Electronic Communications in Probability

Geometry of stochastic delay differential equations with jumps in manifolds

Paulo R. Ruffino and Leandro Morgado

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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


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