2.1 The Financial Market

We consider a financial market in which transaction amounts are so small that we can consider they have no influence on the prices, and the market with no arbitrage, frictionless, and continuously open as well.

Without loss of generality, we assume that the financial market is composed of two kinds of assets: Cash and equities. For the sake of simplicity, we will only consider one equity asset which can indeed represent the index of a stock market.

The cash asset gives an instantaneous risk-free rate r(t), which diffusion is an Ho and Lee model[7]

where W^(t) is a standard Brownian motion, a(t) > 0 is a real-valued function and the constant b > 0. Here, we also can use the Vasicek model[8] to get completely similar results.

With an initial value of 1, the price S₀(t) of the cash asset is then given by the following equation:

Here r(t) is the stochastic interest rate described by (1).

The stock market index price S(t) at time t ≥ 0 can be described by the model

where u(t) is another positive real-valued function, and the constant v > 0 denotes the volatility rate of risk asset. Two Brownian motions satisfying E[W^(t),W~(t)]=ρ2⋅t,ρ2∈[−1,1] is the correlation coefficient of W^(t) and W~(t).

In the financial model (3), u(t) > 0 is actual a vital hypothesis, which shows that the development of economy is in good condition. In addition, u(t) > 0 means that the investment on stock can make more money than investing on cash asset, which is consistent with the economic principle of high-risk high-yield. Only when u(t) > 0, more people are willing to invest on equities.

2.2 The Surplus Process of Insurer

In this paper, we use the classical Cramer–Lunderburg model to describe the surplus process of insurer.

where N(t) is a homogeneous Poisson process with intensity λ, the claim sizes {Y_i, i ≥ 1} are independent and identically distributed positive random variables with E[Y_i] = μ and E[Yi2]=σ2. According to the expected value premium principle, the pure premium rate of insurer is c = (1 + θ) λ μ and θ > 0 is the safety loading of insurer.

The diffusion approximation of the compound poisson process d(∑i=1N(t)Yi) is given by

where W(t) is a standard Brownian motion and independent with W~(t). But E[W(t),W^(t)] = ρ₁ ⋅ t, ρ₁ ∈[−1, 1] is the correlation coefficient of W(t) and W^(t).

In order to avoid unbearable risk, the insurance company can buy proportional reinsurance to transfer part of the potential risk to reinsurance company. The insurer’s retention proportion is p ∈ [0, 1], the remaining 1 − p shall be borne by the reinsurer. η > 0 is the safety loading of reinsurer.

2.3 The Insurer’s Wealth Process with Promotion Budget

The proportion of wealth invested on stock asset denoted by π(t) which is a progressively measurable real function, X(t) with X(0) = x₀ represents the wealth of the insurer, then we have

Insurer not only seek to maximize their own profit by investment, but also surrender part of the profits to policy holder in the form of budgetary promotion resources, the direct way is to reduce the insurance premiums or hold promotional activities. Thus, the insurer took a fixed share of wealth as a promotional budget, for promotional activities or reduce the insurance premiums directly, to attract more customers. The promotion budget at a fixed experience share deducted from premium income at each time t.

We always take η = θ in this paper. The promotional budget deducted directly from the insurer’s premium income at time t, but the reinsurance premium calculation is based on the premium not deducted the promotional budget. Reinsurance premiums were paid according to the expected premium principle, However, the insurer’s premium income is obviously lower than the premium level based on the expected premium principle. Although the safe loading of insurer and reinsurer fixed at the same level, the insurance premium in this paper is also worthy of the phrase: “Cheap reinsurance” is no longer really cheap.

A strategy α = (p(t), π(t)) is said to be admissible, if ∀ t ∈ [0, T], α is ℱ(t) progressively measurable and E[∫0∞π2(t)X2(t)]<∞,p(t)∈[0,1],π(t)∈[0,+∞).Λ denotes the set of all admissible strategies and (8) has a unique (strong) solution. Suppose the insurer has a utility function U(x) which is strictly concave and continuously differentiable on (−∞, +∞) and aims to maximize the expected utility of his/her terminal wealth.

3.1 An Insurer with Promotion Budget Under the Stochastic Interest Rate

In this subsection, we consider the reinsurance-investment problem with stochastic interest rate. Following Merton, we use an expected utility maximization criterion and assume that the insurer’s objective is to maximize the expected utility from terminal wealth, where the utility U(x) is a power utility (constant relative risk aversion) function

the constant δ ∈ (0, 1) is the relative risk aversion coefficient of a risk averse insurer. This utility function plays a vital role in actuarial mathematics and insurance practice. For an admissible strategy α = (p(t), π (t)), the value function J^α(t, x, r) from state (x, r) at time t is defined as

and the optimal value function is given by

with the boundary condition

The goal of the insurer is to find an optimal strategy α^* = (p^*(t), π^*(t)) such that J^α*(t, x, r) = H(t, x, r), where p^*(t) is called the optimal reinsurance strategy and π^*(t) is called the optimal investment strategy.

For any H(t, x, r) ∈ C^{1, 2, 2}([0, T] × R₊ × R₊), define the generator is a variational operator as the following:

where H_t, H_x, H_r, H_xx, H_rr and H_xr denote the corresponding first and second-order partial derivatives with respect to (w.r.t.) the corresponding variables t, x, r, respectively. Similar to Fleming and Soner[9], we can derive the HJB equation for problem (10):

with the boundary condition (11).

In the remainder of this section, we try to find the explicit solutions to problem (10) with stochastic interest by stochastic control technique.

3.2 A Special Case of an Insurer with Promotion Budget Under a Constant Interest Rate

In this subsection, we investigate a special case that the interest rate r(t) is a constant r. All assumptions are the same as above except that r(t)≡ r throughout this part, thus we have

and

where r, v are positive constants, u(t) is a positive real-valued function. The surplus processes with claim and reinsurance are the same as (5) and (6), W(t) and W~(t) are independent standard Brownian motions. So we get the wealth process with promotion budget and investing in the financial market as follows:

we consider the utility U(x) is a power utility (constant relative risk aversion) function

to maximize the expected utility of terminal wealth, where the constant δ ∈ (0, 1) is the relative risk aversion coefficient of a risk averse insurer. For an admissible strategy α = (p(t), π (t)), the value function J^α(t, x, r) from state x at time t is defined as

and the optimal value function is given by

with the boundary condition H(T, x, r) = x^δ.

Applying the classical tools of stochastic optimal control, if the optimal value function H(t, x) ∈ C^1,2([0, T] × R), then H satisfies the following HJB equation:

According to the first order necessary condition of extreme, we obtain that

and

Conjecture of the solution has the following form

The derivatives of the conjecture with respect to t, x, respectively.

Plugging these derivatives into the HJB equation, we have

The solution of the above ordinary differential equation is

We obtain the value function

and the optimal strategy α^* = (p^*, π^*) as follows:

and