The first passage time density of Ornstein-Uhlenbeck process with continuous and impulsive excitations

The first passage time of the Ornstein-Uhlenbeck process plays a prototype role in various noise-induced escape problems. In order to calculate the first passage time density of the Ornstein-Uhlenbeck process modulated by continuous and impulsive periodic excitations using the second kind Volterra integral equation method, we adopt an approximation scheme of approaching Dirac delta function by alpha function to transform the involved discontinuous dynamical threshold into a smooth one. It is proven that the first passage time of the approximate model converges to the first passage time of the original model in probability as the approximation exponent alpha tends to infinity. For given parameters, our numerical realizations further demonstrate that good approximation effect can be achieved when the approximation exponent alpha is 10.