Characterizing the path-independent property of the Girsanov density for degenerated stochastic differential equations

TL;DR

This paper establishes a characterization theorem for the path-independent property of Girsanov densities in degenerated SDEs, extending previous results to more general cases including non-Lipschitz SDEs with jumps and degenerated diffusion.

Contribution

It generalizes the characterization of Girsanov density path-independence to degenerated and non-Lipschitz SDEs with jumps, broadening theoretical understanding.

Findings

Derived a new characterization theorem for degenerated SDEs.

Extended previous results to non-Lipschitz SDEs with jumps.

Generalized the understanding of Girsanov density properties.

Abstract

In this paper, we derive a characterization theorem for the path-independent property of the density of the Girsanov transformation for {\it degenerated} stochastic differential equations (SDEs), extending the characterization theorem of \cite{twwy} for the non-degenerated SDEs. We further extends our consideration to non-Lipschitz SDEs with jumps and with degenerated diffusion coefficients, which generalizes the corresponding characterization theorem established in \cite{hqwu}.