Fractional Brownian motion as a weak limit of Poisson shot noise processes—with applications to finance

We consider Poisson shot noise processes that are appropriate to model stock prices and provide an economic reason for long-range dependence in asset returns. Under a regular variation condition we show that our model converges weakly to a fractional Brownian motion. Whereas fractional Brownian motion allows for arbitrage, the shot noise process itself can be chosen arbitrage-free. Using the marked point process skeleton of the shot noise process we construct a corresponding equivalent martingale measure explicitly.