Stationary distribution of the surplus process in a risk model with a continuous type investment

Abstract

In this paper, we stochastically analyze the continuous time surplus process in a risk model which involves a continuous type investment. It is assumed that the investment of the surplus to other business is continuously made at a constant rate, while the surplus process stays over a given sufficient level. We obtain the stationary distribution of the surplus level and/or its moment generating function by forming martingales from the surplus process and applying the optional sampling theorem to the martingales and/or by establishing and solving an integro-differential equation for the distribution function of the surplus level.

An optimal continuous type investment policy for the surplus in a risk model

“…In this section, we summarize several interesting characteristics in Cho et al (2016), which are necessary to study the optimal investment policy. Cho et al (2016) decomposed {U(t), t ≥ 0} into two processes {U 1 (t), t ≥ 0} and {U 2 (t), t ≥ 0}.…”

“…Cho et al (2016) decomposed {U(t), t ≥ 0} into two processes {U 1 (t), t ≥ 0} and {U 2 (t), t ≥ 0}. U 1 (t) is formed by separating the periods where U(t) ≥ V from the original process and connecting them together.…”

“…Cho et al (2016) obtained the stationary distribution of the surplus level by forming martingales from the surplus process and applying the optional sampling theorems to the martingales. They also obtained the moment generating function of the stationary distribution by establishing and solving an integrodifferential equation for the distribution function of the surplus level.…”

“…In this paper, we study an optimal investment policy for the surplus in the risk model introduced by Cho et al (2016). After assigning, to the risk model, two costs which are the penalty per unit time while the level of the surplus being under V > 0 and the opportunity cost per unit time by keeping a unit amount of the surplus, we show that there exists an optimal investment rate a * > 0 which minimizes the long-run average cost per unit time.…”

“…In Section 2, we review some interesting characteristics of the risk model obtained by Cho et al (2016) which are needed for the optimization. In Section 3, we assign, to the risk model, two costs which are the penalty per unit time while the level of the surplus being under V > 0 and the opportunity cost per unit time by keeping a unit amount of the surplus, and calculate the long-run average cost per unit time.…”