Convergence Rates

Convergence Rates of Turnpike Theorems for Portfolio Choice in Stochastic Factor Models ArXiv ID: 2512.00346 “View on arXiv” Authors: Hiroki Yamamichi Abstract Turnpike theorems state that if an investor’s utility is asymptotically equivalent to a power utility, then the optimal investment strategy converges to the CRRA strategy as the investment horizon tends to infinity. This paper aims to derive the convergence rates of the turnpike theorem for optimal feedback functions in stochastic factor models. In these models, optimal feedback functions can be decomposed into two terms: myopic portfolios and excess hedging demands. We obtain convergence rates for myopic portfolios in nonlinear stochastic factor models and for excess hedging demands in quadratic term structure models, where the interest rate is a quadratic function of a multivariate Ornstein-Uhlenbeck process. We show that the convergence rates are determined by (i) the decay speed of the price of a zero-coupon bond and (ii) how quickly the investor’s utility becomes power-like at high levels of wealth. As an application, we consider optimal collective investment problems and show that sharing rules for terminal wealth affect convergence rates. ...

An Efficient Multi-scale Leverage Effect Estimator under Dependent Microstructure Noise ArXiv ID: 2505.08654 “View on arXiv” Authors: Ziyang Xiong, Zhao Chen, Christina Dan Wang Abstract Estimating the leverage effect from high-frequency data is vital but challenged by complex, dependent microstructure noise, often exhibiting non-Gaussian higher-order moments. This paper introduces a novel multi-scale framework for efficient and robust leverage effect estimation under such flexible noise structures. We develop two new estimators, the Subsampling-and-Averaging Leverage Effect (SALE) and the Multi-Scale Leverage Effect (MSLE), which adapt subsampling and multi-scale approaches holistically using a unique shifted window technique. This design simplifies the multi-scale estimation procedure and enhances noise robustness without requiring the pre-averaging approach. We establish central limit theorems and stable convergence, with MSLE achieving convergence rates of an optimal $n^{"-1/4"}$ and a near-optimal $n^{"-1/9"}$ for the noise-free and noisy settings, respectively. A cornerstone of our framework’s efficiency is a specifically designed MSLE weighting strategy that leverages covariance structures across scales. This significantly reduces asymptotic variance and, critically, yields substantially smaller finite-sample errors than existing methods under both noise-free and realistic noisy settings. Extensive simulations and empirical analyses confirm the superior efficiency, robustness, and practical advantages of our approach. ...

Error Analysis of Deep PDE Solvers for Option Pricing ArXiv ID: 2505.05121 “View on arXiv” Authors: Jasper Rou Abstract Option pricing often requires solving partial differential equations (PDEs). Although deep learning-based PDE solvers have recently emerged as quick solutions to this problem, their empirical and quantitative accuracy remain not well understood, hindering their real-world applicability. In this research, our aim is to offer actionable insights into the utility of deep PDE solvers for practical option pricing implementation. Through comparative experiments in both the Black–Scholes and the Heston model, we assess the empirical performance of two neural network algorithms to solve PDEs: the Deep Galerkin Method and the Time Deep Gradient Flow method (TDGF). We determine their empirical convergence rates and training time as functions of (i) the number of sampling stages, (ii) the number of samples, (iii) the number of layers, and (iv) the number of nodes per layer. For the TDGF, we also consider the order of the discretization scheme and the number of time steps. ...