Abstract
We show the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite q-variation with q∈[1,2). In contrast to previous work, we apply a direct fixpoint argument and do not rely on any type of flow decomposition. The resulting object is an effective tool to study semilinear rough partial differential equations via a Feynman–Kac type representation.
Highlights
Stochastic differential equations (SDEs) driven by Brownian motion W and an additional deterministic path η of low regularity have been well-studied
In (Guerra and Nualart 2008), the well-posedness of such SDEs is established if η has finite q-variation with q ∈ [1, 2). 1 The integral with respect to the latter is handled via fractional calculus
In Section “Main result”, we state and prove this main result. It is well-known that Backward stochastic differential equations (BSDEs) provide a stochastic representation for solutions to semi-linear parabolic partial differential equations (PDEs), in what is sometimes called the “nonlinear Feynman–Kac formula” (Pardoux and Peng 1992)
Summary
Stochastic differential equations (SDEs) driven by Brownian motion W and an additional deterministic path η of low regularity (so called “mixed SDEs”) have been well-studied. Keywords Rough paths theory · Young integration · BSDE · rough PDE Under appropriate conditions on f and ξ , they showed the existence of a unique solution to such an equation. It is well-known that BSDEs provide a stochastic representation for solutions to semi-linear parabolic partial differential equations (PDEs), in what is sometimes called the “nonlinear Feynman–Kac formula” (Pardoux and Peng 1992).
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