# On inference in partially observed Markov models using sequential Monte Carlo methods

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## Abstract

This thesis concerns estimation in partially observed continuous and discrete time Markov models and focus on both inference about the conditional distribution 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 NIG-CIR models, using options data observed through bid-ask 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 EM-algorithm 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 time-uniform 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 constructed 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 approximation for continuous time models.

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