Residual-Bridge Constructs for Conditioned Diffusions

On residual and guided proposals for diffusion bridge simulation

Recently Whitaker et al. (2017) considered Bayesian estimation of diffusion driven mixed effects models using data-augmentation. The missing data, diffusion bridges connecting discrete time observations, are drawn using a "residual bridge construct". In this paper we compare this construct (which we call residual proposal) with the guided proposals introduced in Schauer et al. 2017. It is shown that both approaches are related, but use a different approximation to the intractable stochastic differential equation of the true diffusion bridge. It reveals that the computational complexity of both…

Efficient sampling of conditioned Markov jump processes

We consider the task of generating draws from a Markov jump process (MJP) between two time-points at which the process is known. Resulting draws are typically termed bridges and the generation of such bridges plays a key role in simulation-based inference algorithms for MJPs. The problem is challenging due to the intractability of the conditioned process, necessitating the use of computationally intensive methods such as weighted resampling or Markov chain Monte Carlo. An efficient implementation of such schemes requires an approximation of the intractable conditioned hazard/propensity functio…

On the Exact Simulation of (Jump) Diffusion Bridges

In this paper we outline methodology to efficiently simulate (jump) diffusion bridge sample paths without discretisation error. We achieve this by considering the simulation of conditioned (jump) diffusion bridge sample paths in light of recent work developing a mathematical framework for simulating finite dimensional sample path skeletons (which flexibly characterise the entirety of sample paths).

Improved bridge constructs for stochastic differential equations

We consider the task of generating discrete-time realisations of a nonlinear multivariate diffusion process satisfying an Itô stochastic differential equation conditional on an observation taken at a fixed future time-point. Such realisations are typically termed diffusion bridges. Since, in general, no closed form expression exists for the transition densities of the process of interest, a widely adopted solution works with the Euler-Maruyama approximation, by replacing the intractable transition densities with Gaussian approximations. However, the density of the conditioned discrete-time pro…

区间约束布朗路径：精确插值与外推

Interval-Constrained Brownian Paths: Exact Interpolation and Extrapolation

本文研究区间约束布朗运动与布朗桥的精确插值和外推，提出统一采样套件，含边界端点并自动切换密度表示，提升接受率，并扩展至多时刻骨架路径模拟。

Simulating bridges using confluent diffusions

Diffusions are a fundamental class of models in many fields, including finance, engineering, and biology. Simulating diffusions is challenging as their sample paths are infinite-dimensional and their transition functions are typically intractable. In statistical settings such as parameter inference for discretely observed diffusions, we require simulation techniques for diffusions conditioned on hitting a given endpoint, which introduces further complication. In this paper we introduce a Markov chain Monte Carlo algorithm for simulating bridges of ergodic diffusions which (i) is exact in the s…