Exact Simulation of Sample Paths

Recently in the mathematical literature [14] an exact algorithm for the simulation of a class of Ito s diffusion is presented, which allows the stochastic differential equation to be simulated without the need for Euler discretisation. If adaptable into the random flights framework, the efiiciency at which simulations of chemical systems can be performed would be greatly increased, allowing much better statistics to be obtained through the use of many more realisations. The general recipe for the exact algorithm proceeds as follows  

Starting with the stochastic differential equation of the form 

The basic idea is to propose sample paths from the biased Brownian motion W conditioned that the end point Wb = h, to allow rejection sampling to be performed. By performing some complex analysis the authors have shown that it is possible to define the Radon-Nikodym derivative as 

Construct a Brownian bridge for the process started at x at time zero and ends on y atS. 

The authors report two further variations of the algorithm described above which relax the restriction of . Unfortunately, the adaption of this algorithm to simulate the stochastic differential equation for charged species is not possible since (p u) = 18u clearly this function is not bounded from below. Further complications also arise, since the function A (m) is not defined at y = 0, making it impossible to simulate a biased diffusion path W. The authors have not yet generalised their procedure to the problem where (m) is bounded purely because of an inner boundary. Such a generalisation will obviously be very useful. 

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