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2012 A quasi-sure approach to the control of non-Markovian stochastic differential equations
Marcel Nutz
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Electron. J. Probab. 17: 1-23 (2012). DOI: 10.1214/EJP.v17-1892

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.

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Marcel Nutz. "A quasi-sure approach to the control of non-Markovian stochastic differential equations." Electron. J. Probab. 17 1 - 23, 2012. https://doi.org/10.1214/EJP.v17-1892

Information

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

zbMATH: 1244.93176
MathSciNet: MR2900464
Digital Object Identifier: 10.1214/EJP.v17-1892

Subjects:
Primary: 93E20
Secondary: 49L20 , 60G44 , 60H10 , 91B30

Keywords: $g$-expectation , non-Markovian SDE , random $G$-expectation , risk measure , second order BSDE , stochastic optimal control , volatility uncertainty

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