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

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.

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

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 kernel-approximation and truncation error estimates and translate them into real-space bounds that are used to establish -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.

Paper Structure

This paper contains 53 sections, 14 theorems, 115 equations, 4 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Modelling
  3. Intervention times.
  4. State variables and log representation.
  5. Increment law and Fourier transform.
  6. Mean--CVaR optimization
  7. Intervention map
  8. Mean--CVaR formulation
  9. Lifted-state recursion (fixed $w$).
  10. Localization and problem statement
  11. Fourier-trained transition kernel approximation
  12. Description
  13. Bounded parameter space and positivity.
  14. Closed--form CF.
  15. $L_2$-approximation error and Fourier-domain invariance.
  16. ...and 38 more sections

Key Result

Lemma 3.2

Fix $\gamma>0$. Then the outer optimization in eq:PCEC_interchange is finite and is attained by some $w^*(x_0)\in[0,\infty)$.

Figures (4)

  • Figure 3.1: Spatial sub-domains at each $t$ (fixed $w\in\Gamma$).
  • Figure 8.1: 2D Kou jump--diffusion test case. Panels (a)--(b) show representative 1D slices of the target CF $G$ and the fitted CF $\widehat{G}$, and panel (c) shows a representative slice of the trained density $\widehat{g}$. To improve conditioning for oscillatory $G$, the CF targets in the loss function are affinely rescaled as in Remark 5.1 of du2025fourier; the plotted slices are shown in the original (unrescaled) coordinates.
  • Figure 8.2: Pre-commitment mean--CVaR optimal control heat map (allocation weight in component $s$; interpreted as equity in the DC illustration).
  • Figure 8.3: Efficient frontier of mean--CVaR with $\alpha=0.05$, computed on the finest refinement level (DC illustration).

Theorems & Definitions (25)

  • Remark 2.1: Examples and scope
  • Remark 3.1: Finiteness of $\mathbb{E}_{\mathcal{U}_0}^{x_0,t_0^-}\lbrack W_T\rbrack$
  • Lemma 3.2: Existence of a finite optimal threshold
  • Proposition 3.3: Equivalence to the pre-commitment problem
  • Lemma 3.4: One–step propagation for exponential functions
  • proof
  • Definition 3.1: Localized Mean--CVaR formulation
  • Remark 3.5: Uniqueness and regularity of the localized inner problem
  • Theorem 4.1: $L_2$-approximation and Fourier-domain invariance; cf. du2025fourier, Thm. 3.4
  • proof
  • ...and 15 more