The Bayes posterior solves the variational problem of maximizing the expected likelihood of the data minus the Kullback–Leibler divergence from the prior distribution. We generalize Bayesian inference from this variational point of view. In particular, we introduce a variational inference (VI) problem whose solutions we call posteriors and show that these posteriors are risk identifiers, i.e., subgradients of risk measures. This insight leads to: (i) rigorous results concerning the existence and asymptotics of posteriors, (ii) the ability to constrain the divergence from the prior—which motivates a measure-valued relaxation of M-estimation that we call coherent inference—and (iii) insights into VI problems derived from ϕ -divergences. These insights include semi-analytical expressions for posteriors and connections with utility theory. We use the semi-analytical expressions for ϕ -divergence posteriors to numerically solve a multi-armed bandit problem.
This paper addresses a key limitation in the real options literature on copper mine valuation, where conventional approaches often rely on geometric Brownian motion (GBM) to model underlying commodity and exchange rate dynamics. Such approaches may overlook discrete jumps and abrupt market shocks that are particularly relevant in emerging markets and commodity-dependent economies. To address this limitation, we first conduct a statistical analysis of copper prices and the Dollar/Rial exchange rate over the period 2013–2025 to identify jump behavior. The empirical evidence supports the presence of significant discontinuous movements in both series. Motivated by these findings, we adopt a Merton Jump–Diffusion (MJD) framework to characterize continuous fluctuations and discrete shocks in the two stochastic variables. Using the estimated jump–diffusion parameters, one-year Monte Carlo simulations of copper prices and exchange rates are generated under the MJD framework. The simulated paths are used to characterize the range of stochastic outcomes and the relevant state-space ranges, rather than being used directly to compute the project value. The Sarcheshmeh copper mine project is then evaluated within a real options framework. The valuation problem is formulated as a two-factor partial differential equation and solved numerically using the Crank–Nicolson finite difference method. To represent the jump component within the valuation PDE, we employ a diffusion-equivalent moment-matching approximation that preserves the first two moments of the jump component. The PDE-based grid solution provides the project-value surface over the relevant state space, which is essential for the subsequent ex-post interpretation of continuation and abandonment regions. This approach is therefore adopted to obtain the complete value surface required for the decision analysis, rather than solely for computational tractability. The approximation does not explicitly capture higher-order distributional characteristics and tail effects associated with extreme jump realizations. A consistent comparison is made between GBM- and MJD-based project valuations, with net present value (NPV) analysis serving as a deterministic benchmark. The results show that incorporating jump risk through the proposed approximation leads to systematically lower project values compared with GBM-based valuation. The analysis examines the effects of jump intensity, jump magnitude, and jump volatility on project value, highlighting the economic significance of discontinuous shocks. The findings indicate that ignoring jump risk may result in the overvaluation of mining projects. The analysis also examines the implications of uncertainty for continuation and abandonment decisions; these operational decisions are inferred ex post from the computed project-value surface rather than modeled as endogenous managerial controls or derived optimal thresholds. Overall, the proposed framework provides a real options approach for evaluating mining investments under commodity price and exchange rate uncertainty, while explicitly recognizing that jump risk in the valuation PDE is represented through a first- and second-moment diffusion-equivalent approximation rather than the full integro-partial differential equation. The Monte Carlo simulations are used to characterize stochastic outcomes and inform the relevant state-space ranges, whereas the PDE solution provides the project-value surface required for the ex-post decision analysis.
We study the existence of equilibrium in dynamic asset markets where agents possess quasi-concave mean-value utility functions. These markets extend the classical Arrow–Debreu framework to the case where consumption sets are unbounded from below and continuous trading occurs over a finite time interval. The dynamic asset market problem is reformulated as a quasi-variational inequality (QVI) with unbounded constraint maps, and sufficient conditions for the existence of solutions to this class of QVI problems are established. The QVI formulation subsequently ensures the existence of equilibrium for the given economic model under a coercivity condition, without relying on traditional no-arbitrage assumptions. Finally, we propose an algorithm for computing an equilibrium of dynamic asset markets by resolving the corresponding quasi-variational inequality into variational inequalities.
This study develops a risk-averse interval bi-level programming framework for sustainable closed-loop production systems operating under dependent uncertainty. Correlated ambiguities in demand, return rates, and operational costs are represented through a structured mean–radius interval formulation with a common dependence factor, while heterogeneous uncertainty attitudes of the leader and follower are incorporated using a preference-based scalarisation approach. The lower-level interval optimisation problem is transformed into a deterministic nonlinear programme via KKT-type optimality conditions, yielding a KKT-based single-level reformulation that is valid under the stated convexity and constraint qualification assumptions. To solve the resulting complementarity-constrained nonlinear model, a hybrid evolutionary–NLP algorithm is proposed that combines global exploration with local refinement. A stylised closed-loop production example is presented to illustrate the applicability of the proposed framework and to examine the influence of uncertainty-width aversion and dependent uncertainty on production, remanufacturing, disposal, and policy decisions. The numerical results indicate that changes in uncertainty attitudes primarily affect the valuation of interval uncertainty, while the system’s operational structure remains comparatively stable across the tested scenarios.
