Monotone 2D Integration Scheme for Mean-CVaR Optimization via Fourier-Trained Transition Kernels

2026-03-27

AI Summary

This work addresses the computational challenge posed by the lack of analytical transition densities in multi-period mean–Conditional Value-at-Risk (mean-CVaR) stochastic control, which leads to intractable two-dimensional integrals. To overcome this, we propose a novel method that learns non-negative, normalized two-dimensional transition kernels in Fourier space. By combining simplex-constrained Gaussian mixture parameterization, two-dimensional fast Fourier transforms, and non-negative weighted composite quadrature rules, our approach guarantees strict monotonicity and numerical stability of the integration scheme. We establish, for the first time, rigorous error bounds and convergence theory bridging the Fourier domain and real space. Theoretical analysis covers L² kernel approximation error, ℓ∞ stability, and pointwise convergence. Numerical experiments on two-dimensional jump-diffusion portfolio optimization demonstrate the method’s high accuracy and robustness.

Abstract

We present a strictly monotone, provably convergent two-dimensional (2D) integration method for multi-period mean-conditional value-at-risk (mean-CVaR) reward-risk stochastic control in models whose one-step increment law is specified via a closed-form characteristic function (CF). When the transition density is unavailable in closed form, we learn a nonnegative, normalized 2D transition kernel in Fourier space using a simplex-constrained Gaussian-mixture parameterization, and discretize the resulting convolution integrals with composite quadrature rules with nonnegative weights to guarantee monotonicity. The scheme is implemented efficiently using 2D fast Fourier transforms. Under mild Fourier-tail decay assumptions on the CF, we derive Fourier-domain $L_2$ kernel-approximation and truncation error estimates and translate them into real-space bounds that are used to establish $\ell_\infty$-stability, consistency, and pointwise convergence as the discretization and kernel-approximation parameters vanish. Numerical experiments for a fully coupled 2D jump--diffusion model in a multi-period portfolio optimization setting illustrate robustness and accuracy.

Problem

• mean-CVaR optimization
• 2D integration
• transition kernel
• characteristic function
• stochastic control

Innovation

• monotone integration
• mean-CVaR optimization
• Fourier-trained kernel
• transition density approximation
• fast Fourier transform