_{1}

In this paper, we consider an insurer who wants to maximize its expected utility of terminal wealth by selecting optimal investment and risk control strategies. The insurer’s risk process is modeled by a jump-diffusion process and is negatively correlated with the returns of securities and derivatives in the financial market. In the financial model, a part of insurers’ wealth is invested into the financial market. Using a martingale approach, we obtain an explicit solution of optimal strategy for the insurer under logarithmic utility function.

In the past two decades, more and more attention has been paid to the problem of optimal investment in financial markets for an insurer. Indeed, this is a very important portfolio selection problem for the insurer from a point of finance theory. Merton (1969) [

For an insurer, since reinsurance is an important tool to manage its risk exposure, optimal reinsurance problem should be considered carefully. This issue implies that the insurer has to select reinsurance payout for certain financial objectives. The classical model for risk in the insurance literatures is Cramer-Lundberg model, which uses a compound Poisson process to measure risk. Based on the limiting process of compound Poisson process, Taksar (2000) [

In this paper, the model and optimization problem are different from others. Firstly, it is not total wealth of insurer invested, but a part of wealth invested. So in this model, we can obtain the optimal property of the total wealth. Secondly, different from Merton’s work, we use a jump-diffusion process to model an insurer’s risk. Lastly, we regulate the insurer’s risk by controlling the number of polices.

This paper is organized as follow. In Section 2, we formulate investment and risk control problem and describe the financial model and risk process model. The explicit solution of optimal investment and risk control strategy for logarithmic utility is derived in Section 3. In Section 4, we conduct a sensitive analysis. Conclusions of the research are reached in Section 5.

Suppose that there are two assets for investment in the financial market. One is a riskless asset with price process

respectively, where

For an insurer, most of its incomes come from writing insurance policies, and we denote the total outstanding number of policies at time t by

A classical risk model for claims is compound Poisson model, in which the claim for per policy is given by

where

where

For an insurer, it should be noted that it is impossible for an insurer to invest its total wealth. At time t, we denote

with initial wealth

Following Stein (2012, chapter 6) [

with initial wealth

In this model, as Zou (2014) [

Define the criterion function as

conditional expectation under probability measure P given

where u will be changed accordingly if the control we choose is

Firstly formulate the stochastic control problem. The problem in this model is to select an admissible control

Furthermore, we assume

As we know that for

that

Applying Ito’s formula to

where

Proposition 3.1. The associated optimal terminal wealth

Proof. According to (2.5) and the Doleans-Dade exponential formula, we have that

From (3.4), to prove

ty 1. For any

Next, we will use the martingale method to get an optimal control for the SDE (2.5). To begin with, we give two important Lemmas. Lemma 3.1 gives the condition that optimal control must satisfy while Lemma 3.2 is a generalized version of martingale representation theorem.

Lemma 3.1. (Wang (2007) [

stant over all admissible controls, then

Lemma 3.2. (Wang (2007) [

Now for the value function

Step 1: Conjecture candidates for optimal control

Following the definition in Zou (2014) [

for any stopping time

From the SDE (2.4), we have

From the above expression of X and Lemma 3.1, for all admissible strategies, we have

is constant.

Define

According to Lemma 3.2, there exists a predictable process

From the Doleans-Dade exponential formula, we can obtain that

Through Girsanov’s Theorem,

For a stopping time

control. By substituting this control into (3.8), we have

So

Let

is a Q-martingale, which in turn yields that

From the SDE (2.5), we can derive to

Comparing the

By substituting (3.15) into (3.11) and (3.13), we have that

which the coefficients define as

With the conditions above, solve the Equation (3.17) to get

where

According to the (3.19), we can derive

Then we choose that

so we can have

Step 2: Verify that Z_{T} defined by (3.10) is consistent with its definition, for

First rewrite (3.14) as

where

Then substituting (3.15) back into (3.10), we can obtain

is constant.

According to the (3.6), Z is a P-martingale and

so Z given by (3.10) with

Step 3: Prove that

For any

From the Equation (3.13), the above

which make possible for us to conclude that the family

In this part, we analyze the influence of the market parameters on the optimal control. To simplify the analysis, we assume that the coefficients are constant in the financial market and the parameters are given in

Firstly, we fix

Secondly, we fix

a | b | p | r | |||
---|---|---|---|---|---|---|

0.08 | 0.1 | 0.15 | 0.01 | 0.05 | 0.25 | 0.1 |

In our model, the insurer’s risk process obeys a jump-diffusion process, and it is not total its capital to invest but a part of wealth to invest in financial market. Besides, we consider that an insurer wants to maximize its expected utility of terminal wealth by selecting optimal control. According to the sensitive analysis, we know that the optimal total proportion of wealth invested

The limitation in this paper is that the liability is an average for per policy, which conflicts with the fact that the premium for per policy changes all the time for different insurance. Therefore we can research the liability described with a linear function in the model. Besides, the model proposed in the paper can be further explored in another ways as well, something we plan to do in future work.

The author thanks to the editor and the reviewers for their thoughtful comments that help the author improve a prior version of this article.

Tingyun Wang, (2016) Optimal Investment and Risk Control Strategy for an Insurer under the Framework of Expected Logarithmic Utility. Open Journal of Statistics,06,286-294. doi: 10.4236/ojs.2016.62024