Abstract
In this paper, we discuss the solvability of backward stochastic differential equations (BSDEs) with superquadratic generators. We first prove that given a superquadratic generator, there exists a bounded terminal value, such that the associated BSDE does not admit any bounded solution. On the other hand, we prove that if the superquadratic BSDE admits a bounded solution, then there exist infinitely many bounded solutions for this BSDE. Finally, we prove the existence of a solution for Markovian BSDEs where the terminal value is a bounded continuous function of a forward stochastic differential equation.
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References
Ben-Artzi M., Souplet P., Weissler F.B.: The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces. J. Math. Pures Appl. 81, 343–378 (2002)
Bismut J.M.: Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404 (1973)
Briand P., Hu Y.: BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. Fields 136, 604–618 (2006)
Briand P., Hu Y.: Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Relat. Fields 141, 543–567 (2008)
Cheridito P., Soner H.M., Touzi N., Victoir N.: Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60, 1081–1110 (2007)
Crandall M.G., Ishii H., Lions P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)
Delbaen, F., Peng, S., Rosazza Gianin, E.: Representation of the penalty term of dynamic concave utilities. Finance Stoch. (to appear)
El Karoui N., Peng S., Quenez M.C.: Backward stochastic differential equations in finance. Math. Finance 7, 1–71 (1997)
Föllmer H., Schied A.: Convex measures of risk and trading constraints. Finance Stoch. 6, 429–447 (2002)
Gilding B.H., Guedda M., Kersner R.: The Cauchy problem for u t = Δu + |∇u|q. J. Math. Anal. Appl. 284, 733–755 (2003)
Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou property. Advances in Mathematical Economics, vol. 9, pp. 49–71, Adv. Math. Econ., vol. 9. Springer, Tokyo (2006)
Kobylanski M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28, 558–602 (2000)
Pardoux E., Peng S.: Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14, 55–61 (1990)
Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic partial differential equations and their applications (Charlotte, NC, 1991), pp. 200–217. Lecture Notes in Control and Inform. Sci., vol. 176. Springer, Berlin (1992)
Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability, 2nd edn. Lecture Notes in Mathematics, vol. 1364. Springer, Berlin (1993)
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Delbaen, F., Hu, Y. & Bao, X. Backward SDEs with superquadratic growth. Probab. Theory Relat. Fields 150, 145–192 (2011). https://doi.org/10.1007/s00440-010-0271-1
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DOI: https://doi.org/10.1007/s00440-010-0271-1