A Stochastic Dynamical Theory of Optimism: Environmental Feedback, Adaptive Learning, and Optimal

Optimism is commonly represented as a bias in beliefs about the probability of favorable outcomes. Recent computational work has formalized this bias within active inference and shown how a latent optimism state can influence belief updating and choice. We develop a complementary mathematical theory in which optimism is a continuously evolving learning state coupled to the environment that an agent helps create through its decisions. The central object is a controlled stochastic dynamical system for an optimism variable, an environmental state, and a decision rule. The agent’s optimism changes its action probabilities; actions alter environmental exposure and opportunities; and the resulting observations update optimism. This closed feedback loop provides a mathematical account of how optimistic and pessimistic learning regimes may persist or change.We first define a bounded stochastic model and establish well-posedness and state-space invariance under explicit conditions. We then derive a local feedback criterion for equilibrium stability. In a two-dimensional mean-field reduction, the product of the learning-to-environment and environment-to-learning sensitivities determines whether feedback is stabilising or destabilising. We identify conditions under which nonlinear feedback can generate multiple stable learning regimes. We next formulate a stochastic optimal-control problem in which interventions can alter learning and environmental affordances, derive the associated Hamilton–Jacobi–Bellman equation, and state conditions for the value function to solve it in the viscosity sense. Finally, in a one-dimensional small-noise reduction, we derive the Freidlin–Wentzell action governing rare transitions between pessimistic and optimistic basins. The theory yields testable predictions: environmental affordances can change the stability of optimism, interventions that change opportunity may be more effective than interventions that change beliefs alone, and identical external conditions can support different long-run optimism levels when feedback is sufficiently nonlinear.The results are mathematical consequences of the proposed model, not empirical claims that human optimism necessarily exhibits bistability. Establishing the model’s empirical adequacy requires longitudinal measurements of beliefs, choices, experienced outcomes, and environmental conditions.