Electronic Journal of Probability

A quasi-sure approach to the control of non-Markovian stochastic differential equations

Marcel Nutz

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Abstract

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.

Article information

Source
Electron. J. Probab. Volume 17 (2012), paper no. 23, 23 pp.

Dates
Accepted: 19 March 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062345

Digital Object Identifier
doi:10.1214/EJP.v17-1892

Mathematical Reviews number (MathSciNet)
MR2900464

Zentralblatt MATH identifier
1244.93176

Subjects
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

Keywords
Stochastic optimal control non-Markovian SDE second order BSDE $G$-expectation random $G$-expectation volatility uncertainty risk measure

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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.


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