Maximum likelihood estimation of stochastic differential equations with random effects driven by fractional Brownian motion

•We use random effects to explain the dependence between variables and the variation of variables over time. We consider stochastic differential equations with random effects driven by fractional Brownian motion with long memory characteristics.•We make use of the maximum likelihood estimation to obtain the exact likelihood function.•We calculate out the estimates of the unknown mean and variance assuming that obeys Gaussian distribution.•Our work provides a good solution for the long memory phenomenon in the fields of biomedicine and physics. Stochastic differential equations and stochastic dynamics are good models to describe stochastic phenomena in real world. In this paper, we study N independent stochastic processes Xi(t) with real entries and the processes are determined by the stochastic differential equations with drift term relying on some random effects. We obtain the Girsanov-type formula of the stochastic differential equation driven by Fractional Brownian Motion through kernel transformation. Under some assumptions of the random effect, we estimate the parameter estimators by the maximum likelihood estimation and give some numerical simulations for the discrete observations. Results show that for the different H, the parameter estimator is closer to the true value as the amount of data increases.