Abstract
We introduce the use of the Zig-Zag sampler to the problem of sampling conditional diffusion processes (diffusion bridges). The Zig-Zag sampler is a rejection-free sampling scheme based on a non-reversible continuous piecewise deterministic Markov process. Similar to the Lévy–Ciesielski construction of a Brownian motion, we expand the diffusion path in a truncated Faber–Schauder basis. The coefficients within the basis are sampled using a Zig-Zag sampler. A key innovation is the use of the fully local algorithm for the Zig-Zag sampler that allows to exploit the sparsity structure implied by the dependency graph of the coefficients and by the subsampling technique to reduce the complexity of the algorithm. We illustrate the performance of the proposed methods in a number of examples.
Highlights
Diffusion processes are an important class of continuoustime probability models which find applications in many fields such as finance, physics and engineering
Note that the only challenge in implementing Algorithm 1 lies on the simulation of the waiting times which correspond to the simulation of the first event time of d inhomogeneous Poisson processes (IPPs) with rates λ1, λ2, . . . , λd which are functions of the state space (ξ, θ ) of the process
The factorisation of the partial derivatives leads to a specific dependency structure of the coefficients under the target diffusion bridge measure: the coefficient ξi, j is conditionally independent of the coefficient ξk,l if Si, j ∩Sk,l = ∅ conditionally on the set of common ancestors
Summary
Diffusion processes are an important class of continuoustime probability models which find applications in many fields such as finance, physics and engineering. Due to the Markov property, the missing paths in between subsequent observations can be sampled independently and each of such segments constitutes a diffusion bridge. As this application requires sampling iteratively many diffusion bridges, it is crucial to have a fast algorithm for this step. The Zig-Zag sampler is an innovative nonreversible and rejection-free Markov process Monte Carlo algorithm which can exploit the structure present in this high-dimensional sampling problem. It is based on simulating a piecewise deterministic Markov process (PDMP). The problem of diffusion bridge simulation has received considerable attention over the past two decades, see, for
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
