Optimized Certainty Equivalent (OCE)

• Optimized Certainty Equivalent (OCE) is a family of convex, cash-additive risk measures that optimizes a deterministic shift of a random variable using a convex loss or concave utility function.
• Its variational structure facilitates dual representations, dynamic programming via state augmentation, and practical computations in stochastic control and robust risk estimation.
• OCE encapsulates popular criteria such as entropic risk, CVaR/AVaR, and mean-variance measures, enabling broad applications in finance, statistical learning, and reinforcement learning.

Optimized certainty equivalent (OCE) is a family of convex, cash-additive, law-invariant risk functionals introduced by Ben-Tal and Teboulle and now used across stochastic control, mathematical finance, statistical learning, and reinforcement learning. In the standard loss-based convention, an OCE evaluates a random loss $X$ by optimizing over a deterministic cash shift before applying a convex loss or disutility function; in a reward-based convention, the same object is written through a concave utility and a supremum. This variational structure makes OCE broad enough to include entropic risk, CVaR/AVaR, and mean-variance-type criteria, while remaining tractable enough to admit duality, dynamic representations, nonasymptotic estimation theory, and algorithmic reductions in several modern settings (Veraguas et al., 2020, Drapeau et al., 2012).

1. Definition and sign conventions

A common loss-based formulation fixes a convex, increasing loss function $\ell:\mathbb{R}\to\mathbb{R}$ and defines

$$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$

Equivalent variants appear throughout the literature, including

$$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$

or, with a sign change, utility-based forms such as

$$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$

where $u$ is concave and nondecreasing. These expressions are not contradictory; they reflect different conventions for whether $X$ is interpreted as loss or reward, and whether the primitive object is a convex disutility or a concave utility (Drapeau et al., 2012, Xu et al., 2023, Mortensen et al., 20 May 2026).

Several papers impose normalization and regularity conditions tailored to their downstream analysis. In continuous-time stochastic control, a typical standing assumption is that $\ell$ is increasing, convex, bounded from below, has at most polynomial growth, satisfies $\ell(0)=0$, $\ell^*(1)=0$, and $\ell:\mathbb{R}\to\mathbb{R}$0 for $\ell:\mathbb{R}\to\mathbb{R}$1 large enough, where $\ell:\mathbb{R}\to\mathbb{R}$2 is the convex conjugate (Veraguas et al., 2020). In statistical-learning and conformal formulations, the analogue is a convex, closed, nondecreasing transfer function $\ell:\mathbb{R}\to\mathbb{R}$3 with $\ell:\mathbb{R}\to\mathbb{R}$4 and $\ell:\mathbb{R}\to\mathbb{R}$5, giving

$\ell:\mathbb{R}\to\mathbb{R}$6

for a loss random variable $\ell:\mathbb{R}\to\mathbb{R}$7 (Yeh et al., 9 Oct 2025).

This scalar shift parameter is the defining feature of OCE. It is an optimized certainty-equivalent offset: the random position is first translated by $\ell:\mathbb{R}\to\mathbb{R}$8, then transformed by $\ell:\mathbb{R}\to\mathbb{R}$9 or $$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$0, and finally recombined with the shift. A plausible implication is that OCE sits exactly at the interface between certainty-equivalent utility theory and convex risk measurement, which explains why it appears in both reward-maximization and loss-minimization literatures (Ghosh et al., 2024, Liang et al., 2023).

2. Structural properties and dual representations

Under the standard assumptions on $$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$1 or $$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$2, OCE satisfies the canonical axioms of a convex monetary risk measure: monotonicity, convexity, and cash additivity or translation invariance. In the loss convention, if $$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$3 almost surely then $$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$4; for any constant $$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$5, $$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$6; and $$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$7. Law invariance also holds in the usual formulations: the value depends only on the law of $$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$8 (Veraguas et al., 2020, Ghosh et al., 2024).

The primal one-dimensional optimization is complemented by a convex dual representation. In a loss-based setting,

$$\rho_\ell(X)=\inf_{m\in\mathbb{R}}\left\{m+\mathbb{E}\big[\ell(X-m)\big]\right\}.$$9

with $$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$0 the convex conjugate. In Orlicz-space formulations the dual domain is written as a family of probability measures or densities with appropriate integrability under $$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$1, but the operative structure is the same: OCE is a penalized worst-case expectation over tilted densities (Veraguas et al., 2020, Drapeau et al., 2012). In Brownian filtrations, dual densities can be represented as stochastic exponentials,

$$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$2

which is the starting point for dynamic programming in the time-inconsistent case (Veraguas et al., 2020, Veraguas et al., 2016).

The optimizer in the primal problem is characterized by first-order or subgradient conditions. If the relevant derivative is continuous, the optimal shift satisfies an equation of the form

$$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$3

depending on notation. For nonsmooth losses, the characterization becomes a pair of subgradient inequalities. This root condition underlies essentially all numerical procedures for OCE, from Fourier methods and SAA to SG-based optimization (Drapeau et al., 2012, Ghosh et al., 2024, Gupte et al., 1 Jun 2025).

A further structural point is that OCE admits meaningful extensions beyond the univariate scalar case. In multivariate form, with $$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$4 interpreted as a profit-and-loss vector, one writes

$$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$5

and retains convexity, monotonicity, cash invariance, continuity, subdifferentiability, and a robust dual representation against vector-valued densities (Kaakai et al., 2022).

3. Canonical examples

The principal appeal of OCE is that many widely used risk criteria are recovered by particular choices of $$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$6, $$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$7, or $$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$8.

Choice	Resulting criterion	Brief note
$$\rho_\ell(X)=\inf_{\eta\in\mathbb{R}}\left\{\mathbb{E}\big[\ell(\eta-X)\big]-\eta\right\},$$9 or $$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$0	Entropic risk	Time-consistent special case in several dynamic settings
$$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$1 or $$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$2	CVaR / AVaR	Piecewise linear, coherent, tail-focused
$$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$3 or $$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$4 on the appropriate domain	Mean-variance type	Produces mean plus or minus variance terms, depending on sign convention
Piecewise linear $$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$5	Mean–CVaR mixture	Interpolates expectation and tail risk

For entropic risk, the OCE collapses to a log-mgf expression. In one common sign convention,

$$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$6

while in reward-based notation the corresponding OCE is

$$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$7

Its conjugate is a scaled relative entropy, and several dynamic theories identify it as essentially the only OCE within the usual class that preserves time consistency without augmentation (Veraguas et al., 2020, Drapeau et al., 2012).

For CVaR or AVaR, the OCE representation is variational: $$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$8 up to the standard notational shift between $$\mathrm{OCE}^u(X)=\sup_{\eta\in\mathbb{R}}\left\{\eta+\mathbb{E}[u(X-\eta)]\right\},$$9 and tail probability level. The optimizer is a quantile under continuity, so the OCE shift becomes a VaR threshold and the OCE value averages the corresponding tail (Drapeau et al., 2012, Ghosh et al., 2024).

Quadratic or truncated-quadratic choices yield mean-variance and monotone mean-variance criteria. In one formulation,

$u$0

whereas in a reward-based formulation with concave quadratic utility one obtains $u$1. This apparent discrepancy is again purely a sign-convention issue: the OCE family contains both the risk measure on losses and the certainty-equivalent functional on rewards (Ghosh et al., 2024, Xie, 2 Dec 2025).

The family also includes coherent piecewise-linear models and hybrid criteria. A piecewise linear utility with slopes $u$2 and $u$3 yields

$u$4

which makes explicit how OCE interpolates between expectation and a coherent tail criterion (Wang et al., 2024).

4. Dynamic representations and time inconsistency

Static OCE is generally time inconsistent. Outside the entropic special case, conditional OCEs do not satisfy a tower property, so Bellman recursion fails in the original state space. This obstacle is central in continuous-time control: controlled diffusions with OCE objectives cannot be treated by standard risk-neutral dynamic programming unless the state is enlarged (Veraguas et al., 2020, Veraguas et al., 2016).

The now-standard resolution is dual-state augmentation. In the continuous-time diffusion setting

$u$5

one introduces an auxiliary density state $u$6 driven by a dual control $u$7,

$u$8

and defines an augmented value function $u$9. This converts the original time-inconsistent risk minimization into a zero-sum stochastic game between the primal control $X$0 and the dual density control $X$1. The augmented problem satisfies a DPP and is characterized by a singular HJBI equation of the form

$X$2

Under uncontrolled uniformly nondegenerate $X$3, continuity of $X$4, viscosity existence, and comparison-based uniqueness are established in important cases, notably when $X$5 is compact, as for AVaR (Veraguas et al., 2020).

Earlier Brownian-filtration work already identified this enlarged-state mechanism as a substitute for time consistency and showed that CVaR and monotone mean-variance can be computed dynamically through singular PDE techniques. That line emphasizes minimal viscosity supersolutions rather than a classical comparison theorem in full generality (Veraguas et al., 2016). In both formulations, the entropic case is exceptional: a logarithmic transform collapses the augmented HJBI to an ordinary risk-neutral HJB, reflecting genuine dynamic consistency (Veraguas et al., 2020, Veraguas et al., 2016).

A distinct dynamic extension is forward OCE in incomplete markets with random endowment. There, the objective is not a backward recursion from a fixed horizon but a maturity-independent forward performance criterion. The theory develops primal and dual FBSDE systems, establishes strong duality, and defines a forward OCE

$X$6

with a dual representation in terms of state-price densities $X$7. In the exponential case, the normalized forward OCE coincides with the negative of a forward entropic risk measure (Liang et al., 2023).

5. Estimation, numerical computation, and optimization

Algorithmically, OCE computation is dominated by the scalar shift parameter. Once the optimizer $X$8 or $X$9 is identified, evaluating the risk reduces to a single expectation. This has enabled several distinct computational paradigms (Drapeau et al., 2012, Gupte et al., 1 Jun 2025).

When the characteristic function or mgf of the loss is known, OCE admits efficient Fourier-transform methods. The core idea is to express $\ell$0 or $\ell$1 as a one-dimensional Fourier integral involving the transform of the loss function and the mgf $\ell$2, then solve the first-order condition by root finding, typically Brent’s method. For CVaR, this yields a method comparable in cost to VaR computation; in the reported Normal Inverse Gaussian experiments at $\ell$3 and $\ell$4, Fourier CVaR required roughly $\ell$5 seconds, compared with about $\ell$6 seconds for VaR and $\ell$7–$\ell$8 seconds for standard CVaR integration (Drapeau et al., 2012).

For sample-based estimation, recent work provides nonasymptotic theory. Under $\ell$9-strong convexity, $\ell(0)=0$0-smoothness, differentiability, and sub-Gaussianity of $\ell(0)=0$1, the empirical optimizer $\ell(0)=0$2 in SAA satisfies

$\ell(0)=0$3

and the OCE value estimate enjoys Bernstein-type concentration. The same paper analyzes Robbins–Monro SA,

$\ell(0)=0$4

and proves $\ell(0)=0$5 for the Polyak–Ruppert average $\ell(0)=0$6, which makes OCE estimation viable in streaming settings (Ghosh et al., 2024).

Optimization with respect to model parameters $\ell(0)=0$7 is likewise tractable. For a smooth parametrization $\ell(0)=0$8, the gradient of the OCE objective takes the envelope form

$\ell(0)=0$9

where $\ell^*(1)=0$0 is the OCE shift solving the derivative constraint. Double-sample SAA gradient estimators and projected stochastic-gradient schemes then admit explicit MAE, MSE, and $\ell^*(1)=0$1-type optimization rates under strong convexity and smoothness assumptions (Gupte et al., 1 Jun 2025).

In multivariate settings, stochastic approximation replaces deterministic quadrature or Monte Carlo root finding. For

$\ell^*(1)=0$2

the optimal allocation $\ell^*(1)=0$3 is characterized by

$\ell^*(1)=0$4

and projected Robbins–Monro updates converge almost surely. Polyak–Ruppert averaging yields asymptotic normality and confidence intervals for allocations and risk values (Kaakai et al., 2022).

6. Generalizations: robust, multivariate, and preference-robust variants

Several extensions preserve the OCE architecture while altering either the uncertainty model or the object being optimized. A prominent example is distributional robustness under optimal-transport ambiguity. If model uncertainty is penalized by a transport cost $\ell^*(1)=0$5 and a convex penalty $\ell^*(1)=0$6, the robust OCE reduces to a finite-dimensional problem: $\ell^*(1)=0$7 where $\ell^*(1)=0$8 is the $\ell^*(1)=0$9-transform of the loss. This reduction yields explicit formulas in important cases, including robust AVaR under linear or quadratic transport costs, and extends to shortfall risk and option pricing (Bartl et al., 2017).

The multivariate extension replaces a scalar cash shift by a vector allocation $\ell:\mathbb{R}\to\mathbb{R}$00, allowing systemic coupling through a convex multivariate loss $\ell:\mathbb{R}\to\mathbb{R}$01. Besides the primal formula already noted, the dual representation becomes

$\ell:\mathbb{R}\to\mathbb{R}$02

and one obtains risk-allocation and marginal-contribution formulae that generalize the univariate case (Kaakai et al., 2022).

A further modification is the modified OCE (MOCE), which replaces the monetary term $\ell:\mathbb{R}\to\mathbb{R}$03 in classical OCE by the utility of present cash, producing

$\ell:\mathbb{R}\to\mathbb{R}$04

This keeps present and future consumption in the same utility units and is invariant, at the level of optimal $\ell:\mathbb{R}\to\mathbb{R}$05, under positive linear rescaling of the utility representation. Preference ambiguity can then be modeled by minimizing over a Kantorovich ball of plausible utility functions, yielding a robust modified OCE solved by alternating linear programs in piecewise-linear utility classes (Wu et al., 2022).

These variants suggest that OCE is less a single risk measure than a design pattern: optimize over a certainty-equivalent shift, then choose how uncertainty enters—through laws, transport neighborhoods, utility ambiguity, or systemic coupling. That interpretive synthesis is consistent with the breadth of the recent literature, although the exact axiomatic behavior depends on the chosen extension (Bartl et al., 2017, Wu et al., 2022, Kaakai et al., 2022).

7. Statistical learning, decision making, and reinforcement learning

In supervised learning, empirical OCE minimization generalizes ERM by replacing expected loss with a risk-sensitive objective. For bounded losses, generalization bounds for the empirical OCE minimizer are controlled by the Rademacher average of the hypothesis class and the Lipschitz constant of the disutility; a complementary variance-based characterization bounds the gap between OCE and mean loss by the loss standard deviation, which suppresses the explicit smoothness dependence in expected-loss guarantees (Lee et al., 2020).

Conformal prediction has recently imported OCE into finite-sample risk control. Conformal risk training extends conformal risk control from expected losses to general OCEs under exchangeability and bounded monotone losses, with guarantees of the form

$\ell:\mathbb{R}\to\mathbb{R}$06

for parameters selected by a monotone calibration rule. The end-to-end training variant differentiates through conformal OCE control and reported reductions of false positive rate by $\ell:\mathbb{R}\to\mathbb{R}$07–$\ell:\mathbb{R}\to\mathbb{R}$08 in FNR-controlled segmentation, as well as profit improvements of $\ell:\mathbb{R}\to\mathbb{R}$09–$\ell:\mathbb{R}\to\mathbb{R}$10 in CVaR-controlled battery operation (Yeh et al., 9 Oct 2025). A related development, OCE-RCPS, replaces average-case RCPS guarantees by high-probability guarantees for OCE risk via WSR-style UCBs; in the reported segmentation experiments, OCE-RCPS met target satisfaction rates where OCE-CRC did not, including average satisfaction about $\ell:\mathbb{R}\to\mathbb{R}$11 versus $\ell:\mathbb{R}\to\mathbb{R}$12 for CVaR and about $\ell:\mathbb{R}\to\mathbb{R}$13 versus $\ell:\mathbb{R}\to\mathbb{R}$14 for entropic risk in the stated settings (Huang et al., 14 Feb 2026).

In individualized decision-making, OCE has been extended to a covariate-dependent equivalent (CDE),

$\ell:\mathbb{R}\to\mathbb{R}$15

which supports risk-sensitive individualized decision rules in precision medicine. For decomposable $\ell:\mathbb{R}\to\mathbb{R}$16, the optimal rule maximizes the conditional OCE action by action; in the reported heavy-tail simulations, the method improved lower-tail performance over several competing IDR estimators (1908.10742).

Reinforcement learning has developed two distinct OCE paradigms. Recursive OCE imposes a Bellman-style nonlinear certainty-equivalent at each transition, leading to tabular regret bounds, Bellman contractions, and sample-complexity analyses in discounted and episodic MDPs (Xu et al., 2023, Mortensen et al., 20 May 2026). A key impossibility result states that worst-case PAC learnability for discounted recursive OCE holds exactly for full-domain utilities $\ell:\mathbb{R}\to\mathbb{R}$17 with $\ell:\mathbb{R}\to\mathbb{R}$18; when $\ell:\mathbb{R}\to\mathbb{R}$19, the problem is not PAC-learnable in the worst case, and for $\ell:\mathbb{R}\to\mathbb{R}$20 the correct dependence on $\ell:\mathbb{R}\to\mathbb{R}$21 in the lower bound is $\ell:\mathbb{R}\to\mathbb{R}$22 (Mortensen et al., 20 May 2026). Static trajectory-level OCE, by contrast, may require genuinely history-dependent optimal policies. A reductions approach handles this by passing to an augmented MDP with a budget state $\ell:\mathbb{R}\to\mathbb{R}$23, obtaining

$\ell:\mathbb{R}\to\mathbb{R}$24

and thereby leveraging risk-neutral optimism or policy-gradient oracles in the augmented model (Wang et al., 2024). Recent extensions include OCE-based constrained RL with strong partial duality and PPO-style wrappers (Lee et al., 23 Oct 2025), and continuous-time risk-sensitive Q-learning in augmented SDE environments, where OCE restores Markovian optimality after state augmentation and leads to the CT-RS-q algorithm (Xie, 2 Dec 2025).

Across these domains, a recurring theme is that OCE preserves enough convex-analytic structure to permit estimation, optimization, and control, yet is flexible enough to encode materially different attitudes toward dispersion, tails, and rare catastrophic events. The main source of technical difficulty is not static tractability but dynamic inconsistency; much of the modern theory can be read as a sequence of increasingly sophisticated ways to recover dynamic structure—via dual densities, augmented states, FBSDEs, or conformal calibration—without abandoning the original certainty-equivalent variational form (Veraguas et al., 2020, Wang et al., 2024).