Numerical computation of Gerber-Shiu function for insurance surplus process with additional investment

This paper studies the Gerber-Shiu function for the insurance surplus process with additional investment under the Bachelier model. The Gerber-Shiu function allows us to study the moments of the time of ruin, which is the first time that the surplus is negative. First, we use the martingale theory in deriving the integro-differential equation of the Gerber-Shiu function. Then, we give the exact solution of the ruin probability in case the amount of claims follows the exponential distribution. Under a general distribution case, we propose a numerical method of the Gerber-Shiu function using the finite differential method based on the integro-differential equation. Then, numeric illustrations are provided to study the effect of the parameters on the Gerber-Shiu function.