Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs

TL;DR

This paper develops a decoupling method on the Wiener space to define new anisotropic Besov spaces, which are used to analyze regularity properties of solutions to backward stochastic differential equations (BSDEs) without relying on explicit Malliavin derivatives.

Contribution

It introduces a novel decoupling approach to define broad classes of Besov spaces applicable to BSDE analysis, including spaces that characterize directional derivatives without explicit computation.

Findings

New Besov spaces include traditional and novel anisotropic types.

Established regularity and $L_p$-variation bounds for BSDE solutions.

Improved bounds in harmonic analysis inequalities.

Abstract

We introduce a decoupling method on the Wiener space to define a wide class of an\-iso\-tro\-pic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin derivatives explicitly. Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of the initial data, in particular upper bounds for their $L_p$-variation, where the generator might be of quadratic type and where no structural assumptions, for example in terms of a forward diffusion, are assumed. As an example we treat sub-quadratic BSDEs with unbounded terminal conditions. Among other tools, we use methods from harmonic analysis. As a by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the optimality of the bound we established.