This work exploits the auxiliary-threshold representation of CVaR to establish the existence of an optimal strategy and strong duality without requiring market completeness, and proves that the resulting strategies converge to the optimal control as the number of iterations tends to infinity.
Abstract
We study continuous-time dynamic portfolio optimization under a Conditional Value-at-Risk (CVaR) constraint on the investor's terminal loss. For a general class of convex trading objectives, we exploit the auxiliary-threshold representation of CVaR to establish the existence of an optimal strategy and strong duality without requiring market completeness. These results motivate a dual-based nested bisection--golden-search algorithm over the threshold and Lagrangian multiplier, where the inner iterations reduce to standard unconstrained stochastic control problems. We prove that the resulting strategies converge to the optimal control as the number of iterations tends to infinity. Numerical experiments recover the Merton policy when the risk constraint is nonbinding. When the constraint is binding, the optimal strategy becomes state dependent: the investor reduces risky exposure following adverse outcomes but preserves, and near maturity may increase, exposure following favorable outcomes. Thus, a terminal CVaR constraint produces an asymmetric reallocation across states rather than uniform de-risking. Nontraded endowment risk amplifies the conservative adjustment, whereas price impact lowers desired positions and adjustment speeds.
We study a multi-period portfolio optimization problem with finite-scenario Value-at-Risk (VaR) constraints, transaction costs, and diversification regularization. Using a finite-scenario VaR--CVaR identity, we derive a penalized difference-of-convex (DC) formulation over the underlying convex portfolio set. To solve the resulting nonsmooth and nonconvex problem, we propose a projected inertial Boosted Difference-of-Convex Functions Algorithm (iBDCA) that combines inertial extrapolation, an objective safeguard, and a boosted line search. We prove that the method is well defined, generates a monotonically decreasing objective sequence, and has only critical accumulation points for the penalized DC problem. Under a local no-ties condition at an accumulation point, we further establish whole sequence convergence with a local \(R\)-linear rate. Numerical experiments compare iBDCA with DCA and standard BDCA on matched problem instances. Out-of-sample backtests additionally include equal-weight and buy-and-hold benchmarks to illustrate the trade-offs among realized return, risk, transaction costs, and empirical VaR control.
Modern portfolio theory identifies diversification as the primary tool for risk reduction. However, under model uncertainty, this cornerstone may no longer remain optimal. This paper investigates the tension between portfolio diversification and concentration under dependence uncertainty. In the absence of model uncertainty, we employ the framework of the majorization order and doubly stochastic matrices to formalize the degree of diversification, and prove that quasi-convexity is a necessary and sufficient property for a risk functional to be weakly consistent with the majorization order. We further derive worst-case risk measure inequalities and solve robust portfolio selection problems for a broad class of risk measures, including VaR, ES, Range-VaR (RVaR), and standard deviation (SD). Our results reveal a''concentration paradox''for many widely-used risk functionals: when the dependence structure is fully ambiguous, robust optimization often recommends concentrating investment in a single asset to hedge against the worst-case dependence scenario. As an application, we propose a weighted robustness formulation that interpolates between a reference dependence structure and the worst-case structure. The formulation is structurally analogous to the constrained/unconstrained Expected Shortfall blend in the Fundamental Review of the Trading Book (FRTB) and provides a theoretical foundation for balancing diversification against robustness in the presence of model uncertainty.
In portfolio optimization, a cardinality constraint, which limits the number of assets held, plays a key role in cutting down monitoring and transaction costs. However, the resulting problem is NP-hard and becomes computationally difficult to solve globally as the number of candidate assets grows. Safe screening addresses this difficulty by fixing decision variables before optimization without excluding any globally optimal solution, thereby reducing the problem size while preserving optimality guarantees. We propose safe screening rules for cardinality-constrained portfolio optimization with a convex quadratic objective function and linear constraints. Using a perspective relaxation of the L2-regularization term and Fenchel duality, we derive asset-specific scores that incorporate the Lagrange multipliers of the linear constraints. Combined with a relaxation-based lower bound and a feasible-solution upper bound, these scores safely fix binary asset-selection variables to zero or one. Experiments on S&P 500 and Russell 2000 datasets show substantial computational improvements on challenging cases, particularly under moderate or strong regularization and less stringent return requirements. These results demonstrate the effectiveness of safe screening as an optimality-preserving preprocessing technique that greatly boosts computational efficiency in large-scale cardinality-constrained portfolio optimization.
Nanari Wada, Shunnosuke Ikeda, Yuichi Takano et al.· 0 citations
In mean-variance portfolio analysis, the efficient frontier represents the optimal trade-off between expected return and risk, assuming stable underlying parameters. This paper investigates portfolio fragility: the instability of optimal weights, risk-adjusted performance, and diversification when model inputs and construction choices are jointly perturbed. Combining constrained Markowitz optimization with variance-based global sensitivity analysis (Sobol indices), we map out how input uncertainty and portfolio-construction choices propagate along the target-return dimension. Using an empirical universe of multi-asset exchange-traded funds (ETFs), we find a distinct transition in the sensitivity structure: in the baseline experiment, lower target returns are dominated by l2 regularization, whereas aggressive return requirements become increasingly sen- sitive to the weight cap and expected-return perturbations. This shift coincides with a sharp drop in effective diversification and a rise in weight dispersion. We extend the analysis to a multi-universe fragility atlas, showing that under a com- mon absolute concentration rule, the smallest universe is weight-cap-driven in the aggressive return region, while larger sampled universes remain more often regularization-driven. The fragile frontier serves as a direct diagnostic tool to evaluate the structural robustness of constrained optimizers without altering the underlying allocation rule.
Stefano Pellegrino, Giulia Vannucci, R. Siciliano· 0 citations
This paper investigates optimal portfolio choice for a risk-averse investor who is operating in a
financial market characterized by continuous time usage and with explicit attention being paid to
the investor's sensitivity to market movements. The investor's preferences are described by a
power utility function of constant relative risk aversion, which promotes economically interesting
behavior over wealth levels. The market consists of a risk-free asset and many risky assets,
evolving under stochastic differential equations. By allowing the investor to adjust portfolio
positions according to the changes in the processes of risky assets, the model can extend
traditional portfolio optimization frameworks. Aiming to self-assemble against dynamic
programming and the Hamilton-Jacobi-Bellman equation, exploiting Itô's calculus, we got
analytical representations of the optimal portfolio strategy and its expected terminal utility. The
quantification of the explicit sensitivity effect parameter allowed the activation of market
responsiveness to additional advantage. Numerical simulations, with Python, demonstrate the
role of market signals on a market-responsive portfolio. Two- and three-dimensional numerical
figure analyses also depict the behavior, in terms of the optimal investment decisions, of risk
aversion, asset volatility, and sensitivity parameters.
C. Achudume· International Journal of Com...· 0 citations
We develop simulation-based policy iteration for continuous-time portfolio choice with predictable returns and convex constraints. Each outer step re-evaluates a fixed-latent open-loop backpropagation-through-time (OL-BPTT) adjoint after deployment and solves the constrained update. Shifted-adjoint cancellation controls the adjoint--HJB Hamiltonian-gradient discrepancy by the policy-improvement residual. For CRRA portfolios, exact HJB policy iteration identifies the optimal reduced value factor, while population OL-BPTT iteration converges globally when the adjoint update is directionally improving and approximate stationarity is asymptotically HJB-compatible. A theorem-matched occupation audit yields maximal $95\%$ upper endpoints of $0.066$ for the primitive directional ratio and $0.074$ for a stronger norm-relative ratio, both against the half-step threshold $0.75$. In the high-precision $50$--$50$ occupancy/broad-anchor design of a three-factor, fifty-asset benchmark, current-policy re-evaluation outperforms matched pooled refinement under the on-policy and broad evaluation laws.
Jeonggyu Huh, Yeoneung Kim, Seungwon Jeong· 0 citations
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