References

Abstract

This article develops a class of Monte Carlo (MC) methods for simulating conditioned diffusion sample paths, with special emphasis on importance sampling schemes. We restrict attention to a particular type of conditioned diffusions, the so-called diffusion bridge processes. The diffusion bridge is the process obtained by conditioning a diffusion to start and finish at specific values at two consecutive times t 0 < t 1 .Diffusion bridge simulation is a highly non-trivial problem. At an even more elementary level unconditional simulation of diffusions, that is without fixing the value of the process at t 1 , is difficult. This is a simulation from the transition distribution of the diffusion which is typically intractable. This intractability stems from the implicit specification of the diffusion as a solution of a stochastic differential equation (SDE). Although the unconditional simulation can be carried out by various approximate schemes based on discretizations of the SDE, it is not feasible to devise similar schemes for diffusion bridges in general. This has motivated active research in the last 15 years or so for the development of MC methodology for diffusion bridges.

Guided proposals for simulating multi-dimensional diffusion bridges

“…The simulation of diffusion bridges has received much attention over the past decade, see for instance the papers Elerian et al [11], Eraker [12], Roberts and Stramer [21], Durham and Gallant [10], Stuart et al [22], Beskos and Roberts [5], Beskos et al [3], Beskos et al [4], Fearnhead [13], Papaspiliopoulos and Roberts [20], Lin et al [17], Bladt and Sørensen [6], Bayer and Schoenmakers [2] to mention just a few. Many of these papers employ accept-reject-type methods.…”

“…We will consider so-called guided proposals, according to the terminology suggested in Papaspiliopoulos and Roberts [20]. This means that our proposals are realizations of a process X • that solves an SDE of the form (1.1) as well, but with a drift term that is adapted in order to force the process X • to hit the point v at time T .…”

Bayesian estimation of incompletely observed diffusions

“…This implies that the algorithm is valid when taking this limit. We refer to [23] to a discussion on additional advantages of the simulation-projection strategy, which we will employ in this paper.…”

“…The approach taken in [11,12] is to approximate this stochastic process by a finite-dimensional vector and next carry out simulation. [23] call this the projection-simulation strategy and advocate the simulation-projection strategy where an appropriate Monte-Carlo scheme is designed that operates on the infinitely-dimensional space of diffusion bridges. For practical purposes it needs to be discretised but the discretisation error can be eliminated by letting the meshwidth tend to zero.…”

Variational Characterization of Free Energy: Theory and Algorithms

“…To keep the presentation easily accessible, we present only a vanilla version of the theorem based on finite-dimensional Gaussian measures, partly following an idea in [43].…”