Abstract
We extend the results of the FBSDE theory in order to construct a probabilistic representation of a viscosity solution to the Cauchy problem for a system of quasilinear parabolic equations. We derive a BSDE associated with a class of quasilinear parabolic system and prove the existence and uniqueness of its solution. To be able to deal with systems including nondiagonal first order terms along with the underlying diffusion process, we consider its multiplicative operator functional. We essentially exploit as well the fact that the system under consideration can be reduced to a scalar equation in an enlarged phase space. This allows to obtain some comparison theorems and to prove that a solution to FBSDE gives rise to a viscosity solution of the original Cauchy problem for a system of quasilinear parabolic equations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Belopolskaya, Ya., Dalecky, Yu.: Investigation of the Cauchy problem for systems of quasilinear equations via Markov processes. Izv. VUZ Mat. 12, 6–17 (1978)
Belopolskaya, Ya., Dalecky, Yu.L.: Stochastic Equations and Differential Geometry. Kluwer, Boston (1990)
Belopolskaya, Ya., Dalecky, Yu.: Markov processes associated with nonlinear parabolic systems. DAN SSSR 250(3), 268–271 (1980)
Pardoux, E., Peng. S.: Adapted solutions of backward stochastic equations. Syst. Control Lett. 14, 55–61 (1990)
Pardoux, E., Peng, S.: Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations. Lecture Notes in CIS, vol. 176, pp. 200–217. Springer, New York (1992)
Peng, S.: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep. 37, 61–74 (1991)
Ma, J., Yong, J.: Forward–Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Mathematics, vol. 1702. Springer, New York (1999)
Albeverio, S., Belopolskaya, Ya.: Probabilistic approach to systems of nonlinear pdes and vanishing viscosity method. Markov Process. Relat. Filelds 12(1), 59–94 (2006)
Crandall, M., Ishii, H., Lions, P.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)
Pardoux, E.: Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. Stochastic Analysis and Relates Topics: The Geilo Workshop, pp. 79–127. Birkhäuser, Boston (1996)
Pardoux, E., Tang, S.: Forward–backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Relat. Fields 114(2), 123–150 (1999)
Acknowledgements
Financial support of grant RFBR 12-01-00427-a and the Minobrnauki project 1.370.2011 is gratefully acknowledged
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Belopolskaya, Y.I. (2014). Probabilistic Counterparts of Nonlinear Parabolic Partial Differential Equation Systems. In: Korolyuk, V., Limnios, N., Mishura, Y., Sakhno, L., Shevchenko, G. (eds) Modern Stochastics and Applications. Springer Optimization and Its Applications, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-03512-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-03512-3_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03511-6
Online ISBN: 978-3-319-03512-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)