Explicit error bounds for multivariate nonlinear filtering in stochastic volatility models with spread variance estimation

Jin Zou; Jin Zou

doi:10.3934/math.20261168

AIMS Mathematics

2026, Volume 11, Issue 9: 29399-29429. doi: 10.3934/math.20261168

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Explicit error bounds for multivariate nonlinear filtering in stochastic volatility models with spread variance estimation

Jin Zou ^,

School of Artificial Intelligence, Lishui University, Lishui, Zhejiang, China

Received: 01 April 2026 Revised: 01 September 2026 Accepted: 03 September 2026 Published: 11 September 2026
MSC : 93E11, 91G70, 60G35

We considered real-time state estimation for multiple assets in a financial market, where the hidden state consists of logarithmic prices and stochastic volatilities, and spreads were derived as linear combinations of estimated prices. This led to a continuous-time nonlinear filtering problem with state-dependent diffusion. We adopted the improved Yau–Yau algorithm of Yau and Niu, which combines low-discrepancy sampling, high-order kernel approximations, and a local resampling-restart mechanism to mitigate dimension-dependent degradation under the tested low-effective-dimensional configurations. Our main theoretical contribution was an explicit upper bound for the Hardy–Krause variation of the kernel and likelihood function in terms of model parameters; as a model-specific corollary, we derived an explicit error bound for posterior spread-variance estimators. We also provided empirical guidance for selecting the resampling interval. Numerical experiments exhibited a near-$ N^{-1} $ randomized quasi-Monte Carlo trend for the tested integrand, supported the qualitative polynomial dependence of the error bound, and demonstrated the importance of correctly specifying the correlation matrix for spread variance estimation. An illustrative case study on crude oil futures demonstrated the practical feasibility.
- nonlinear filtering,
- Yau–Yau algorithm,
- stochastic volatility,
- quasi-Monte Carlo,
- pairs trading
Citation: Jin Zou. Explicit error bounds for multivariate nonlinear filtering in stochastic volatility models with spread variance estimation[J]. AIMS Mathematics, 2026, 11(9): 29399-29429. doi: 10.3934/math.20261168

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Abstract

We considered real-time state estimation for multiple assets in a financial market, where the hidden state consists of logarithmic prices and stochastic volatilities, and spreads were derived as linear combinations of estimated prices. This led to a continuous-time nonlinear filtering problem with state-dependent diffusion. We adopted the improved Yau–Yau algorithm of Yau and Niu, which combines low-discrepancy sampling, high-order kernel approximations, and a local resampling-restart mechanism to mitigate dimension-dependent degradation under the tested low-effective-dimensional configurations. Our main theoretical contribution was an explicit upper bound for the Hardy–Krause variation of the kernel and likelihood function in terms of model parameters; as a model-specific corollary, we derived an explicit error bound for posterior spread-variance estimators. We also provided empirical guidance for selecting the resampling interval. Numerical experiments exhibited a near-$ N^{-1} $ randomized quasi-Monte Carlo trend for the tested integrand, supported the qualitative polynomial dependence of the error bound, and demonstrated the importance of correctly specifying the correlation matrix for spread variance estimation. An illustrative case study on crude oil futures demonstrated the practical feasibility.

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