arXiv subjects

Masahiro Kurisaki

Publications and source records attributed to Masahiro Kurisaki.

7 recordsLinked to original sources

arXiv2026

A Second-Order Asymptotic Expansion for Nonlinear Filtering in the Lorenz--63 System

A challenge in nonlinear filtering is to incorporate nonlinear effects into approximations of the conditional mean while retaining a finite-dimensional recursive algorithm. We derive a second-order asymptotic expansion of the filtering mean with respect to the system-noise intensity. We apply this expansion to the Lorenz--63 model, a prototypical three-dimensional chaotic dynamical system. The first-order approximation is computed using a Kalman--Bucy filter for the system linearized around the deterministic reference trajectory. The second-order correction incorporates the Hessian of the Lorenz--63 drift and is computed through a closed recursive system with 3+9+18=30 independent scalar components. The full system, including the reference trajectory and the first-order mean and covariance, has 42 independent scalar components. To support stable tracking of chaotic trajectories over long time intervals, we introduce recentering of the expansion and redistribution of Gaussian components between expansion orders. In the numerical experiments, we compare the proposed algorithm with an ensemble transform Kalman filter (ETKF) as a practical benchmark and a particle filter (PF) as a numerical reference. Under the full-observation settings, the proposed algorithm achieves a lower root-mean-square error relative to the truth and a smaller distance to the 10,000-particle PF reference than the 20-member ETKF. These results suggest that the proposed second-order extension of Kalman filtering has the potential to balance filtering accuracy and computational cost.

nlin.CD↗

arXiv2026

Finite-Dimensional Recursions for Small-Noise Expansions in Nonlinear Filtering

This paper provides a recursive formula for computing the coefficients in a small-system-noise asymptotic expansion for nonlinear filtering. The expansion, obtained from the Kallianpur--Striebel formula, was justified in the author's previous work. Our main contribution is to reduce the coefficient calculation to a finite-dimensional system extending the Kalman--Bucy filter by applying Fubini's theorem and Wick's formula and differentiating the resulting terms. For each fixed expansion order, the number of variables grows at most polynomially, rather than exponentially, with the system dimension. To justify the construction, we define the required non-adapted integrals as limits of discrete sums and establish a generalized Ito formula. We also extend the expansion from conditional expectations to conditional characteristic functions and provide a numerical illustration of the method.

math.PR↗

arXiv2026

Asymptotic Expansion of the Kallianpur-Striebel Formula under Perturbations of Linear Models

In this paper, we study a nonlinear state-space model represented as a perturbation of a linear model and investigate an asymptotic expansion of the nonlinear filter with respect to the perturbation parameter. The conditional expectation of the hidden state given the observations admits a closed-form representation via the Kallianpur-Striebel formula. We provide a rigorous justification of the expansion of this representation in probability under very general assumptions. Our framework applies not only to nearly linear models but also to more general situations, including nonlinear systems with small system noise. In these cases, the coefficients of the resulting expansion can be computed through systems of ordinary differential equations. Although the explicit computational procedure will be presented in a subsequent paper, the present work establishes the theoretical foundation of our asymptotic expansion approach to nonlinear filtering.

math.PR↗

arXiv2026

Pathwise Representation of the Smoothing Distribution in Continuous-Time Linear Gaussian Models

We study the filtering and smoothing problem for continuous-time linear Gaussian systems. While classical approaches such as the Kalman-Bucy filter and the Rauch-Tung-Striebel (RTS) smoother provide recursive formulas for the conditional mean and covariance, we present a pathwise perspective that characterizes the smoothing error dynamics as an Ornstein-Uhlenbeck process. As an application, we show that standard filtering and smoothing equations can be uniformly derived as corollaries of our main theorem. In particular, we provide the first mathematically rigorous derivation of the Bryson-Frazier smoother in the continuous-time setting. Beyond offering a more transparent understanding of the smoothing distribution, our formulation enables pathwise sampling from it, which facilitates Monte Carlo methods for evaluating nonlinear functionals.

math.ST↗

arXiv2025

Asymptotic Expansion for Nonlinear Filtering in the Small System Noise Regime

We propose a new asymptotic expansion method for nonlinear filtering, based on a small parameter in the system noise. The conditional expectation is expanded as a power series in the noise level, with each coefficient computed by solving a system of ordinary differential equations. This approach mitigates the trade-off between computational efficiency and accuracy inherent in existing methods such as Gaussian approximations and particle filters. Moreover, by incorporating an Edgeworth-type expansion, our method captures complex features of the conditional distribution, such as multimodality, with significantly lower computational cost than conventional filtering algorithms.

eess.SP↗

arXiv2025

A New Proof for the Linear Filtering and Smoothing Equations, and Asymptotic Expansion of Nonlinear Filtering

In this paper, we propose a new asymptotic expansion approach for nonlinear filtering based on a small parameter in the system noise. This method expresses the filtering distribution as a power series in the noise level, where the coefficients can be computed by solving a system of ordinary differential equations. As a result, it addresses the trade-off between computational efficiency and accuracy inherent in existing methods such as Gaussian approximations and particle filters. In the course of our derivation, we also show that classical linear filtering and smoothing equations, namely Kalman-Bucy filter and Rauch-Tung-Striebel smoother, can be obtained in a unified and transparent manner from an explicit formula for the conditional distribution of the hidden path.

eess.SP↗

arXiv2022

Parameter estimation for ergodic linear SDEs from partial and discrete observations

We consider a problem of parameter estimation for the state space model described by linear stochastic differential equations. We assume that an unobservable Ornstein-Uhlenbeck process drives another observable process by the linear stochastic differential equation, and these two processes depend on some unknown parameters. We construct the quasi-likelihood estimator (QMLE) of the unknown parameters and show asymptotic properties of the estimator.

math.ST↗