On the stop-loss transform and order for the surplus process perturbed by diffusion

In this paper we extend the results obtained in [Chang, Y., Pai, J.S., 2003. On the nth stop-loss transform order of ruin probability. Insurance: Mathematics and Economics 32, 51-60] to ones based on the surplus process perturbed by diffusion. We first study the stop-loss transforms of the random variables for the maximal aggregate loss. Then we propose theorems for the order of stop-loss transforms of ruin probabilities for two surplus processes perturbed by diffusion associated with different claim size random variables of same mean. We also find that an exponentially distributed random variable is less, in the meaning of the first stop-loss order, than a mixture of two exponentials with the same mean. Finally, a numerical example is given to illustrate these theorems.