The paper contains a generalization of Girsanov’s theorem for following multi-dimensional problem \[ \min \{EL(\gamma^ Tx(t))/dx=A(t)x+B(t)[u(t)+Z]dt+C(t)dw(t) \]
\[ x(0)=x_ 0,\quad f(t)\leq u(t)\leq g(t),\quad t\in [0,T]\} \] where \[ \dim x=\dim u=\dim z=\dim w, \] L is even continuously differentiable; \(L'(x)\geq 0\), \(x>0\), and L is of polynomial growth as \(x\to \infty\); z is a constant random vector; w(t) is a standard Wiener process. It is shown that the optimal solution has the form \[ u_ 0(t)=2^{-1}[f(t)+g(t)]+2^{-1}[g(t)-f(t)]sign \psi (t) \] where \(\psi\) (t) can be calculated.