Title: Continuous-discrete smoothing of diffusions
Abstract: Suppose X is a multidimensional diffusion taking values in Rd with drift b and dispersion coefficient σ, satisfying the stochastic differential equation (SDE)
dXt =b(t,Xt)dt+σ(t,Xt)dWt. (1)
The process W is a vector valued process consisting of independent Brownian motions. times by We assume the process is incompletely observed at times 0 = t0 < t1 < ··· < tn and the observations are given by
Vi =LiXti +ηi, i=0,…,n.
Here, for 0 ≤ i ≤ n, Li is a mi × d-matrix with mi ≤ d. The random variables η1,…,ηn are
independent random variables, independent of the diffusion process X.
Reconstructingthepath(Xt,t∈[0,tn])basedonV0,…,Vn isknownascontinuous-discrete smoothing. In case ηi ≡ 0 and Li = Id (for all i), the problem boils down to sampling independent diffusion bridges. As the transition densities of the diffusion are intractable, we propose to simulate diffusion bridge proposals generated from the SDE
dXt◦ =b(t,Xt◦)dt+q(t,Xt◦)dt+σ(t,Xt◦)dWt. (2)
and subsequently accept/reject these using a Metropolis-algorithm step. The “guiding term” q in (2) is chosen to ensure the process X◦ ends up in the right endpoint of the bridge. An effective way for constructing this term was introduced in [1].
It turns out that this framework can be extended to the general setting. The resulting algorithm does not assume a Gaussian approximation to the smoothing distribution. From a computational point of view, it can be shown that two backward ordinary differential equations need to be solved once, and at each step of the Metropolis-Hastings algorithm the process X◦ has to be simulated forward. The performance of the proposed algorithm is illustrated using numerical examples (including the Lorenz attractor).
This is joint work with Moritz Schauer (Leiden University) .
[1] M.R. Schauer, F.H. van der Meulen and J.H. van Zanten. Guided proposals for simulating multi-dimensional diffusion bridges. Bernoulli 23(4A), 2917–2950, 2017
[2] F.H. van der Meulen and M.R. Schauer. Bayesian estimation of incompletely observed diffu- sions. Stochastics, 1–22, 2017.