How risky is a random process?

Abstract

The riskiness of random processes is compared by (a) employing a decision-theoretic equivalence between processes and lotteries on pathspaces to identify the riskiness of the former with that of the latter, and (b) using the theory of comparative riskiness of lotteries over vector spaces to compare the riskiness of lotteries on a given path-space. We derive the equivalence used in step (a) and contribute a new criterion to the theory applied in step (b). The new criterion, involving a generalized form of second order stochastic dominance, is shown to be valid by establishing its equivalence to the standard decision-theoretic criterion. We demonstrate its tractability via diverse economic applications featuring risk embodied in random processes.

The decision‐making of optimal equity and capital structure based on dynamical risk profiles: A Langevin system framework for SME growth

“…It is synthetically regarded as internal risk, and additively affects the system. [33][34][35] ξ t ( ) is the asymmetric dichotomous process satisfying:…”

“…It mimics the stochastic fluctuation from environment uncertainty, including configuration instability, internal stimulation mechanism, macro policy fluctuation, even the volatility of bilateral efforts from subjective uncertainty to technology or marketing prospect. It is synthetically regarded as internal risk , and additively affects the system 33–35 is the asymmetric dichotomous process satisfying: It describes the external risk introduced by the outward investment behavior with capital , and it causes the damping coefficient to show a certain time dependence: .…”

“…It mimics the stochastic fluctuation from environment uncertainty, including configuration instability, internal stimulation mechanism, macro policy fluctuation, even the volatility of bilateral efforts from subjective uncertainty to technology or marketing prospect. It is synthetically regarded as internal risk , and additively affects the system 33–35 …”