A Quasi-Sure Approach to the Control of Non-Markovian Stochastic Differential Equations

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.

27 pages

References in corpus (2)

• A theoretical framework for the pricing of contingent claims in the presence of model uncertainty
• Note on Viscosity Solution of Path-Dependent PDE and G-Martingales

Cited by in corpus (10)

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• Optimal stopping under adverse nonlinear expectation and related games
• Second-order BSDEs with jumps: Formulation and uniqueness
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• Dual and backward SDE representation for optimal control of non-Markovian SDEs
• A stochastic recursive optimal control problem under the G-expectation framework
• Measurability of Semimartingale Characteristics with Respect to the Probability Law
• A note on pricing of contingent claims under G-expectation
• A pseudo-Markov property for controlled diffusion processes