On the Cauchy problem for backward stochastic partial differential equations in H\"{o}lder spaces

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions in H"{o}lder spaces for linear and semi-linear backward stochastic PDEs of super-parabolic type, introducing new inequalities and functional frameworks.

Contribution

It develops a novel functional H"{o}lder space framework for BSPDEs and establishes fundamental properties including existence, uniqueness, and regularity, with new results even for deterministic PDEs.

Findings

Established existence and uniqueness of solutions in H"{o}lder spaces.

Proved regularity results for solutions of BSPDEs.

Derived new inequalities among H"{o}lder norms for these equations.

Abstract

This paper is concerned with solution in H\"{o}lder spaces of the Cauchy problem for linear and semi-linear backward stochastic partial differential equations (BSPDEs) of super-parabolic type. The pair of unknown variables are viewed as deterministic spatial functionals which take values in Banach spaces of random (vector) processes. We define suitable functional H\"{o}lder spaces for them and give some inequalities among these H\"{o}lder norms. The existence, uniqueness as well as the regularity of solutions are proved for BSPDEs, which contain new assertions even on deterministic PDEs.