Abstract
AbstractThe problem of filtering of unobservable components x(t) of a multidimensional continuous diffusion Markov process zt=xt,yt, given the observations of the (multidimensional) process y(t) taken at discrete consecutive times with small time steps, is analytically investigated. On the base of that investigation the new algorithms for simulation of unobservable components, x(t), and the new algorithms of nonlinear filtering with the use of sequential Monte Carlo methods, or particle filters, are developed and suggested. The analytical investigation of observed quadratic variations is also developed. The new closed-form analytical formulae are obtained, which characterize dispersions of deviations of the observed quadratic variations and the accuracy of some estimates for x(t). As an illustrative example, estimation of volatility (for the problems of financial mathematics) is considered. The obtained new algorithms extend the range of applications of sequential Monte Carlo methods, or particle filters,...
Highlights
In the present work the precise, explicit and compact analytical formulae and algorithms for simulation of the sample sequences xi(t)|n0, when y(t)|n0 is given, and for recursive calculations of Pi(n), weights, and estimates are obtained for the general case of a multidimensional continuous diffusion Markov process (x(t), y(t))
Using the Theorem on Normal Correlation, we find the following expressions for the first and second moments of the conditional probability distribution of the increments Δxα (with (α = 1, ..., m) provided that the increments Δyρ (with ρ = (m + 1), ..., p) and the value z(t) = z are given (Khazen, 2009, Chapter 3, Sections 3.1.2 and 3.3.1, pages 79–81, 101–106): E Δx |Δy, z = A (z, t)Δt + b (z, t)c (z, t) Δy − A (z, t)Δt + o(Δt), (14)
The dimensionality of the unobservable process increased for the model of description (59) in comparison with the model (55), (56), the problem of nonlinear filtering will be solved effectively with the use of the new Monte Carlo estimates (23)–(27) and with the use of the new algorithm of simulation (17), derived in the present paper
Summary
In the last two decades, a great deal of works has been devoted to the development and the investigation of particle filters, or sequential Monte Carlo algorithms, for filtering of an unobservable process x(t) def. In the present work the precise, explicit and compact analytical formulae and algorithms for simulation of the sample sequences xi(t)|n0, when y(t)|n0 is given, and for recursive calculations of Pi(n), weights, and estimates are obtained for the general case of a multidimensional continuous diffusion Markov process (x(t), y(t)). The novelty of the estimates (23)–(27) and the simulation (17), obtained in the present paper, is that the simulation of unobservable components and estimates f (̂ x tn ) are obtained in explicit and closed analytical form for the general case of partially observable multidimensional continuous diffusion Markov processes, and the obtained Monte Carlo estimate (26) converges with probability one to the sought posterior expectation of f (x tn ) given y(t)|n0, if N → ∞.
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