Nonlinear filtering via stochastic PDE projection on mixture manifolds in $$L^2$$ L 2 direct metric

Proc. London Math. Soc.

Rossi

2018

Self Cite

We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Itô‐vector and Itô‐jet projections. This allows one to systematically develop low‐dimensional approximations to high‐dimensional SDEs using differential geometric techniques. The approach generalizes the notion of projecting a vector field onto a submanifold in order to derive approximations to ordinary differential equations, and improves the previous Stratonovich projection method by adding optimality analysis and results. Indeed, just as in the case of ordinary projection, our definitions of projection are based on optimality arguments and give in a well‐defined sense ‘optimal’ approximations to the original SDE in the mean‐square sense over small times. We also explain how the Stratonovich projection satisfies an optimality criterion that is more ad hoc and less appealing than the criteria satisfied by the Itô projections we introduce. As an application, we consider approximating the solution of the non‐linear filtering problem with a Gaussian distribution. We show how the newly introduced Itô projections lead to optimal approximations in the Gaussian family and briefly discuss the optimal approximation for more general families of distributions. We perform a numerical comparison of our optimally approximated filter with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.

Section: The Kushner Stratonovich Equationmentioning

confidence: 99%

“…We suppose that the state

X_{t} \in {double-struckR}^{n}

of a system evolves according to the equation:

d X_{t} = f false(X_{t}, t false) d t + σ false(X_{t}, t false) d W_{t},

where

f

and

σ

are smooth

R^{n}

valued functions and

W_{t}

is a Brownian motion. One typically adds growth conditions to ensure a global existence and uniqueness result for the signal equation, see for example and references therein for the details.…”

Section: Application Of the Projection To Non‐linear Filteringmentioning

confidence: 99%

See 1 more Smart Citation

Optimal approximation of SDEs on submanifolds: the Itô‐vector and Itô‐jet projections

Armstrong

Proc. London Math. Soc.

Rossi

2018

Self Cite

“…This is a very general setup and includes, for example, approximating the density using piecewise linear functions to derive a finite difference approximation or approximating the density with Hermite polynomials to derive a spectral method. Other examples include exponential families (considered in [7,8]) and mixture families (considered in [3,4]). …”

Section: The Kushner Stratonovich Equationmentioning

confidence: 99%

“…No optimality result has been derived for the Stratonovich projection, it has simply been derived heuristically from the deterministic case. Nevertheless, it appears to be a good approximation in practice and it has been used to find good quality numerical solutions to the non-linear filtering problem (See [8], [7], [3], [4]). …”

Section: Introductionmentioning

confidence: 99%

Extrinsic Projection of Itô SDEs on Submanifolds with Applications to Non-linear Filtering

Armstrong

Computational Information Geometry

2016

Self Cite

Abstract. We define the notion of the extrinsic Itô projection of a stochastic differential equation (SDE) on a submanifold. This allows one to systematically develop low dimensional approximations to high dimensional SDEs in a differential geometric setting. We consider the example of approximating the non-linear filtering problem with a Gaussian distribution and show how the Itô projection leads to improved approximations in the Gaussian family. We briefly discuss the approximations for more general families of distribution. We perform a numerical comparison of our projection filters with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.

Dimensionality Reduction for Measure Valued Evolution Equations in Statistical Manifolds

Computational Information Geometry

Pistone

2016

Self Cite

We propose a dimensionality reduction method for infinite-dimensional measurevalued evolution equations such as the Fokker-Planck partial differential equation or the Kushner-Stratonovich resp. Duncan-Mortensen-Zakai stochastic partial differential equations of nonlinear filtering, with potential applications to signal processing, quantitative finance, heat flows and quantum theory among many other areas. Our method is based on the projection coming from a duality argument built in the exponential statistical manifold structure developed by G. Pistone and co-authors. The choice of the finite dimensional manifold on which one should project the infinite dimensional equation is crucial, and we propose finite dimensional exponential and mixture families. This same problem had been studied, especially in the context of nonlinear filtering, by D. Brigo and co-authors but the L 2 structure on the space of square roots of densities or of densities themselves was used, without taking an infinite dimensional manifold environment space for the equation to be projected.Here we re-examine such works from the exponential statistical manifold point of view, which allows for a deeper geometric understanding of the manifold structures at play. We also show that the projection in the exponential manifold structure is consistent with the Fisher Rao metric and, in case of finite dimensional exponential families, with the assumed density approximation. Further, we show that if the sufficient statistics of the finite dimensional exponential family are chosen among the eigenfunctions of the backward diffusion operator then the statistical-manifold or Fisher-Rao projection provides the maximum likelihood estimator for the Fokker Planck equation solution. We finally try to clarify how the finite dimensional and infinite dimensional terminology for exponential and mixture spaces are related. arXiv:1601.04189v3 [math.PR]