Paper: arXiv 2603.26291

Authors: Duy-Minh Dang, Hao Zhou

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical foundation establishing monotonicity, stability, and convergence proofs for a 2D mean-CVaR optimization framework using Fourier-trained kernels. However, it relies heavily on synthetic numerical experiments for jump-diffusion models rather than extensive real-world backtesting against empirical market data.

Research Flowchart

  flowchart TD
    A["Research Goal: Monotone & convergent 2D integration for multi-period Mean-CVaR control via closed-form CFs"] --> B["Inputs: Process Characteristic Function (CF) & composite quadrature grids"]
    B --> C["Kernel Learning: Parameterize nonnegative 2D transition kernel via Fourier-space Gaussian mixtures"]
    C --> D["Discretization & Computation: Apply composite quadrature & 2D FFT for efficient convolution"]
    D --> E["Theoretical Bounds: Establish L2 Fourier error estimates, l-infinity stability, and consistency"]
    E --> F["Numerical Validation: Solve multi-period portfolio optimization in coupled 2D jump-diffusion model"]
    F --> G["Key Outcomes: Guaranteed monotonicity, provable convergence, and robust high-dimensional performance"]