H\"ormander-Type Theorem for It\^o Processes and Related Backward SPDEs

TL;DR

This paper proves a H"ormander-type theorem for It ext{"o} processes and backward SPDEs, providing new gradient estimates and a concise proof for the $L^2$-theory of linear degenerate BSPDEs.

Contribution

It introduces a H"ormander-type theorem for It ext{"o} processes and BSPDEs, along with a self-contained proof and novel gradient estimates for degenerate cases.

Findings

Established a H"ormander-type theorem for It ext{"o} processes and BSPDEs.

Provided a new self-contained proof for the $L^2$-theory of linear degenerate BSPDEs.

Derived new gradient estimates for degenerate BSPDEs.

Abstract

A H\"ormander-type theorem is established for It\^o processes and related backward stochastic partial differential equations (BSPDEs). A short self-contained proof is also provided for the $L^2$-theory of linear, possibly degenerate BSPDEs, in which new gradient estimates are obtained.