Weak Differentiability of Product Measures

In this paper, we study cost functions over a finite collection of random variables. For these types of models, a calculus of differentiation is developed that allows us to obtain a closed-form expression for derivatives where “differentiation” has to be understood in the weak sense. The technique for proving the results is new and establishes an interesting link between functional analysis and gradient estimation. The key contribution of this paper is a product rule of weak differentiation. In addition, a product rule of weak analyticity is presented that allows for Taylor series approximations of finite products measures. In particular, from characteristics of the individual probability measures, a lower bound (i.e., domain of convergence) can be established for the set of parameter values for which the Taylor series converges to the true value. Applications of our theory to the ruin problem from insurance mathematics and to stochastic activity networks arising in project evaluation review techniques are provided.