Adjoint equation and Lyapunov regularity for linear stochastic differential algebraic equations of index 1

Abstract

We introduce a concept of adjoint equation and Lyapunov regularity of a stochastic differential algebraic Equation (SDAE) of index 1. The notion of adjoint SDAE is introduced in a similar way as in the deterministic differential algebraic equation case. We prove a multiplicative ergodic theorem for the adjoint SDAE and the adjoint Lyapunov spectrum. Employing the notion of adjoint equation and Lyapunov spectrum of an SDAE, we are able to define Lyapunov regularity of SDAEs. Some properties and an example of a metal oxide semiconductor field-effect transistor ring oscillator under thermal noise are discussed.

Keywords::

• stochastic differential algebraic equation
• adjoint equation, Lyapunov regularity
• Lyapunov exponent
• induced two-parameter stochastic flow

2000 Mathematics Subject Classification::

• 34A09
• 34F05
• 60H10
• 93E03

Notes

^1. Email: [email protected]

^2. Email: [email protected]

Additional information

Funding

The work of N.D. Cong and N.T. The was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.02-2011.47]. N.T. The was also supported by the European Commission programme Erasmus Mundus External Cooperation Windows – Lot 12.