## The Annals of Probability

### Backward stochastic differential equations and partial differential equations with quadratic growth

Magdalena Kobylanski

#### Abstract

We provide existence, comparison and stability results for one- dimensional backward stochastic differential equations (BSDEs) when the coefficient (or generator) $F(t,Y, Z)$ is continuous and has a quadratic growth in $Z$ and the terminal condition is bounded.e also give, in this framework, the links between the solutions of BSDEs set on a diffusion and viscosity or Sobolev solutions of the corresponding semilinear partial differential equations.

#### Article information

Source
Ann. Probab. Volume 28, Number 2 (2000), 558-602.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160253

Digital Object Identifier
doi:10.1214/aop/1019160253

Mathematical Reviews number (MathSciNet)
MR1782267

Zentralblatt MATH identifier
1044.60045

#### Citation

Kobylanski, Magdalena. Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 (2000), no. 2, 558--602. doi:10.1214/aop/1019160253. https://projecteuclid.org/euclid.aop/1019160253

#### References

• [1] Barles, G., Buckdahn, R. and Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60 pp. 57-83.
• [2] Barles, G. and Lesigne, E. (1997). Sde, bsde and pde. In Backward Stochastic Differential Equations. Research Notes in Math. 364 47-80. Pitman, London.
• [3] Barles, G. and Murat, F. (1995). Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions. Arch. Rational Mech. Anal. 133 77-101.
• [4] Boccardo, L., Murat, F. and Puel, J.-P. (1982). Existence de solutions non born´ees pour certaines ´equations quasi-lin´eaires. Portugal Math. 41 507-534.
• [5] Boccardo, L., Murat, F. and Puel, J.-P. (1983). Existence de solutions faibles pour des ´equations elliptiques quasi-lin´eaires a croissance quadratique. In Nonlinear Partial Differential Equations and Their Applications. Research Notes in Math. 84 19-73. Pitman, London.
• [6] Boccardo, L., Murat, F. and Puel, J.-P. (1988). Existence results for some quasilinear parabolic equations. Nonlinear Anal. 13 373-392.
• [7] Crandall, M., Ishii, H. and Lions, P.-L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 1-67.
• [8] Crandall, M. and Lions, P.-L. (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 1-42.
• [9] Donsker, M. and Varadhan, S. (1975). On the principle eigenvalue of second-order elliptic differential operators. Comm. Pure Appl. Math. 29 595-621.
• [10] El Karoui, N. and Maziak, L., eds. (1997). Backward Stochastic Differential Equations. Pitman, London.
• [11] El Karoui, N. Peng, S. and Quenez, M.-C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71.
• [12] Gilbarg, D. and Trudinger, N. (1983). Elliptic Partial Differential Equations of the Second Order. Springer, New York.
• [13] Karatzas, I. and Shreve, S. (1994). Brownian Motion and Stochastic Calculous, 2nd ed. Springer, New York.
• [14] Lepeltier, J.-P. and San Martin, J. (1997). Backward stochastic differential equations with continuous coefficients. Statist Probab. Lett. 32 425-430.
• [15] Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic dfferential equation. Systems Control Lett. 14 55-61.
• [16] Pardoux, E. and Peng, S. (1992). Bsdes and quasilinear parabolic pdes. Lecture Notes in Control and Inform. Sci. 176 200-217. Springer, New York.