Stochastic differential-algebraic equations of index 1 (Q614279)

The authors study the following class of stochastic differential equations: \[ A(t)dx+[B(t)x+f(t)]dt+G(t,x)dW_t=0,\quad t\in J:=[0,T], \] where \(A,B: J\to L(\mathbb{R}^n,\mathbb{R}^n)\) are continuous \(b\times n\)-matrix functions, rank \(A(t)=r\), \(r\) is a fixed integer, \(r<n\) (i.e., the matrix \(A(t)\) is singular for every \(t\in J\)), \(f:J\to \mathbb{R}^n\), \(G:J\times \mathbb{R}^n\to \mathbb{R}^{n\times m}\) are continuous functions (such equations are called stochastic differential algebraic equations of index I and are a stochastic generalization of the differential algebraic equations (DAEs) [see for example \textit{E. Griepentrog, M. Hanke} and \textit{R. März}, Semin.ber., Humboldt-Univ. Berl., Fachbereich Math. 92--1, 2--13 (1992; Zbl 0749.34003)]). After a short introduction of deterministic DAEs (Section 2) and an introduction of general (ordinary) stochastic differential equations (SDEs) (Section 3), the authors try to give some adequate definitions for the solution to a stochastic DAE and to a tractable stochastic DAE with index 1, extending the corresponding definitions for deterministic DAEs. The main results, which concern the existence and uniqueness of a solution to a stochastic DAE, are given in Theorem 3.7 (stochastic DAEs) and Theorem 3.14 (case of stochastic DAEs with index 1). An example ends the paper. The obtained results extend other similar result for DAEs [\textit{E. Griepentrog, M. Hanke} and \textit{R. März}, loc. cit.; \textit{E. Griepentrog} and \textit{R. März}, Differential-algebraic equations and their numerical treatment. Leipzig: BSB B. G. Teubner Verlagsgesellschaft (1986; Zbl 0629.65080), \textit{R. März}, Appl. Numer. Math. 18, No.~1--3, 267--292 (1995; Zbl 0840.65071)] or stochastic DAEs [\textit{O. Schein} and \textit{G. Denk}, J. Comput. Appl. Math. 100, No.~1, 77--92 (1998; Zbl 0928.65014), \textit{R. Winkler}, J. Comput. Appl. Math. 157, No.~2, 477--505 (2003; Zbl 1043.65010)].