A stochastic Taylor expansion generalizes a Taylor series to a stochastic differential equation by repeatedly expanding its drift and diffusion terms and evaluating the resulting iterated stochastic integrals. For the scalar autonomous Itô equation
 |
the first terms over a step of length 
are
![X_(t+h)=X_t+a(X_t)h+b(X_t)DeltaW+1/2b(X_t)b^'(X_t)[(DeltaW)^2-h]+...,](/images/equations/StochasticTaylorExpansion/NumberedEquation2.svg) |
where 
and 
is the derivative of 
. Different truncations lead to strong and weak approximation
schemes of different orders, with higher-order terms indexed by multiple stochastic
integrals.
