Robust sensitivity analysis for stochastic systems (Q2833103)

Let \(X_1,\dots,X_T\) be a sequence of indipendent identically distributed random variables, \(h(.)\) be a real-valued function and \(T\) the time horizon. Let \(h\) be evaluated given to its arguments, but it does not necessarily have the closed form. Moreover, assume a baseline model \(P_0\) approximately describing each \(X_i\), so that the corresponding baseline performance measure is \(E_0\,h(X_T)\). On the other hand, let \(P_f\) denote the distribution that governs the true model, which is unknown, and \(E_f\) the orresponding expectation. The author is interested in the worst (best) case optimization problem NEWLINE\[NEWLINE\max/\min\;E_f\,h(X_T) \quad st\quad D(P_f||P_0)\leq\eta,\;NEWLINE\]NEWLINE where \(D(P_f||P_0)\) denotes the Kullback-Leibler divergence. The main result of this paper stipulates that, when letting \(\eta\to0\), then under mild technical assumptions on~\(h\) the optimal values of the problem can be written in the form NEWLINE\[NEWLINE \max/\min\;E_f\,h(X_T)=E_0\,h(X_T) + \zeta_1(P_0,h)\sqrt{\eta}+\zeta_2(P_0,h)\eta+\dots NEWLINE\]NEWLINE where \(\zeta_1(P_0,h),\,\zeta_2(P_0,h)\) is a sequence of coefficients that can be written explicitly in terms of~\(h\) and~\(P_0\).