  Multiplicative cascades have been introduced in turbulence to generate random
or deterministic fields having intermittent values and long-range power-law
correlations. Generally this is done using discrete construction rules leading
to discrete cascades. Here a causal log-normal stochastic process is
introduced; its multifractal properties are demonstrated together with other
properties such as the composition rule for scale dependence and stochastic
differential equations for time and scale evolutions. This multifractal
stochastic process is continuous in scale ratio and in time. It has a simple
generating equation and can be used to generate sequentially time series of any
length.
