Abstract
We relax conditions on coefficients given in [7] for the existence of solu-tions to backward stochastic differential equations (BSDE) with jumps. Counter examples are given to show that such conditions can not be weakened further in some sense. The existence of a solution for some continuous BSDE with coefficients b(t, y, q) having a quadratic growth in q, having a greater than linear growth in y, and are unbounded in y belonging to a finite interval, is also obtained. Then we obtain an existence and uniqueness result for the Sobolev solution to some integro-differential equation (IDE) under weaker conditions. Some Markov properties for solutions to BSDEs associated with some forward SDEs are also discussed and a Feynman-Kac formula is also obtained. Finally, we obtain probably the first results on the existence of non-Lipschitzian optimal controls for some special stochastic control problems with respect to such BSDE systems with jumps, where some optimal control problem is also explained in the financial market.
This work is supported in part by The National Natural Science Foundation of China No. 79790130.
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
Preview
Unable to display preview. Download preview PDF.
References
Bardhan, I. and Chao, X. (1993). Pricing options on securities with discontinuous returns. Stochastic Process. Appl.48, 123–137.
Bensoussan, A. and J.L. Lions. (1984). Impulse Control and Quasi-Variational Inequalities. Gautheir-Villars.
El Karoui, N., Peng, S., and Quenez, M.C. (1997). Backward stochastic differential equations in finance. Math. Finance7, 1–71.
Ladyzenskaja, O. A., Solonnikov, V. A. and Uralceva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Translation of Monographs 23, AMS Providence, Rode Island.
Peng, S. (1993). Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim.27, 125–144.
Situ Rong (1996). On comparison theorem of solutions to backward stochastic differential equations with jumps and its applications. Proc. of the 3rd CSIAM Conf. on Systems and Control, LiaoNing, China, 46–50.
Situ Rong. (1997). On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process. Appl.66, 209–236.
Situ Rong. (1999). Comparison theorem of solutions to BSDE with jumps and viscosity solution to a generalized Hamilton-JacobiBellman equation. In Control of Distributed Parameter and Stochastic Systems, eds: S. Chen, X. Li, J. Yong, X.Y. Zhou, Kluwer Acad. Pub., Boston, 275–282.
Situ Rong (1999). On comparison theorems and existence of solutions to backward stochastic differential equations with jumps and with discontinuous coefficients. The 26th Conference on Stochastic Processes and their Applications, 14–18, June, Beijing.
Situ Rong (1999). Reflecting Stochastic Differential Equations with Jumps and Applications. CRC Research Notes in Mathematics 408, Chapman Sc Hall/CRC.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Kluwer Academic Publishers
About this chapter
Cite this chapter
Rong, S. (2002). On Solutions of Backward Stochastic Differential Equations with Jumps and Stochastic Control. In: Hou, Z., Filar, J.A., Chen, A. (eds) Markov Processes and Controlled Markov Chains. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0265-0_19
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0265-0_19
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7968-3
Online ISBN: 978-1-4613-0265-0
eBook Packages: Springer Book Archive