[Paper Review] A new proof for the convergence of Picard's filter using partial Malliavin calculus

Hideyuki Tanaka|arXiv (Cornell University)|Nov 24, 2013

Stochastic processes and financial applications8 references3 citations

TL;DR

This paper presents a new proof for the convergence of Picard's filter in nonlinear filtering using partial Malliavin calculus, establishing an $ L^p $-convergence rate of $ O(1/n) $ for $ p > 2 $, even when the function $ g $ is irregular. The method avoids standard strong and weak convergence arguments by leveraging duality and conditional expectations in Hilbert space-valued functionals on Wiener space.

ABSTRACT

The discrete-time approximation for nonlinear filtering problems is related to both of strong and weak approximations of stochastic differential equations. In this paper, we propose a new method of proof for the convergence of approximate nonlinear filter analyzed by Jean Picard (1984), and show a more general result than the original one. For the proof, we develop an analysis of Hilbert space valued functionals on Wiener space.

Motivation & Objective

• To generalize Picard's original $ L^2 $-convergence result for nonlinear filtering to $ L^p $-norms with $ p > 2 $, improving the robustness of error estimates.
• To establish convergence under weaker regularity assumptions on the function $ g $, including cases where $ g $ is not smooth, by avoiding reliance on standard Euler scheme error analysis.
• To develop a novel analytical framework using partial Malliavin calculus to handle conditional expectations involving stochastic integrals under $ \mathcal{F}_T^Y $.
• To clarify the mechanism behind the $ O(1/n) $ convergence rate by analyzing the duality of stochastic integrals and Skorohod integrals in the conditional expectation setting.
• To lay the foundation for extending the method to Lévy-driven SDEs and non-Markovian or dependent signal-observation processes in future work.

Proposed method

• Utilizes partial Malliavin calculus to analyze Hilbert space-valued functionals on Wiener space, particularly focusing on conditional expectations given the observation filtration $ \mathcal{F}_T^Y $.
• Applies the Clark-Ocone formula and duality relations for Skorohod integrals to decompose and estimate the error in the approximation of the Radon-Nikodym derivative $ \Phi_T $.
• Employs the Kallianpur-Striebel formula to express the conditional expectation $ E[g(X_T)\Phi_T|\mathcal{F}_T^Y] $, enabling analysis of the approximation error in $ \tilde{\Phi}_T $.
• Derives a key identity for the error term $ E_1(\rho, Z) $ by expressing the difference between stochastic integrals as a sum of Skorohod integrals and iterated integrals involving conditional expectations.
• Uses Lemma 3.2 to control the $ L^p $-norm of the error via estimates on the Malliavin derivative and conditional expectations, leading to $ O(1/n^p) $ bounds.
• Applies Lemma 3.9 to bound the term involving the Malliavin derivative $ D_r^Y f_s $, enabling the final $ O(1/n) $ convergence rate in $ L^p $.

Experimental results

Research questions

• RQ1 Can the $ L^2 $-convergence result of Picard's filter be extended to $ L^p $-norms with $ p > 2 $, and what conditions are required?
• RQ2 What is the convergence rate of Picard's filter when the function $ g $ is not smooth, and how can this be proven without relying on strong or weak convergence of the Euler scheme?
• RQ3 How does partial Malliavin calculus enable a more refined analysis of conditional expectations involving stochastic integrals in nonlinear filtering?
• RQ4 What is the role of duality between stochastic integrals and Skorohod integrals in deriving the $ O(1/n) $ convergence rate?
• RQ5 Can the proof technique be extended to non-Brownian or Lévy-driven SDEs, and what new tools are needed for such generalizations?

Key findings

• The paper establishes a new $ L^p $-convergence rate of $ O(1/n) $ for Picard’s filter approximation, valid for $ p > 2 $, generalizing Picard’s original $ L^2 $-result.
• The convergence holds even when $ g $ is irregular (not necessarily smooth), provided $ h \in C_b^2 $, $ b $, $ \sigma $ are Lipschitz, and $ h $ is bounded.
• The error estimate is derived via a novel application of partial Malliavin calculus, particularly the duality of stochastic integrals and Skorohod integral representations.
• The key technical step involves bounding the $ L^p $-norm of the error term $ E_1(\rho, Z) $ using estimates on the Malliavin derivative and conditional expectations, yielding $ \|E_1(\rho,Z)\|_p^p \leq \frac{C_3}{n^p}\|Z\Gamma_T(\rho)\|_p^p + \frac{C_4}{n^p}E^{\mathcal{W}_Y}\left[\left(\int_0^T \operatorname{ess\,sup}_{0\leq r\leq T}|E^{\mathcal{W}_B}[(D_r^Y f_s)\theta_s]|^2 ds\right)^{p/2}\right] $.
• The proof shows that the $ O(1/n) $ rate arises from the averaging effect of the observation process $ Y $, despite the $ O(1/n) $ weak error in $ h(X_s) - h(X_{\eta(s)}) $, due to the structure of conditional expectations.
• The method is robust to irregular $ g $, and the result is optimal in the sense that $ \alpha = 1 $ cannot be achieved with a non-degenerate limit, suggesting $ O(1/n) $ is the best possible rate under the current framework.