Electronic Journal of Probability
- Electron. J. Probab.
- Volume 17 (2012), paper no. 23, 23 pp.
A quasi-sure approach to the control of non-Markovian stochastic differential equations
We study stochastic differential equations (SDEs) whose drift and diffusion coefficients are path-dependent and controlled. We construct a value process on the canonical path space, considered simultaneously under a family of singular measures, rather than the usual family of processes indexed by the controls. This value process is characterized by a second order backward SDE, which can be seen as a non-Markovian analogue of the Hamilton-Jacobi Bellman partial differential equation. Moreover, our value process yields a generalization of the $G$-expectation to the context of SDEs.
Electron. J. Probab., Volume 17 (2012), paper no. 23, 23 pp.
Accepted: 19 March 2012
First available in Project Euclid: 4 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 93E20: Optimal stochastic control
Secondary: 49L20: Dynamic programming method 60H10: Stochastic ordinary differential equations [See also 34F05] 60G44: Martingales with continuous parameter 91B30: Risk theory, insurance
This work is licensed under aCreative Commons Attribution 3.0 License.
Nutz, Marcel. A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17 (2012), paper no. 23, 23 pp. doi:10.1214/EJP.v17-1892. https://projecteuclid.org/euclid.ejp/1465062345