On stochastic equations with respect to semimartingales I.†

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Abstract
Two types of equations are considered: where A is a predictable increasing process, M is a locally square integrable martingale taking values in a Hilbert space, q is a stochastic martingale measure, a,b,c are random functions continuous in x which satisfy natural measurability properties, a kind of monotonity condition and a condition on growth in x. (Thaese are weaker than the usual Lipschitz condition and the condition of linear growth in x, respectively.) A uniqueness and exixtence theorem is proved for the solutions (which take values in Rd) of Eq. (1). It is shown that Eq. (2) Can be rewritten into the form of Eq.(1), and so the uniqueness and existence theorem is obtained for Eq.(2) as well. Further, the dependence of the solutions on parameters and initial values are investigated. The proffs are elementary and are based on the methods used in [8].