Derivatives of Probability Measures

Direct Differentiability

Inverse Differentiability

Simulation of Derivatives

Process Derivatives

Distributional Derivatives

See also

References

For stochastic optimization problems of the form

(1)

where H(x, v) is a cost function, μ _x a family of probability measures indexed by x and F(x) the objective value function (OVF), the necessary condition ∇ _x F(x) = 0 must be expressed in terms of the derivatives of H(x, ·) and μ _x w.r.t. x. In particular, concepts of differentiability of probability measures are needed.

Suppose that the family (μ _x ) is dominated , i. e. there is a Borel measure ν such that the densities exist for all x. Then the differentiability of the measures may be defined by the differentiability of the densities.