Author: Situ Rong
Language: English
Published on: 2000-01
Hardcover
In this book we discuss the existence and uniqueness of solutions of backward stochastic differential equatios (BSDE) with jumps under Lipschitzian and non-Lipschitzian conditions and with bounded and unbounded stopping times as terminal times. We also discuss the reflecting BSDE (RBSDE) with jumps and forward-backward SDE (FBSDE) with jumps, which are proved to be useful in applications of stochastic control and financial market. We research on its applications to integro-differential equations (IDE), where a new Feynman-Kac formula for the solution to IDE and the existence of the Sobolev solution to the IDE with non-Lipschitzian force are obtained. Furthermore, one kind of Hamilton-Jacobi-Bellman equation with integral term, which is proved to be important and useful in the optimal control and optimal consumption in finance, is also researched and its viscosity solution is expressed by the solution of some BSDE with jumps. Moreover, the applications to backward stochastic partial differential equations (BSPDE), to stochastic control, and to stochastic Riccati equations, all with jumps, are developed. Three Appendices: a martingale representation theorem, a generalization of Kunita-Ito's formula and the stochastic quadratic control, all for systems with jumps, are provided.
Language: English
Published on: 2000-01
Hardcover
In this book we discuss the existence and uniqueness of solutions of backward stochastic differential equatios (BSDE) with jumps under Lipschitzian and non-Lipschitzian conditions and with bounded and unbounded stopping times as terminal times. We also discuss the reflecting BSDE (RBSDE) with jumps and forward-backward SDE (FBSDE) with jumps, which are proved to be useful in applications of stochastic control and financial market. We research on its applications to integro-differential equations (IDE), where a new Feynman-Kac formula for the solution to IDE and the existence of the Sobolev solution to the IDE with non-Lipschitzian force are obtained. Furthermore, one kind of Hamilton-Jacobi-Bellman equation with integral term, which is proved to be important and useful in the optimal control and optimal consumption in finance, is also researched and its viscosity solution is expressed by the solution of some BSDE with jumps. Moreover, the applications to backward stochastic partial differential equations (BSPDE), to stochastic control, and to stochastic Riccati equations, all with jumps, are developed. Three Appendices: a martingale representation theorem, a generalization of Kunita-Ito's formula and the stochastic quadratic control, all for systems with jumps, are provided.
Part I: BSDE with Jumps
1. BSDE with Jumps in Rd
2. RBSDE with Jumps in Half Space of Rd
3. FBSDE with Jumps
Part II: Applications
1. Application to Integro-Differential Equations
2. Application to Hamilton-Jacobi-Bellman Equations with Integral Terms
3. Application to Backward Stochastic PDE with Jumps
4. Application to Financial Market with Jumps
5. Application to Stochastic Control
Appendix A: Martingale Representation Theorem
Appendix B: A Generalization of Kunita-Ito's Formula
Appendix C: Stochastic Quadratic Control
References
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