Guiding techniques for conditioning Markov processes
Authors:   , Corstanje Marc Amand 
Published: October 2024

This dissertation focusses on the simulation of conditioned Markov processes, a topic with many applications across various fields. Among these applications, statistical inference for discretely observed stochastic processes is our primary focus. For statistical inference for discretely observed Markov processes a closed form expression of the transition density is often required, which is rarely known in closed form. This can be circumvented through simulating conditioned processes and subsequently performing inference on the simulated process.

Markov processes forced to satisfy a conditioning at a fixed time T are obtained through a change of measure that requires a function that is typically dependent on the transition density of the process. A way to circumvent this dependence on the transition density is by introducing guided processes, where one replaces the unknown function by a tractable substitute. We demonstrate various ways of obtaining such a substitute in various contexts. It is usually not difficult to show that the laws of the conditioned and guided process are absolutely continuous when restricted to the filtration at time t<T. However, absolute continuity at T is usually difficult to prove.

In Chapter 3, we present conditions that ensure absolute continuity. This extends earlier work on guided processes to general Markov processes, while also weakening the conditions for equivalence for hypo-elliptic SDEs. This result was applied to inhomogeneous Poisson processes, jump processes on a Poisson-Delaunay graph and hypo-elliptic SDEs. For the latter we provide a proof of equivalence which we applied to deformations of landmarks.

We also apply the guiding technique to chemical reaction processes in Chapter 4. We show that the guiding function can be scaled using any function differentiable in time and applied this property to simulate a guided jump process in a discrete state space using a guiding function obtained from a scaled Brownian motion. This can be extended to processes with monotone components, by utilising a constant rate Poisson process as one component for an auxiliary process.We also present a method for exact simulation of guided chemical reaction processes using Poisson thinning and monotonicity of the jump intensities. We complete the chapter by showing results for a bridge sampler for different types of chemical reaction processes and compared performance to existing methods.

In Chapter 5 we consider manifold-valued diffusion processes. First, we guide using the transition density of Brownian motion on the manifold. This approach is invariant under diffeomorphisms, meaning we can guide processes using a guiding term obtained from Brownian motion on a diffeomorphic manifold. We can therefore construct guided processes on compact manifolds on which the heat kernel is known in closed form or manifolds diffeomorphic to a manifold on which the heat kernel in closed form. Additionally, we provide algorithms for bridge sampling and parameter inference on manifolds. We finish the chapter by demonstrating the simulation of diffusion bridges on the 2-torus and used simulated bridges to estimate a vector field on the torus using discrete observations of a Brownian motion on the torus which is affected by this vector field.