arXiv subjects

Duy-Minh Dang

Publications and source records attributed to Duy-Minh Dang.

10 recordsLinked to original sources

arXiv2026

Money-Back Tontines for Retirement Decumulation: Neural-Network Optimization under Systematic Longevity Risk

Money-back guarantees (MBGs) address bequest concerns in pooled retirement income products by returning the initial purchase price through withdrawals or, after early death, through a benefit to the member's beneficiaries or estate. We study the distinct actuarial problem created by adding an MBG to an individual-account tontine with dynamic withdrawals, investment in domestic and foreign assets, and systematic longevity risk. The retiree trades expected withdrawals (EW) against the lower-tail Conditional Value-at-Risk (CVaR) of terminal wealth under a fixed-horizon plan-to-live convention. Neural networks parameterize admissible withdrawal and rebalancing controls; the MBG is then valued ex post under the learned policy through an equivalent up-front load combining the expected payout with an upper-tail CVaR prudential buffer. We also approximate the effect of contract pooling on per-contract tail risk and pricing. Using long-horizon market and mortality data calibrated for an Australian retiree, we find that expected MBG payouts are modest, while the representative-contract payout has a severe upper tail. Contract pooling substantially reduces the upper-tail exposure on a per-contract basis and lowers the approximate pooled-contract load. International diversification materially improves the EW--CVaR retirement-income trade-off, while affecting the MBG payout distribution and equivalent load only modestly. Stochastic mortality likewise has a modest effect on the efficient frontier and MBG pricing.

q-fin.PM↗

arXiv2026

A data-driven Fourier-mixture neural-network method for density estimation

We propose a data-driven Fourier-trained neural-network method for estimating fixed-horizon probability densities from empirical characteristic-function (CF) information. The estimator is a positive Gaussian--Laplace mixture with closed-form CF, so training can be performed directly in Fourier space while preserving nonnegativity and unit mass. We consider two sampling settings. In the direct i.i.d. sampling setting, the method is trained against an empirical CF constructed from i.i.d. samples. In the resampling-based pseudo-sampling setting, it is trained against an empirical pseudo-CF constructed from dependent data by resampling. For the direct i.i.d. case, we derive an expected squared $L_2$ density-error bound that separates Fourier truncation, empirical training error, discretization, and CF sampling error, together with a corresponding bound for the expected $L_2$ norm. For the pseudo-sampling case, we obtain conditional counterparts with an additional pseudo-law discrepancy term. We develop a multidimensional extension of the framework and analyze its computational complexity. Numerical experiments show competitive performance relative to Expectation--Maximization on Gaussian-mixture benchmarks and clear gains on heavy-tailed targets. A controlled comparison with exact-CF training quantifies the additional density error introduced by empirical-CF training; further experiments show that the $L_2$ norm decays approximately as the inverse square root of the sample size when the remaining error terms are controlled, and demonstrate the method on a one-year Australian equity return law estimated from resampled dependent data.

stat.ML↗

arXiv2026

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

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.

math.OC↗

arXiv2026

Convergence of Neural Network Policies for Risk--Reward Optimization

We develop a neural-network framework for multi-period risk--reward stochastic control problems with constrained two-step feedback policies that may be discontinuous in the state. We allow a broad class of objectives built on a finite-dimensional performance vector, including terminal and path-dependent statistics, with risk functionals admitting auxiliary-variable optimization representations (e.g.\ Conditional Value-at-Risk and buffered probability of exceedance) and optional moment dependence. Our approach parametrizes the two-step policy using two coupled feedforward networks with constraint-enforcing output layers, reducing the constrained control problem to unconstrained training over network parameters. Under mild regularity conditions, we prove that the empirical optimum of the NN-parametrized objective converges in probability to the true optimal value as network capacity and training sample size increase. The proof is modular, separating policy approximation, propagation through the controlled recursion, and preservation under the scalarized risk--reward objective. Numerical experiments confirm the predicted convergence-in-probability behavior, show close agreement between learned and reference control heat maps, and demonstrate out-of-sample robustness on a large independent scenario set.

q-fin.CP↗

arXiv2026

Multi-period Mean-Buffered Probability of Exceedance in Defined Contribution Portfolio Optimization

We investigate multi-period mean-risk portfolio optimization for long-horizon Defined Contribution plans, focusing on buffered Probability of Exceedance (bPoE), a more intuitive, dollar-based alternative to Conditional Value-at-Risk (CVaR). We formulate both pre-commitment and time-consistent Mean-bPoE and Mean-CVaR portfolio optimization problems under realistic investment constraints (e.g., no leverage, no short selling) and jump-diffusion dynamics. These formulations are naturally framed as bilevel optimization problems, with an outer search over the shortfall threshold and an inner optimization over rebalancing decisions. We establish an equivalence between the pre-commitment formulations through a one-to-one correspondence of their scalarization optimal sets, while showing that no such equivalence holds in the time-consistent setting. We develop provably convergent numerical schemes for the value functions associated with both pre-commitment and time-consistent formulations of these mean-risk control problems. Using nearly a century of market data, we find that time-consistent Mean-bPoE strategies closely resemble their pre-commitment counterparts. In particular, they maintain alignment with investors' preferences for a minimum acceptable terminal wealth level-unlike time-consistent Mean-CVaR, which often leads to counterintuitive control behavior. We further show that bPoE, as a strictly tail-oriented measure, prioritizes guarding against catastrophic shortfalls while allowing meaningful upside exposure, making it especially appealing for long-horizon wealth security. These findings highlight bPoE's practical advantages for Defined Contribution investment planning.

q-fin.PM↗

arXiv2025

A monotone piecewise constant control integration approach for the two-factor uncertain volatility model

Option contracts on two underlying assets within uncertain volatility models have their worst-case and best-case prices determined by a two-dimensional (2D) Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) with cross-derivative terms. This paper introduces a novel ``decompose and integrate, then optimize'' approach to tackle this HJB PDE. Within each timestep, our method applies piecewise constant control, yielding a set of independent linear 2D PDEs, each corresponding to a discretized control value. Leveraging closed-form Green's functions, these PDEs are efficiently solved via 2D convolution integrals using a monotone numerical integration method. The value function and optimal control are then obtained by synthesizing the solutions of the individual PDEs. For enhanced efficiency, we implement the integration via Fast Fourier Transforms, exploiting the Toeplitz matrix structure. The proposed method is $\ell_{\infty}$-stable, consistent in the viscosity sense, and converges to the viscosity solution of the HJB equation. Numerical results show excellent agreement with benchmark solutions obtained by finite differences, tree methods, and Monte Carlo simulation, highlighting its robustness and effectiveness.

q-fin.CP↗

arXiv2025

Numerical analysis of American option pricing in a two-asset jump-diffusion model

This paper addresses an important gap in rigorous numerical treatments for pricing American options under correlated two-asset jump-diffusion models using the viscosity solution framework, with a particular focus on the Merton model. The pricing of these options is governed by complex two-dimensional (2-D) variational inequalities that incorporate cross-derivative terms and nonlocal integro-differential terms due to the presence of jumps. Existing numerical methods, primarily based on finite differences, often struggle with preserving monotonicity in the approximation of cross-derivatives, a key requirement for ensuring convergence to the viscosity solution. In addition, these methods face challenges in accurately discretizing 2-D jump integrals. We introduce a novel approach to effectively tackle the aforementioned variational inequalities while seamlessly handling cross-derivative terms and nonlocal integro-differential terms through an efficient and straightforward-to-implement monotone integration scheme. Within each timestep, our approach explicitly enforces the inequality constraint, resulting in a 2-D Partial Integro-Differential Equation (PIDE) to solve. Its solution is expressed as a 2-D convolution integral involving the Green's function of the PIDE. We derive an infinite series representation of this Green's function, where each term is non-negative and computable. This facilitates the numerical approximation of the PIDE solution through a monotone integration method. To enhance efficiency, we develop an implementation of this monotone scheme via FFTs, exploiting the Toeplitz matrix structure. The proposed method is proved to be both $\ell_{\infty} $-stable and consistent in the viscosity sense, ensuring its convergence to the viscosity solution of the variational inequality. Extensive numerical results validate the effectiveness and robustness of our approach.

q-fin.CP↗

arXiv2024

Fourier Neural Network Approximation of Transition Densities in Finance

This paper introduces FourNet, a novel single-layer feed-forward neural network (FFNN) method designed to approximate transition densities for which closed-form expressions of their Fourier transforms, i.e. characteristic functions, are available. A unique feature of FourNet lies in its use of a Gaussian activation function, enabling exact Fourier and inverse Fourier transformations and drawing analogies with the Gaussian mixture model. We mathematically establish FourNet's capacity to approximate transition densities in the $L_2$-sense arbitrarily well with finite number of neurons. The parameters of FourNet are learned by minimizing a loss function derived from the known characteristic function and the Fourier transform of the FFNN, complemented by a strategic sampling approach to enhance training. We derive practical bounds for the $L_2$ estimation error and the potential pointwise loss of nonnegativity in FourNet for $d$-dimensions ($d\ge 1$), highlighting its robustness and applicability in practical settings. FourNet's accuracy and versatility are demonstrated through a wide range of dynamics common in quantitative finance, including Lévy processes and the Heston stochastic volatility models-including those augmented with the self-exciting Queue-Hawkes jump process.

q-fin.CP↗

arXiv2023

A semi-Lagrangian $ε$-monotone Fourier method for continuous withdrawal GMWBs under jump-diffusion with stochastic interest rate

We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump-diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no-arbitrage GMWB pricing problem as a time-dependent Hamilton-Jacobi-Bellman (HJB) Quasi-Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi-Lagrangian method and the Green's function of an associated linear partial integro-differential equation, we develop an $ε$-monotone Fourier pricing method, where $ε> 0$ is a monotonicity tolerance. Together with a provable strong comparison result for the HJB-QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB-QVI as $ε\to 0$. We present a comprehensive study of the impact of simultaneously considering jumps in the sub-account process and stochastic interest rate on the no-arbitrage prices and fair insurance fees of GMWBs, as well as on the holder's optimal withdrawal behaviors.

q-fin.CP↗

arXiv2023

A monotone numerical integration method for mean-variance portfolio optimization under jump-diffusion models

We develop a efficient, easy-to-implement, and strictly monotone numerical integration method for Mean-Variance (MV) portfolio optimization in realistic contexts, which involve jump-diffusion dynamics of the underlying controlled processes, discrete rebalancing, and the application of investment constraints, namely no-bankruptcy and leverage. A crucial element of the MV portfolio optimization formulation over each rebalancing interval is a convolution integral, which involves a conditional density of the logarithm of the amount invested in the risky asset. Using a known closed-form expression for the Fourier transform of this density, we derive an infinite series representation for the conditional density where each term is strictly positive and explicitly computable. As a result, the convolution integral can be readily approximated through a monotone integration scheme, such as a composite quadrature rule typically available in most programming languages. The computational complexity of our method is an order of magnitude lower than that of existing monotone finite difference methods. To further enhance efficiency, we propose an implementation of the scheme via Fast Fourier Transforms, exploiting the Toeplitz matrix structure. The proposed monotone scheme is proven to be both $\ell_{\infty}$-stable and pointwise consistent, and we rigorously establish its pointwise convergence to the unique solution of the MV portfolio optimization problem. We also intuitively demonstrate that, as the rebalancing time interval approaches zero, the proposed scheme converges to a continuously observed impulse control formulation for MV optimization expressed as a Hamilton-Jacobi-Bellman Quasi-Variational Inequality. Numerical results show remarkable agreement with benchmark solutions obtained through finite differences and Monte Carlo simulation, underscoring the effectiveness of our approach.

q-fin.CP↗