On inference in partially observed Markov models using sequential Monte Carlo
methods
Jonas Ströjby
Centre for Mathematical Sciences
Mathematical Statistics
Lund University
2010
ISBN 9789174730203
LUTFMS10362010

Abstract:

This thesis concerns estimation in partially observed continuous and discrete
time Markov models and focus on both inference about the conditional
distributions of the unobserved process as well as parameter inference
for the dynamics of the unobserved process.


Paper A concerns calibration of advanced stock price models, in particular
the Bates and NIGCIR models, using options data observed through bidask
spreads. The parameter estimation problem is recast as a filtering problem
and time dependent parameter estimates are obtained through the use of the
iterated Kalman filter. This proves to be both faster and more stable than
the nonlinear least squares used in practice.


Paper B and C treats an extension to the sequential Monte Carlo framework
allowing closed form transition kernels in the algorithm to be replaced by
random approximations. The resulting method is coined random weight particle
filters and have many applications for partially and discretely observed
continuous time models, in particular ones modeled by stochastic differential
equations. The random weight filter is extended to a random weight smoother
and a random formulation of the intermediate quantity in the EMalgorithm
and used to perform parameter inference. Asymptotic consistency of the random
weight filter and the intermediate quantity is proved. In addition, for the
random weight particle filter, asymptotic normality is shown as well as finite
sample expected moment bounds. These are extended to timeuniform results
under standard assumptions.


Paper D and E concerns the construction of an estimate of the optimal particle
filter through the use of parametric approximations of the joint transition
kernel. It is argued that by using a flexible class of approximations, so
called 'mixture of experts', an arbitrarily good approximation can be obstructed
efficiently using an offline stochastic approximation algorithm. This
approximation is used to calculate optimal proposal kernels in the particle
filter and optimal adjustment weights, using a novel stochastic approximation
based estimation procedure, whose convergence is proved. Also, through extending
the state space, the method is used to provide the basis for simulation based
transition density approximations for continuouos time models.



Key words:

Filtering, particle filter, sequential Monte Carlo, hidden Markov models,
continuous time processes, adaption








