Diffusion Approximation of a Poisson Model for Cumulative Excess Returns

The paper analyzes the regressive equation for cumulative excess returns with residual conditional variances as a GARCH(1,1) process and statistical uncertainty as an AR(1) Gaussian process with correlation parameter ρ. Under assumption that the lengths of time intervals between transactions are independent exponentially distributed random variables with sufficiently small mean h, we derive diffusion approximation equations. The continuous time limit equation allows concluding that a stationary conditional variance exists. Moreover, we derive this stationary distribution as inverse gamma distribution and analyze the dependence of this distribution on the correlation parameter