4,364 research outputs found
A filtering approach to tracking volatility from prices observed at random times
This paper is concerned with nonlinear filtering of the coefficients in asset
price models with stochastic volatility. More specifically, we assume that the
asset price process is given by where
is a Brownian motion, is a positive function, and
is a c\'{a}dl\'{a}g strong Markov process. The
random process is unobservable. We assume also that the asset price
is observed only at random times This is an
appropriate assumption when modelling high frequency financial data (e.g.,
tick-by-tick stock prices).
In the above setting the problem of estimation of can be approached
as a special nonlinear filtering problem with measurements generated by a
multivariate point process . While quite natural,
this problem does not fit into the standard diffusion or simple point process
filtering frameworks and requires more technical tools. We derive a closed form
optimal recursive Bayesian filter for , based on the observations
of . It turns out that the filter is
given by a recursive system that involves only deterministic Kolmogorov-type
equations, which should make the numerical implementation relatively easy
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
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