Asymptotics of Parisian ruin of Brownian motion risk model over an infinite-time horizon

Let B(t), t ≥ 0 be a standard Brownian motion. In this paper, we derive the asymptotics of the probability of Parisian ruin over an infinite time horizon for the following risk process (Formula presented.) where u ≥ 0 is the initial reserve, δ ≥ 0 is the force of interest, c > 0 is the rate of premium and σ > 0 is a volatility factor. It turns out that the Parisian ruin probability decays exponentially as u tends to infinity and is a decreasing function of the force of interest for u large. Moreover, we obtain the approximations of Parisian ruin time.