Linear stochastic differential equations in the dual of a multi-Hilbertian space

We prove the existence and uniqueness of strong solutions for linear stochastic differential equations in the space dual to a multi–Hilbertian space driven by a finite dimensional Brownian motion under relaxed assumptions on the coefficients. As an application, we consider equtions in $S'$ with coefficients which are differential operators violating the typical growth and monotonicity conditions.