 79-127, Progr. Probab. 42 (1998) Birkhäuser Boston
• &Eacute.; Pardoux. BSDEs, weak convergence and homogenization of semilinear PDEs. Nonlinear analysis, differential equations and control (Montreal, QC, 1998), 503–549, NATO Sci. Ser. C Math. Phys. Sci. 528 (1999) Kluwer Acad. Publ.
• D. Revuz, M. Yor. Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften 293 (1999). Springer-Verlag.
• M. Royer. BSDEs with a random terminal time driven by a monotone generator and their links with PDEs. Stoch. Stoch. Rep. 76 (2004), 281-307.