Stochastic delay differential equations with jumps in differentiable manifolds

TL;DR

This paper introduces a geometric model for stochastic delay differential equations with jumps on differentiable manifolds, incorporating parallel transport and horizontal lifts within the manifold's structure.

Contribution

It develops a novel geometric framework for SDDEJs on manifolds, including their lift to the linear frame bundle with respect to a compatible connection.

Findings

Modeling SDDEJs on manifolds with jumps and delays.

Horizontal lift of solutions remains an SDDEJ in the frame bundle.

Utilizes geometric tools like parallel transport and connections.

Abstract

In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold $M$ endowed with a connection $\nabla$. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (c\`adl\`ag) noise, without delay. The jumps occur along adopted differentiable curves with some dynamical relevance (with fictitious time) which allow to take parallel transport along them. Using a geometrical approach, in the last section, we show that the horizontal lift of the solution of an SDDEJ is again a solution of an SDDEJ in the linear frame bundle $BM$ with respect to a connection $\nabla^H$ in $BM$.