Time-fractional geometric Brownian motion from continuous time random walks

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We construct a time-fractional geometric Fokker–Planck equation from the diffusion limit of a continuous time random walk with a power law waiting time density, and a biased multiplicative jump length density dependent on the particles’ current position. The bias is related to a force, and an associated potential, through a steady state Boltzmann distribution. The limit of the random walk, with a force derived from a logarithmic potential, defines a stochastic process that is a fractional generalization of geometric Brownian motion. We have investigated the moments for this process. In geometric Brownian motion the expectation of the logarithm of the position of the particle scales linearly with time. In the fractional generalization, this scales as a sub-linear power law in time, similar to anomalous scaling of the mean square displacement in subdiffusion. In financial applications this could be observed as a sub-linear scaling in the logarithmic return.