Geometry of stochastic delay differential equations with jumps in manifolds

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 (càdlàg) 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. In the last section, using a geometrical approach, 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 horizontal connection $\nabla ^H$ in $BM$.