Linear and nonlinear filtering in mathematical finance: a review

Copyright @ The Authors 2010This paper presents a review of time series filtering and its applications in mathematical finance. A summary of results of recent empirical studies with market data are presented for yield curve modelling and stochastic volatility modelling. The paper also outlines different approaches to filtering of nonlinear time series

The Hitchhiker's Guide to Nonlinear Filtering

Nonlinear filtering is the problem of online estimation of a dynamic hidden variable from incoming data and has vast applications in different fields, ranging from engineering, machine learning, economic science and natural sciences. We start our review of the theory on nonlinear filtering from the simplest `filtering' task we can think of, namely static Bayesian inference. From there we continue our journey through discrete-time models, which is usually encountered in machine learning, and generalize to and further emphasize continuous-time filtering theory. The idea of changing the probability measure connects and elucidates several aspects of the theory, such as the parallels between the discrete- and continuous-time problems and between different observation models. Furthermore, it gives insight into the construction of particle filtering algorithms. This tutorial is targeted at scientists and engineers and should serve as an introduction to the main ideas of nonlinear filtering, and as a segway to more advanced and specialized literature.Comment: 64 page

Forecasting trends with asset prices

In this paper, we consider a stochastic asset price model where the trend is an unobservable Ornstein Uhlenbeck process. We first review some classical results from Kalman filtering. Expectedly, the choice of the parameters is crucial to put it into practice. For this purpose, we obtain the likelihood in closed form, and provide two on-line computations of this function. Then, we investigate the asymptotic behaviour of statistical estimators. Finally, we quantify the effect of a bad calibration with the continuous time mis-specified Kalman filter. Numerical examples illustrate the difficulty of trend forecasting in financial time series.Comment: 26 pages, 11 figure

A partially linearized sigma point filter for latent state estimation in nonlinear time series models

A new technique for the latent state estimation of a wide class of nonlinear time series models is proposed. In particular, we develop a partially linearized sigma point filter in which random samples of possible state values are generated at the prediction step using an exact moment matching algorithm and then a linear programming-based procedure is used in the update step of the state estimation. The effectiveness of the new ¯ltering procedure is assessed via a simulation example that deals with a highly nonlinear, multivariate time series representing an interest rate process

Multilevel ensemble Kalman filtering for spatio-temporal processes

We design and analyse the performance of a multilevel ensemble Kalman filter method (MLEnKF) for filtering settings where the underlying state-space model is an infinite-dimensional spatio-temporal process. We consider underlying models that needs to be simulated by numerical methods, with discretization in both space and time. The multilevel Monte Carlo (MLMC) sampling strategy, achieving variance reduction through pairwise coupling of ensemble particles on neighboring resolutions, is used in the sample-moment step of MLEnKF to produce an efficient hierarchical filtering method for spatio-temporal models. Under sufficient regularity, MLEnKF is proven to be more efficient for weak approximations than EnKF, asymptotically in the large-ensemble and fine-numerical-resolution limit. Numerical examples support our theoretical findings.Comment: Version 1: 39 pages, 4 figures.arXiv admin note: substantial text overlap with arXiv:1608.08558 . Version 2 (this version): 52 pages, 6 figures. Revision primarily of the introduction and the numerical examples sectio

Deterministic Mean-field Ensemble Kalman Filtering

The proof of convergence of the standard ensemble Kalman filter (EnKF) from Legland etal. (2011) is extended to non-Gaussian state space models. A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence $\kappa$ between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF when the dimension $d<2\kappa$. The fidelity of approximation of the true distribution is also established using an extension of total variation metric to random measures. This is limited by a Gaussian bias term arising from non-linearity/non-Gaussianity of the model, which exists for both DMFEnKF and standard EnKF. Numerical results support and extend the theory

Non-linear minimum variance estimation for discrete-time multi-channel systems

A nonlinear operator approach to estimation in discrete-time systems is described. It involves inferential estimation of a signal which enters a communications channel involving both nonlinearities and transport delays. The measurements are assumed to be corrupted by a colored noise signal which is correlated with the signal to be estimated. The system model may also include a communications channel involving either static or dynamic nonlinearities. The signal channel is represented in a very general nonlinear operator form. The algorithm is relatively simple to derive and to implement

Higher order sigma point filter: A new heuristic for nonlinear time series filtering

In this paper we present some new results related to the higher order sigma point filter (HOSPoF), introduced in [1] for filtering nonlinear multivariate time series. This paper makes two distinct contributions. Firstly, we propose a new algorithm to generate a discrete statistical distribution to match exactly a specified mean vector, a specified covariance matrix, the average of specified marginal skewness and the average of specified marginal kurtosis. Both the sigma points and the probability weights are given in closed-form and no numerical optimization is required. Combined with HOSPoF, this random sigma point generation algorithm provides a new method for generating proposal density which propagates the information about higher order moments. A numerical example on nonlinear, multivariate time series involving real financial market data demonstrates the utility of this new algorithm. Secondly, we show that HOSPoF achieves a higher order estimation accuracy as compared to UKF for smooth scalar nonlinearities. We believe that this new filter provides a new and powerful alternative heuristic to existing filtering algorithms and is useful especially in econometrics and in engineering applications