Linear stochastic differential-algebraic equations with constant coefficients. (English) Zbl 1135.60033

Alabert, Aureli; Ferrante, Marco

Electron. Commun. Probab. 11, 316-335 (2006).

The authors of the present paper study linear stochastic differential-algebraic equations (DAEs) with additive noise of the form \(A\dot{X}_t+BX_t=f(t)+\Lambda\dot{W}_t,\, t\geq 0\), where \(W\) is an \(m\)-dimensional Brownian motion defined over some probability space \((\Omega,{\mathcal F},P)\) and \(\dot{W}\in L^0(\Omega; D'(R_+))\) is the associated white noise; recall that \(D'(R_+)\) is the space of distributions over \(R_+\), i.e., the dual space of \(D(R_+)=C_c^{\infty}(R_+)\). Under the assumption that the matrices \(A,B\in R^{n\times n}\) form a regular matrix pencil, i.e., that there is some \(\lambda\in R\) such that det\((\lambda A+B)\neq 0\), the authors show that there is a unique \(D'(R_+)\)-valued solution \(X\). For this the authors consider the pathwise DAE \(A\dot{X}_t(\omega)+BX_t(\omega)=f(t)+\Lambda\dot{W}_t(\omega),\, t\geq 0\), which, for almost all \(\omega\in\Omega\) can be solved in \(D'(R_+)\). For proving the measurability of the such got solution \(X(\omega)\) with respect to \(\omega\), the authors reduce their pathwise DAEs to the associated Kronecker Canonical Form in order to get with the help of a variation of constants argument a nearly explicit expression of the solution. After the authors study sufficient conditions for the absolute continuity of the probability law of the solution with respect to the Lebesgue measure. The last section of the paper discusses an example arising from a problem of electrical circuit simulation.

Reviewer: Rainer Buckdahn (Brest)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
65L80	Numerical methods for differential-algebraic equations
34A09	Implicit ordinary differential equations, differential-algebraic equations

Keywords:

Differential-algebraic equation; stochastic differential-algebraic equation; additive white noise.