We extend the notion of risk measures generated by distortion functions to the dynamic discrete time setup. Consequently, using dual representations, we define coherent acceptability indices – a special class of performance measures - generated by families of dynamic risk measures generated by distortion functions. In this talk:

- We present a characterization of these risk and performance measures in terms of dynamic weighted value at risk measures.

- We discuss several widely used classes of distortion functions and derive some new representations of these distortions.

- In the context of intertemporal decision making, we study the time consistency of these classes of risk and performance measures, beyond the classical notion of strong time consistency.

- We formulate and solve a diversification optimization problem, using these risk measures. We show that the underlying stochastic control problem can be formulated as a vector-optimization problem, and we prove that a version of a set-valued Bellman principle is satisfied.

- We derive an explicit policy gradient formula and implement a deep neural network to solve numerically the diversification optimization problem. The proposed deep learning technique allows to overcome computation difficulty caused by the non-convexity of the vector optimization problem.

This is joint work with T. R. Bielecki and H. Liu