Simulation of diffusion bridges for stochastic differential equations

Abstract: The simulation of diffusion bridges is a difficult statistical problem, and methods for this are attracting a great deal of attention. Analytically, the conditioning on a future pointintroducesasecondtermtothedriftoftheSDEthatincludesthetransitiondensity of the unconditioned process. However, transition densities of diffusion processes are often intractable, whereby other, often computationally intensive methods are necessary. A useful method is Monte Carlo simulation, the computational cost of which, however, is often prohibitively high, necessitating the use of some variance reduction techniques. In this thesis, a novel method for simulating discrete time realisations of diffusion processes satisfying some stochastic differential equation (SDE), and conditioned on hitting some end-point is proposed. This new method is based on the partitioning of the process into one deterministic and one random part, where the deterministic part accounts for the drift of the process, leaving the residual, random process a Brownian bridge. The difficulty then lies in finding a good approximation of the deterministic process, and it is herein proposed that already accepted realisations are used to adaptively improve an estimate of the optimal deterministic process, which is used for simulation. The proposed sampler uses the linear noise approximation (LNA) of the process as initial distribution. Taking the mean of previous realisations ensures convergence to the proper expected path, but may initially produce distorted approximations. During an initial burn-in period, the proposed sampler therefore uses recursively formulated regression based on a suitably chosen number of sine basis functions to approximate the remaining dynamics and forms the approximation of the expected path as the sum of the LNA and the approximation of the remainder. Choosing a suitably low number of basis functions accords the regressive approximation with a resilience towards the erraticbehaviourofindividualrealisationsbutintroducesbias. Therefore,afteraburnin period, the sampler switches to taking the mean, ensuring proper convergence. The proposed sampler is tested on two different models: the Cox-Ingersoll-Ross model and the stochastic Lorenz model, and its performance compared to existing approaches, showing similar results for easy cases and a significant increase for difficult ones.