Risk-Averse Stochastic Programming: Time Consistency and Optimal Stopping

Abstract

This paper addresses time consistency of risk-averse optimal stopping in stochastic optimization. It is demonstrated that time-consistent optimal stopping entails a specific structure of the functionals describing the transition between consecutive stages. The stopping risk measures capture this structural behavior and allow natural dynamic equations for risk-averse decision making over time. Consequently, associated optimal policies satisfy Bellman’s principle of optimality, which characterizes optimal policies for optimization by stating that a decision maker should not reconsider previous decisions retrospectively. We also discuss numerical approaches to solving such problems.

Discrete-time risk-aware optimal switching with non-adapted costs

“…We thus may have, but do not limit ourselves to, the situation where is the natural filtration generated by . To our knowledge, the necessary and sufficient conditions we provide for an optimal switching strategy in this infinite-horizon setting under general filtration are novel and extend results in, for example, [1, 4, 8, 11, 15].…”

“…Note that we are considering a setting where each of the costs gi,j (t) : i, j ∈ I t∈T is measurable with respect to the σ -algebra F and G is any filtration with G t ⊂ F for all t ∈ T. We thus may have, but do not limit ourselves to, the situation where G is the natural filtration generated by {g i,j (t) : i, j ∈ I} t∈T . To our knowledge, the necessary and sufficient conditions we provide for an optimal switching strategy in this infinite-horizon setting under general filtration are novel and extend results in, for example, [1,4,8,11,15]. The rest of the paper is structured as follows.…”

“… ). Let be a -conditional risk mapping: a family of mappings , , satisfying normalisation, conditional translation invariance, and monotonicity (see Appendix A.1). For with , let be the finite-horizon aggregated (or nested) risk mapping generated by ([5, 14, 15, 20, 21]; see also [2]): that is, and All inequalities are interpreted in the -almost-sure sense. …”

“…• For s, t ∈ T with s ≤ t, let ρ s,t be the finite-horizon aggregated (or nested) risk mapping generated by ρ ( [5,14,15,20,21]; see also [2]): that is, ρ t,t (W t ) = ρ t (W t ) and…”

Equilibrium transport with time-inconsistent costs: An application to matching problems in the job market

“…There are empirical findings that human behavior may contradict the axiom of rationality and deviate from the previous optimal strategy at later stages. This is observed and studied in various areas such as non-exponential discounting (Laibson, 1997), state-dependent preferences and mean-variance portfolio selection Björk et al, , 2017Han et al, 2021;Dai et al, 2021;Kováčová and Rudloff, 2021), optimal stopping (Bayraktar et al, 2021;Pichler et al, 2022), robust decision-making (Epstein and Ji, 2022;Arora and Gao, 2022), prospect theory (Kahneman and Tversky, 1979), probability distortion (Ma et al, 2021), and gambling behaviors (Barberis, 2012).…”