A piecewise deterministic Monte Carlo method for diffusion bridges

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.

Figures (11)

• Figure 1: Lévy-Ciesielski construction of a Brownian motion on $(0,1)$. On the left the Faber-Schauder basis functions up to level $N =3$, on the top-right the values of the corresponding coefficients located at the peak of their relative FS basis function and on the bottom-right the resulting approximated Brownian path $X^N$ (black line) compared with a finer approximation (red line). The truncated sum defines the process in $2^{N+1} + 1$ finite dyadic points (black dots) with linear interpolation in between points. A finer approximation corresponds to Brownian fill-in noise between any two neighboring dyadic points.
• Figure 2: One dimensional Zig-Zag targeting a Gaussian random variable $\mathcal{N}(0,1)$. Left: $t\mapsto \xi(t)$, right: $t\mapsto \theta(t)$.
• Figure 3: 100 samples from the Brownian bridge measure starting at $0$ and hitting $0$ at time $1$ obtained by one run of the Zig-Zag sampler targeting the coefficients relative to the measure expanded with the Faber-Schauder basis. The resolution level is fixed to $N = 6$ and the Zig-Zag clock to $\tau_{\text{final}} = 500$ and initial burn in $\tau_{\text{burn-in}} = 10$.
• Figure 4: Support of the Faber-Schauder functions $(\phi_{i,j} : i \in \{0,1,\dots,N\}, \, j = \{0,1,\dots,2^i-1\}$ with $N =3$. The coefficient $\xi_{i,j}$ is independent of the coefficient $\xi_{k,l}$ conditionally on the set of common ancestors $(\xi_{m,n} \colon S_{m,n} \cap S_{i,j}\ne \emptyset \wedge S_{m,n} \cap S_{k,l} \ne \emptyset)$ if $S_{i,j} \cap S_{k,l} = \emptyset$.
• Figure 5: Simulation of the diffusion bridge measure (100 samples) given by equation (\ref{['linear equation']}) starting at $-1.0$ and conditioned to hit $2.0$ at $T=10$. $\alpha = -5.0, \beta = -1.0$ which is equivalent to a mean reverting process with mean reversion at $x = -5$ (straight line). The truncation level is $N = 6$, final clock $\tau_{\text{final}} = 1000$ and burn-in $\tau_{\text{burn-in}} = 10$.

Theorems & Definitions (19)

• Example 2.1
• Theorem 3.2
• proof
• Definition 3.3
• Theorem 3.5
• proof
• Remark 3.6
• proof
• Remark 3.7
• proof