## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 37, 9 pp.

### Geometry of stochastic delay differential equations with jumps in manifolds

Paulo R. Ruffino and Leandro Morgado

#### 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 $BM$.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 37, 9 pp.

**Dates**

Received: 16 December 2015

Accepted: 15 April 2016

First available in Project Euclid: 28 April 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1461868666

**Digital Object Identifier**

doi:10.1214/16-ECP4764

**Mathematical Reviews number (MathSciNet)**

MR3510245

**Zentralblatt MATH identifier**

1338.60150

**Subjects**

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 34K50: Stochastic functional-differential equations [See also , 60Hxx] 53C05: Connections, general theory

**Keywords**

stochastic geometry stochastic delay differential equations parallel transport linear frame bundle stochastic differential equations with jumps

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Ruffino, Paulo R.; Morgado, Leandro. Geometry of stochastic delay differential equations with jumps in manifolds. Electron. Commun. Probab. 21 (2016), paper no. 37, 9 pp. doi:10.1214/16-ECP4764. https://projecteuclid.org/euclid.ecp/1461868666