Abstract
Backward stochastic differential equations extend the martingale representation theorem to the nonlinear setting. This can be seen as path-dependent counterpart of the extension from the heat equation to fully nonlinear parabolic equations in the Markov setting. This paper extends such a nonlinear representation to the context where the random variable of interest is measurable with respect to the information at a finite stopping time. We provide a complete wellposedness theory which covers the semilinear case (backward SDE), the semilinear case with obstacle (reflected backward SDE), and the fully nonlinear case (second order backward SDE).
Highlights
Let (Ω, F, {Ft}t≥0, P) be a filtered probability space, supporting a d-dimensional Brownian motion W
Second order backward SDE with random terminal time reduces to the case ξ = g(Wt0, . . . , Wtn ) for an arbitrary partition 0 = t0 < . . . < tn = T of
In view of the formulation of second order backward SDEs as backward SDEs holding simultaneously under a non-dominated family of singular measures, we review –and complement– the corresponding theory of backward SDEs, and we develop the theory of reflected backward SDEs, which is missing in the literature, and which plays a crucial role in the wellposedness of second order backward SDEs
Summary
Let (Ω, F , {Ft}t≥0, P) be a filtered probability space, supporting a d-dimensional Brownian motion W. Our main interest in this paper is on the extension to the fully nonlinear second order parabolic equations, as initiated in the finite horizon setting by Soner, Touzi & Zhang [40], and further developed by Possamaï, Tan & Zhou [34], see the first attempt by Cheridito, Soner, Touzi & Victoir [9], and the closely connected BSDEs in a nonlinear expectation framework of Hu, Ji, Peng & Song [17, 18] (called GBSDEs) This extension is performed on the canonical space of continuous paths with canonical process denoted by X.
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