Finite horizon stochastic programs typically come in the following general form: (P1) minimize the overall expected cost \(Ef(w,x_ 1(w),\dots,x_ T(w))\) by making, sequentially, at each state \(w\) of the world. The purpose of this paper is to characterize locally optimal solutions to problem (P1) in terms of the so-called Lagrange multipliers or Kuhn- Tucker conditions and to identify the precise nature of multipliers, to guarantee their existence, and finally, to provide constraint qualifications that imply that the multiplier rule has the desired normal form.