  In this chapter we review stochastic modelling methods in climate science.
First we provide a conceptual framework for stochastic modelling of
deterministic dynamical systems based on the Mori-Zwanzig formalism. The
Mori-Zwanzig equations contain a Markov term, a memory term and a term
suggestive of stochastic noise. Within this framework we express standard model
reduction methods such as averaging and homogenization which eliminate the
memory term. We further discuss ways to deal with the memory term and how the
type of noise depends on the underlying deterministic chaotic system. Secondly,
we review current approaches in stochastic data-driven models. We discuss how
the drift and diffusion coefficients of models in the form of stochastic
differential equations can be estimated from observational data. We pay
attention to situations where the data stems from multi scale systems, a
relevant topic in the context of data from the climate system. Furthermore, we
discuss the use of discrete stochastic processes (Markov chains) for e.g.
stochastic subgrid-scale modeling and other topics in climate science.
