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

TL;DR

This paper introduces a new 2D integration method for mean-CVaR stochastic control that guarantees convergence and monotonicity, even when the transition law is learned via Fourier methods.

Contribution

It develops a provably convergent, monotone 2D integration scheme for mean-CVaR optimization, including a Fourier-based kernel learning approach when the transition density is unavailable.

Findings

The scheme guarantees monotonicity and convergence under mild assumptions.

Numerical experiments demonstrate robustness and accuracy in portfolio optimization.

Fourier-based kernel approximation achieves controlled error bounds.

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.