In the risk process that is perturbed by diffusion, the surplus process of an insurance portfolio is given by
where is the initial surplus, is the positive constant premium income
rate, is the aggregate claims process, in which
is the claim number process (denoting the number of claims up to time t), and the interarrival times is a sequence of positive random variables. is a sequence of nonnegative independent identically distributed (i.i.d.) random variables with distribution function and density function, is a standard Brownian motion that is independent of the aggregate claims process, is a positive constant.
2. Improved Risk Model
In this paper, it is assumed that the claim occurrence process to be of the following type: If a claim is larger than a random variable, then the time until the next claim is exponentially distributed with rate, otherwise it is exponentially distributed with rate. The quantities are assumed to be i.i.d. random variables with distribution function. Assuming that
which is the net profit condition.
In the daily operation of insurance company, in addition to the premium income and claim to the operation of spending has a great influence on the outside, and there is also a factor that interest rates should not be neglected. As in  , this paper assume that the risk model Equation (1) is invested in a stochastic interest process which is assumed to be a geometric Brownian motion, where r and σ2 are positive constants, and is a standard Brownian motion independent of. Let denote the surplus of the insurer at time t under this investment assumption. Thus,
Denote T to be the ruin time (the first time that the surplus becomes negative), i.e.,
This article is interested in the expected discounted penalty (Gerber-Shiu) function:
where is the indicator function, is the force of interest and is a nonnegative function of and satisfies.
Furthermore, let be the time when the first claim occurs, and random variable being exponentially distributed with rate. Assuming that
3. Integro-Differential Equations for
In this section, a system of integro-differential equations with initial value conditions satisfied by the Gerber-Shiu function is derived.
Lemma 3.1 Let for. For define the hitting time. Then, for, it can be concluded that
Proof is a reflecting diffusion with generator
acting on functions satisfying the reflecting boundary condition.
and for t > 0, then, according to Itô’s formula is a local mar-
tingale. Using the separation variable technique, we find that
is a solution, where
is a solution of
Using the initial condition for, we get, consequently
Applying the Optional Stopping Theorem, it follows that
This ends the proof of Lemma 3.1.
Similarly, the following lemma can also be obtained.
Lemma 3.2 Let for. For define the hitting time. Then, for, it can be concluded that
Theorem 3.1 Assuming that is second order continuously differentiable functions in u, then satisfies the following integro-differential equation
with the initial value conditions
Proof Let be the time when the first claim occurs which exponentially distributed with rate. Consider the risk process defined by Equation (2) in an infinitesimal time interval. There are three possible cases in as follows.
1) There are no claims in with probability, thus;
2) There is exactly one claim in with probability. According to different of the claim amount, there are three possible cases in this case as follows.
a) The amount of the claim, i.e., ruin does not occur, and thus;
b) The amount of the claim, i.e., ruin occurs due to the claim;
c) The amount of the claim, i.e., ruin occurs due to oscillation (observe that the probability that this case occurs is zero).
3) There is more than one claim in with probability.
Thus, considering the three cases above and noting that is a strong Markov process, we have
By Taylor expansion, we have, thus Equation (11) becomes
Then, by Itô’s formula we have
Therefore, by dividing t on both sides of Equation (12), letting, using Equation (13), we obtain Equation (9), and similarly we can obtain Equation (10).
The condition follows from the oscillating nature of the sample paths of. Now, we prove.
For all, let,. Then, by the strong property of, it can be concluded that
According to Lemma 3.1, it can be concluded that
Thus, , and correspondingly. Similar results can be derived for.
And for all, let, , according to Lemma 3.2 we obtain, thus.
This ends the proof of Theorem 3.1.
4. Differential Equations for
Let and in Equation (3), correspondingly the expected discounted penalty function turns into the ultimate ruin probability.
and is exponentially distributed with rate. Then, we get the following theorem.
Theorem 4.1 Assuming that is second order continuously differentiable functions in u, then satisfies the following integro-differential equation
with the initial value conditions
Proof According to Equation (9), it can be concluded that
By taking the derivative with respect to u on both sides of the above formula, and after some careful calculations, we obtain Equation (14). And similarly we can prove that Equation (15) holds. This ends the proof of Theorem 4.1.
In this paper, we consider a jump-diffusion risk process compounded by a geometric Brownian motion with dependence between claim sizes and claim intervals. We derive the integro-differential equations for the Gerber-Shiu functions and the ultimate ruin probability by using the martingale measure. Further studies are needed for the numerical solution of Equations (9), (10), (14) and (15). The results derived in this paper can be generalized to similar dependence ruin models.
This research was supported by the National Natural Science Foundation of China (No. 11601036), the Natural Science Foundation of Shandong (No. ZR2014GQ005) and the Natural Science Foundation of Binzhou University (No. 2016Y14).
 Dufresne, F. and Gerber, H.U. (1991) Risk Theory for the Compound Poisson Process That Is Perturbed by Diffusion. Insurance: Mathematics and Economics, 10, 51-59.
 Gerber, H.U. and Landry, B. (1998) On the Discounted Penalty at Ruin in a Jump-Diffusion and the Perpetual Put Option. Insurance: Mathematics and Economics, 22, 263-276.
 Tsai, C.C.L. and Willmot, G.E. (2002) A Generalized Defective Renewal Equation for the Surplus Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 30, 51-66.
 Tsai, C.C.L. and Willmot, G.E. (2002) On the Moments of the Surplus Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 31, 327-350.
 Chiu, S.N. and Yin, C.C. (2003) The Time of Ruin, the Surplus Prior to Ruin and the Deficit at Ruin for the Classical Risk Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 33, 59-66.
 Boudreault, M., Cossette, H., Landriault, D. and Marceau, E. (2006) On a Risk Model with Dependence between Interclaim Arrivals and Claim Sizes. Scandinavian Actuarial Journal, 5, 265-285.
 Li, J. (2012) Asymptotics in a Time-Dependent Renewal Risk Model with Stochastic Return. Journal of Mathematical Analysis and Applications, 387, 1009-1023.
 Yong, W. and Xiang, H. (2012) Differential Equations for Ruin Probability in a Special Risk Model with FGM Copula for the Claim Size and the Inter-Claim Time. Journal of Inequalities and Applications, 2012, 156.
 Fu, K.A. and Ng, C.Y.A. (2014) Asymptotics for the Ruin Probability of a Time-Dependent Renewal Risk Model with Geometric Lévy Process Investment Returns and Dominatedly-Varying-Tailed Claims. Insurance Mathematics & Economics, 56, 80-87.
 Zhang, J. and Xiao, Q. (2015 ) Optimal Investment of a Time-Dependent Renewal Risk Model with Stochastic Return. Journal of Inequalities and Applications, 2015,181.
 Cai, J. and Xu, C. (2005) On the Decomposition of the Ruin Probability for a Jump-Diffusion Surplus Process Compounded by a Geometric Brownian Motion. North American Actuarial Journal, 10, 120-132.