Approximation of Markov Jump Processes by Diffusions

Diffusion processes enable realistic and convenient modelling of dynamic systems. They typically arise as approximations of exact but computationally expensive individual-based stochastic models. However, the correct derivation of an appropriate diffusion approximation is often complicated, and hence their utilisation is not widely spread in the applied sciences. Instead, practitioners often favour rather unrealistic deterministic models and their relatively simple analysis. This chapter motivates the application of diffusion approximations and explains their correct derivation. It reviews and develops different approaches and points out differences and correspondences between them. All methods are formulated for multi-dimensional processes and extended to an even more general framework where systems are characterised by multiple size parameters. The chapter addresses mathematicians who are interested in the theory of diffusion approximations and practitioners who wish to apply diffusion models for their specific problems.