Optimized Certainty Equivalents (OCEs)

• OCEs are convex risk measures defined by a variational principle that blends a cash anchor with a transformed loss, unifying metrics like CVaR, entropic risk, and mean-variance.
• They extend to multivariate, conditional, and robust frameworks using dual representations and efficient algorithmic schemes, ensuring practical statistical error bounds.
• OCEs underpin diverse applications in risk management, reinforcement learning, and control by providing a unified, tractable, and theoretically grounded risk quantification.

Optimized Certainty Equivalents (OCEs) are a foundational class of convex risk measures unifying a broad array of risk metrics, including entropic risk, Conditional Value-at-Risk (CVaR), and mean-variance models. OCEs are formulated through a variational principle that expresses risk as the minimum aggregate of a cash anchor and a transformed loss, parametrized by a convex disutility (loss) function. The OCE framework provides a rigorous mathematical backbone for risk quantification in finance, insurance, statistical estimation, optimal control, and reinforcement learning, and its flexible structure accommodates extensions to multivariate, conditional, robust, and model-uncertain settings.

1. Definition and Mathematical Formulation

The optimized certainty equivalent for a real-valued random variable $X$ and convex (increasing) loss function $l$ is

$$\operatorname{OCE}_l(X) := \inf_{m \in \mathbb{R}} \left\{ m + \mathbb{E}[l(-X - m)] \right\}.$$

The associated optimizer $m^*$ is characterized by the subdifferential equation

$1 \in \mathbb{E}[\partial l(-X - m^*)].$

Special choices of $l$ generate common risk measures:

• CVaR at level $\alpha$: $l(x) = x^+/\alpha$,
• Entropic risk: $l(x) = (e^{\lambda x} - 1)/\lambda$,
• Monotone mean-variance: $l(x) = ([1 + x]^+)^\gamma/\gamma - 1/\gamma$.

OCEs admit a dual representation: $$\operatorname{OCE}_l(X) = \sup_{Q \in \mathcal{D}^{l^*}} \left\{ \mathbb{E}_Q[-X] - \mathbb{E}[l^*(dQ/dP)] \right\},$$ where $l^*$ is the convex conjugate of $l$ and the domain $\mathcal{D}^{l^*}$ is determined by the integrability of $l^*(dQ/dP)$.

Key properties established for OCEs:

• Convexity: $$R(\lambda X + (1-\lambda)Y) \leq \lambda R(X) + (1-\lambda) R(Y)$$,
• Monotonicity: If $X \leq Y$ a.s., $R(X) \geq R(Y)$,
• Cash invariance: $R(X + m) = R(X) - m$,
• (Positive) Homogeneity: When $l$ is positively homogeneous, $R(\lambda X) = \lambda R(X)$.

These conditions ensure that OCEs fit into the framework of convex monetary risk measures and, under positive homogeneity (coherence), they align with the axioms of coherent risk measures.

2. Extensions: Multivariate, Conditional, and Robust OCEs

OCEs generalize to multivariate risk measurement, conditional information, and ambiguous models:

Multivariate OCEs

Given $X \in \mathbb{R}^d$, and a multivariate loss $l: \mathbb{R}^d \to (-\infty, \infty]$ (convex, monotone, $l(0)=0$), the risk measure is

$$R(X) = \inf_{w \in \mathbb{R}^d} \left\{ \sum_{i=1}^d w_i + \mathbb{E}[l(-X - w)] \right\},$$

where systemic dependence can be encoded via aggregation terms such as $\Lambda(x)$ added to sums of univariate losses (Kaakai et al., 2022).

Conditional OCEs

For a sub-$\sigma$-algebra $G$, the conditional OCE for $x \in L^\infty(F|G)$ is given by

$$\inf_{v \in M(G)} \left\{ \mathbb{E}_v[x|G] + D_{\phi,G}(v\|P) \right\} = \sup_{a \in L^0(G)} \left( a - \mathbb{E}[\phi^*(a-x) | G] \right),$$

where the minimization is over conditional probabilities $M(G)$, and $D_{\phi,G}$ is a conditional $\phi$-divergence (Principi et al., 2022).

Robust OCEs

Model ambiguity is handled by

$$\mathcal{OCE}_{l,\varphi}(X) = \inf_{m \in \mathbb{R}} \left\{ m + \sup_{\mu} \left( \int l(x - m)d\mu(x) - \varphi(d_c(\mu_X, \mu)) \right) \right\},$$

where $\mu_X$ is the nominal distribution, $d_c$ is an optimal transport cost (e.g., Wasserstein), and $\varphi$ penalizes model distance. This reduces to tractable forms using conjugate duality and dimension reduction (Bartl et al., 2017, Li et al., 2023).

3. Numerical Computation and Algorithmic Schemes

Efficient OCE computation is crucial for practical risk management.

• Univariate OCEs: Fourier-based algorithms exploit the moment generating function and Fourier transforms to convert expectation minimizations to one-dimensional integral computations, achieving computational time on par with Value-at-Risk and outperforming simulation-based methods, particularly for CVaR and mean-variance (Drapeau et al., 2012).
• Multivariate OCEs: Stochastic approximation (SA) schemes (Robbins-Monro style) are designed:

$$m_{n+1} = \Pi_K\left[ m_n + \gamma_n(\nabla l(-X_{n+1} - m_n) - 1) \right],$$

with simultaneous estimation of the risk value $R_n$ (via a companion update). These SA methods provide joint estimation of optimizers and risk values, with finite-sample error control via Central Limit Theorems, robust confidence intervals, and improved scaling in high dimensions relative to Monte Carlo or Fourier methods (Kaakai et al., 2022, Ghosh et al., 31 May 2024).

Comparison of methods:

Method	Computes	Error Estimation	Confidence Intervals	Handles High Dims
Monte Carlo + Root Find	$R(X)$, $m^*$	Only for $R(X)$	No	Moderate
Fourier	$R(X)$, $m^*$	Only for $R(X)$	No	Limited
Stochastic Approx.	$R(X)$, $m^*$	Yes (CLT)	Yes	Yes

4. Statistical Estimation and Concentration Results

Estimation of OCEs from data is addressed using sample average approximation (SAA) and stochastic approximation, with rigorous non-asymptotic error analysis:

• SAA estimator for OCE, using i.i.d. samples $(X_i)$:

$$\hat{\operatorname{OCE}} = \inf_{\xi \in \mathbb{R}} \left\{ \xi + \frac{1}{n} \sum_{i=1}^n \phi(X_i - \xi) \right\}$$

MSE and concentration bounds under strong convexity and smoothness:

$$\mathbb{E}\left[ (\hat{e}_n - e^*)^2 \right] \leq \frac{C}{n},\qquad \mathbb{P}(|\hat{e}_n - e^*| > \epsilon) \leq 2 \exp\left(-\frac{n \mu^2 \epsilon^2}{8L^2 \sigma^2}\right)$$

and sub-exponential tails for the risk value (Ghosh et al., 31 May 2024).

• Streaming (online) estimation via SA achieves the same $O(1/n)$ rate for the minimizer and $O(1/\sqrt{n})$ for the risk value (Ghosh et al., 31 May 2024, Gupte et al., 1 Jun 2025).
• These results generalize to non-Lipschitz losses (e.g., mean-variance), with error rates depending on moments and smoothness of the disutility $\phi$ (Gupte et al., 1 Jun 2025).

5. Applications in Risk Management, Control, and Learning

OCEs provide rigorous foundations and tractable methods for a range of applications:

Portfolio and Financial Risk Aggregation

• Multivariate OCEs accommodate risk allocation and systemic risk, with characterization of risk allocations via minimizer $m^*$, sensitivity analysis, and robust computational schemes for large (possibly dependent) portfolios (Kaakai et al., 2022).

Robust Risk and Option Pricing

• Robust OCEs enable closed-form uncertainty quantification in AVaR and option pricing:

$$\mathcal{AV@R}_\alpha = \mathrm{AV@R}_\alpha(\mu_0) + \frac{\delta}{\alpha}$$

for Wasserstein ball ambiguity radius $\delta$ (Bartl et al., 2017).

Reinforcement Learning and Sequential Decision Making

• Risk-sensitive RL uses OCEs as trajectory-level or recursive risk criteria, providing regret-optimal learning algorithms and enabling history-dependent optimal policies in complex Markov Decision Processes (Xu et al., 2023, Wang et al., 10 Mar 2024, Lee et al., 23 Oct 2025).
• Constrained RL leverages OCEs as both objectives and constraints, with strong duality and convergent minimax algorithms exploiting the risk structure (Lee et al., 23 Oct 2025).

Statistical Learning Theory

• Generalization bounds for empirical risk minimization under OCE risks use Rademacher complexity and variance-based arguments, recovering and sharpening excess risk bounds for CVaR, mean-variance, and entropic loss regimes (Lee et al., 2020).

6. Generalizations, Dualities, and Structural Relationships

OCEs possess dual and variational representations that provide interpretative and computational advantages:

• Dual forms connect to $\phi$-divergence risk measures and entropic penalties, enabling robust, conditional, and law-invariant risk representations (Principi et al., 2022, Bartl et al., 2017).
• Reverse optimization formulas relate mean excess functions and OCEs via Fenchel-Legendre duality, facilitating explicit computation under model uncertainty and for insurance pricing (Guan et al., 2022).
• Preference robustification and set-valued generalizations of OCE to incomplete preferences and multiple utility/prior frameworks are addressed through vector convex optimization (Wu et al., 2022, Rudloff et al., 2019).

7. Theoretical Significance and Methodological Impact

OCEs serve as the backbone of a unified risk modeling paradigm that links convex analysis, stochastic control, robust optimization, and statistical learning. Recent research advances include:

• Restoration of dynamic programming principles for OCE-based control via state space augmentation and duality, leading to viscosity solution characterizations for time-inconsistent risk objectives (Veraguas et al., 2020).
• Modular computational frameworks for risk-aware decision making that admit sharp, finite-sample statistical guarantees and practical error bars.
• Systematic integration of information-theoretic, robust, and distributionally-ambiguous considerations into convex risk assessment.

These properties and developments position OCEs as a central tool of contemporary risk science and quantitative decision theory.