Multiplicative cascades have been introduced in turbulence to generate randomor deterministic fields having intermittent values and long-range power-lawcorrelations. Generally this is done using discrete construction rules leadingto discrete cascades. Here a causal log-normal stochastic process isintroduced; its multifractal properties are demonstrated together with otherproperties such as the composition rule for scale dependence and stochasticdifferential equations for time and scale evolutions. This multifractalstochastic process is continuous in scale ratio and in time. It has a simplegenerating equation and can be used to generate sequentially time series of anylength.
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