Taylor-based pseudo-metrics for random process fitting in dynamic programming: expected loss minimization and risk management

Stochastic optimization is the research of x optimizing E C(x, A), the expectation of C(x, A), where A is a random variable. Typically C(x, a) is the cost related to a strategy x which faces the realization a of the random process. Many stochastic optimization problems deal with multiple time steps, leading to computationally difficult problems ; efficient solutions exist, for example through Bellman’s optimality principle, but only provided that the random process is represented by a well structured process, typically an inhomogeneous Markovian process (hopefully with a finite number of states) or a scenario tree. The problem is that in the general case, A is far from being Markovian. So, we look for A′, "looking like A", but belonging to a given family A′ which does not at all contain A. The problem is the numerical evaluation of "A′ looks like A". A classical method is the use of the Kantorovitch-Rubinstein distance or other transportation metrics [Pflug, 2001], justified by straightforward bounds on the deviation |EC(x,A) − EC(x,A′)| through the use of the Kantorovitch-Rubinstein distance and uniform lipschitz conditions. These approaches might be better than the use of high-level statistics [Keefer, 1994]. We propose other (pseudo)distances, based upon refined inequalities, guaranteeing a good choice of A′. Moreover, as in many cases, we indeed prefer the optimization with risk management, e.g. optimization of EC(x, noise(A)) where noise(.) is a random noise modelizing the lack of knowledge on the precise random variables, we propose distances which can deal with a user-defined noise. Tests on artificial data sets with realistic loss functions show the relevance of the method.