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Exact MCMC inference for Markov switching diffusion models
Keywords: exact algorithms for diffusions; retrospective simulation; MCMC
Wed, 09:25--09:50
  • Latuszynski, Krzysztof (Department of Statistics, University of Warwick, United Kingdom)
  • Palczewski, Jan (University of Leeds)
  • Roberts, Gareth O (University of Warwick)

Consider a stochastic differential equation for $X= X_t$, $t \in [0,T]$, \begin{equation} dX_t = b(X_t, Y_t, \theta) dt + \sigma(X_t, \theta) \gamma (Y_t, \theta) dB_t. \end{equation} where $B = B_t$ is a Brownian motion, $\theta \in \Theta \subset \mathbb{R}^n$ is a multi-dimensional parameter and the switching process $Y = Y_t,$ $t \in [0,T]$ is a continuous-time Markov process on the state space $\mathcal{Y} = \{1, \dots, m\}$ with the intensity matrix $\Lambda = \{\lambda_{i,j}\}$. Models of this type naturally arise in finance and economics, where $Y$ represents unobservable factors that describe the state of the economy (e.g., a business cycle).

We assume that $X$ is observed at discrete time instances and propose an MCMC algorithm that samples from the exact full posterior \begin{equation} \pi(X_M, Y, \theta, \Lambda \;| \; X_D) \;\; \propto \;\; \pi_{\theta}(\theta) \pi_{\Lambda}(\Lambda) \pi( Y \, | \, \Lambda) \pi(X_M , X_D \, |\, Y, \theta) \end{equation} including the infinite dimensional sample path of $X$ and avoiding any discretization error. The approach relies on a generalization of the Exact Algorithm of [1,2] and relies on retrospective sampling ideas.

[1] A. Beskos, O. Papaspiliopoulos, G.O. Roberts (2008). A factorisation of diffusion measure and finite sample path constructions. Methodol. Comput. Appl. Probab. 10(1), 85–104.
[2] A. Beskos, O. Papaspiliopoulos, G.O. Roberts, and P. Fearnhead (2006). Exact and computationally efficient likelihood-based estimation for discreetly observed diffusion processes (with discussion). Journal of the Royal Statistical Society B, 68(3), 333-382.