# Asymptotic Behavior of an Optimal Investment-Reinsurance Problem with General Utility Functions

Authors: Yan Zhang; Peibiao Zhao
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

It is usually extremely difficult to derive an analytical solution to the Hamilton-Jacobi-Bellman (HJB) equation for some optimal control problems under general utility functions, while this paper provides a dual control method to solve the HJB equation for an optimal investment-reinsurance problem with general utility functions. We first change the HJB equation into its dual one. By solving the solution to the dual HJB equation, we derive a classical solution to the primal HJB equation and obtain the expression of the optimal investment-reinsurance strategy, then some examples, which would be difficult to solve with the standard method, are given to demonstrate the usefulness of our dual control method. Moreover, applying the dual method along with the partial differential equation method, we verify the asymptotic behavior of the optimal control strategy, which means that the optimal investment-reinsurance strategy with the general utility functions converges to that with the power utility function at any level of the initial wealth.


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## Body

## 1. Instruction

Reinsurance is an effective way for insurers to avoid risks in a fiercely competitive market, while the insurers would like to invest in the financial market to gain more profits and improve their solvency so as to reduce claim risks. How can the optimal reinsurance and/or investment strategy be found so as to balance their risk and profit under a certain objective criterion? This is of great interest to them. In fact, optimal reinsurance and/or investment problems have been extensively investigated in the past few decades. A number of scholars considered these optimal control problems for various risk models under different optimization objectives. Bai and Zhang [1] studied an optimal mean-variance problem; then Li et al. [2] and Bian et al. [3] investigated the optimal time-consistent reinsurance-investment problem under the mean-variance criterion. Under the objective of minimizing the ruin probability, Schmidli [4]; Promislow and Young [5]; and Liang et al. [6] obtained the optimal reinsurance and/or investment strategy.Besides the above-mentioned optimization objectives, maximizing the expected utility of the terminal wealth is another important optimization objective in the literature due to utility functions characterizing the insurers’ risk preference in economics. Lin and Li [7] and Liang et al. [8] considered exponential utility in the optimal control problem and aimed to maximize the expected exponential utility of the insurer’s terminal wealth and obtained the corresponding optimal reinsurance and investment strategy; moreover, Sun et al. [9] and Zhang and Zhao [10] took ambiguity into account and discussed the robust optimal control problem under the similar criterion. Irgens and Paulsen [11] considered power utility function, logarithmic utility function, and exponential utility function and studied optimal control problem under the objective of maximizing expected utility of assets at a terminal time, respectively.In reality, insurers will choose different utility functions to describe their different degree of risk preference. Hyperbolic Absolute Risk Aversion (HARA) utility function, which contains exponential utility, power utility, and logarithmic utility, has drawn a great deal of attention and is taken into consideration in the optimal control problems. Zhang et al. [12] introduced HARA utility into an optimal reinsurance-investment problem.For the optimal reinsurance-investment problems, in order to obtain the optimal control strategy, most of the abovementioned research papers usually applied the dynamic programming to establish the HJB equation. To solve the HJB equation, a standard method in the literature is to guess the solution, but the analytical solution only can be derived when the utility function is of some specific forms, such as the power utility, exponential utility, and HARA utility and so on. However, if the utility functions chosen by insurers are more complex, specially, if they are more general utility functions, the solution form of the HJB equation may be difficult to guess; as a result, the standard method may be ineffective. Federico et al. [13] dealt with an investment-consumption problem by a dual approach. Bian and Zheng [14] provided a dual method to solve an optimal investment problem with general utility functions and investigated the turnpike property and convergence rate; moreover, Bian and Zheng [15] discussed the turnpike property for optimal investment and consumption problems. Later, Ma et al. [16] employed a monotone numerical scheme to investigate Merton problem under general utility functions.As far as we know, few research papers investigated optimal reinsurance-investment problems with general utility functions, which is just our main interest in this paper. Inspired by Bian and Zheng [14], we consider a reinsurance-investment model and aim to find the optimal control strategy with general utility functions under the objective of maximizing the expected utility of the terminal wealth. We first adopt a dual control method to show the existence of the classical solution to the HJB equation and obtain the expression of this solution in terms of the dual function. Based on the dual control method proposed here, which may be applied to solve many other utility maximization problems, we do not need to guess the solution form of the primal HJB equation while we can find the solution directly and achieve the expression of the optimal investment-reinsurance strategy with general form utility functions. Some examples, which would be difficult with the standard method mentioned above, are also given to illustrate the usefulness of our method. Moreover, the asymptotic behavior is verified, which implies that the limit of the optimal control strategy with the general utility functions is equal to that with the power utility function.This paper proceeds as follows. We give the model formulation in Section2. Section 3 discusses the optimal reinsurance-investment strategy with power utility which plays a key role in verifying the asymptotic behavior of the optimal reinsurance-investment strategy with general utility functions. Section 4 applies the dual control method to solve the HJB equation and achieves the representations of the optimal strategy. In addition, we provide the verification theorem and give some examples to illustrate the usefulness of our method. Section 5 verifies the asymptotic behavior of our optimal strategy with the general utility functions. We conclude this paper in Section 6.


## 2. Model Formulation

Let (Ω,F,P) be a complete probability space with filtration Ft,t∈0,T, where Ft stands for the information of the market available up to time t and positive finite constant T represents the terminal time. All processes in this paper are assumed to be well-defined and adapted processes in this space, and the risky asset is assumed to be traded continuously, without taxes or transaction costs.The reserve processRt of the insurer is described by(1)dRt=cdt−d∑i=1NtXi,where the positive constant c is the premium rate. Xi is the size of the ith claim and Xi,i≥1 are assumed to be i.i.d. positive random variables with common distribution function FX⋅ and EXi=μX>0. The claim number process Nt,t≥0 is assumed to be a Poisson process with intensity λ>0 representing the number of claims occurring in time interval 0,t. Moreover Xi,i≥1, and Nt,t≥0 are mutually independent. In this paper, the insurer’s premium rate is calculated according to the expected value principle, i.e., c=λμX1+θ, where θ>0 is the safety loadings.For eacht∈0,T, the insurer purchases reinsurance with the retention level qt∈0,+∞ in order to transfer claim risks. When qt∈0,1, it represents a proportional reinsurance cover. That is to say, for a claim Xi, the insurer pays qtXi at time t while the reinsurer pays 1−qtXi, and meanwhile, a premium χq should be paid to the reinsurer for sharing claim risks, where(2)χq=1+η1−qtλμX.Andη≥θ is the reinsurer’s safety loading. If qt∈1,+∞, it stands for acquiring new business (see [17]).With reinsurance, the wealth process of the insurer is(3)dXqt=λμXθ−η+λμXqt1+ηdt−qtd∑i=1NtXi.According to Grandell [18] (pages 15–17), the wealth process (3) can be approximated by the following diffusion model:(4)dXqt=λμXθ−η+λημXqtdt+λσXqtdW0t,where σX2=EXi2,W0tt≥0 is a standard Ftt≥0-adapted Brownian motion, independent of Xi,i≥1 and Nt,t≥0.The insurer is allowed to invest in a risk-free asset (bond or bank account) whose price process satisfies(5)dBt=rBtdt,B0=1,and a risky asset (stock) whose price process is described by the geometric Brownian motion(6)dSt=Stbdt+σdW1t,t∈0,T,S0=S0,where r>0 is the interest rate of the bond, b>r is the expected instantaneous rate of the risky asset, and σ>0 represents the volatility of the risky asset price. W1tt≥0 is a standard Ftt≥0-adapted Brownian motion, independent of W0tt≥0Xi,i≥1 and Nt,t≥0.The insurer, starting from an initial capitalx0 at time 0, is allowed to dynamically purchase proportional reinsurance and invest in the financial market described above. A trading strategy is denoted by two-dimensional stochastic process qt,πt, where πt is the total amount of money invested in the risky asset at time t. The amount of money invested in the risk-free asset at time t is Xt−πt, where Xt is the wealth process associated with strategy qt,πt. In this paper, short-selling is allowed and the insurer is allowed to borrow for the investment in the risky asset, i.e., πt∈R, where R is the set of real number. We call qt,πt an admissible strategy if it is Ft-progressively measurable and satisfies(7)qt≥0,E∫0tqs2ds<∞,andE∫0tπs2ds<∞,for all t≥0. Denote the set of all admissible strategies by Π.Then the wealth processXt is given by(8)dXt=rXt+b−rπt+λμXηqt+λμXθ−ηdt+σπtdW1t+λσXqtdW0t.X0=x0.Assume now that the insurer is interested in maximizing the expected utility of terminal wealth, say at timeT. Let ux be a continuous utility function with u′>0 and u″<0. The objective function is(9)Vt,x=supq,π∈∏EuXT|Xt=x.Our objective is to find an optimal strategyq∗,π∗ and the value function Vt,x. Let C1,2 be the space of ϕt,x such that ϕ and its partial derivatives ϕt,ϕx,ϕxx are continuous on 0,T×R+. To solve the above problem, we use the dynamic programming approach described in Fleming and Soner [19]. From standard arguments, we see that if the value function V∈C1,2, then Vt,x satisfies the following HJB equation:(10)Vt+supq,π∈Πrx+b−rπt+λμXηqt+λμXθ−ηVx+12σ2πt2+λσX2qt2Vxx=0,for t<T with the boundary condition(11)VT,x=ux.


## 3. Optimal Investment-Reinsurance Strategy with Power Utility

Since the optimal investment-reinsurance strategy with power utility plays a key role in verifying the asymptotic behavior of the optimal strategy with general utility functions, in this section, we first derive the optimal control strategy with power utility for problem (9).The power utility function is given by(12)ux=xkk,0<k<1.Theorem 1.
Let(13)x1∗=λμXη−θr1−e−rT,t1∗=T+1rln1−rxλμXη−θ.For problem (9) under the power utility (12), the optimal investment strategy is given by(14)πpow∗t=b−rσ21−kx+λμXθ−ηr1−e−rT−t,t∈0,T.And the optimal reinsurance strategyqpow∗t is given as follows:(1)
ifx>x1∗, for t∈0,T, the optimal reinsurance strategy is given by(15)qpow∗t=ημXσX21−kx+λμXθ−ηr1−e−rT−t;(2)
ifx≤x1∗, for t1∗<t≤T, (15) is the optimal reinsurance strategy; for 0≤t≤t1∗, we have qpow∗t=0.Proof.
If (10) attains its maximum, then q and π must satisfy the following expressions:(16)q∗t=−ημXσX2VxVxx,π∗t=−b−rσ2VxVxx.
Substituting (16) into (10), we derive(17)Vt+rx+λμXθ−ηVx−12λμX2η2σX2+b−r2σ2Vx2Vxx=0.
To solve (17), we try to construct a solution of the following form:(18)Vt,x=gtx−htkk,where gt and ht are two suitable functions such that (18) is a solution to (9). And the boundary condition (11) implies that gT=1 and hT=0.
From (18), we get(19)Vt=Vt,xg′tgt−kh′tx−ht,Vx=Vt,xkx−ht,Vxx=Vt,xkk−1x−ht2.
Putting (19) into (17) and rearranging terms, we get(20)kx−htrht+λμXθ−η−h′t+g′tgt+rk−12kk−1λμX2η2σX2+b−r2σ2=0.
Eliminating the dependence onh, we get two equations as follows:(21)rht+λμXθ−η−h′t=0,(22)g′tgt+rk−12kk−1λμX2η2σX2+b−r2σ2=0.
According togT=1 and hT=0, we obtain the solutions to (21) and (22) given by(23)ht=λμXθ−ηre−rT−t−1,(24)gt=exp−12λμX2η2σX2+b−r2σ2−rkT−t.
From (16), (19), (23), and (24), we arrive at the optimal investment strategy πpow∗t and the optimal reinsurance strategy qpow∗t given by Theorem 1. This completed the proof of the Theorem 1.
Applying the standard method in Fleming and Soner [19], we have the following verification theorem.Theorem 2.
LetWt,x∈C1,2 be a classical solution to (10) with power utility and satisfy the boundary condition (11); then we have Vt,x=Wt,x. Furthermore, qqow∗t,πqow∗t∈Π makes the HJB equation (10) hold for all t,x∈0,T×R+, and then the strategy qqow∗t,X∗t,πqow∗t,X∗t is indeed optimal, where X∗t is the reserve process under the optimal strategy.Remark 1.
Compared with Zhang et al. [12], the optimal reinsurance-investment strategy with power utility in Theorem 1 is consistent with Corollary 1 in Zhang et al. [12].


## 4. Optimal Investment-Reinsurance Strategy with General Utility Functions

The safety loading of reinsurerη is usually higher than that of the insurer θ, i.e., η>θ. In fact, when the insurer has a promotional budget, the insurer’s actual premium is lower than that charged from the reinsurer, which is for the reason that promotional budget has been deducted from the insurer’s actual premium income, so it is not fair to assume that η>θ, and in this case, it is reasonable to consider θ=η. Hence this section discusses the optimal control problem (9), with θ=η, and aims to investigate the optimal strategy with general utility functions and then we discuss the asymptotic behavior of the optimal control strategy.Whenθ=η, the wealth process (8) can be rewritten as(25)dXt=rXt+b−rπt+λμXηqtdt+σπtdW1t+λσXqtdW0t,X0=x0,And the value function defined by (9) with θ=η satisfies the following HJB equation:(26)Vt+supq,π∈∏rx+b−rπt+λμXηqtVx+12σ2πt2+λσX2qt2Vxx=0,with the boundary condition VT,x=ux. According to (16), it leads to(27)Vt+rxVx−12λμX2η2σX2+b−r2σ2Vx2Vxx=0.Suppose that the utility functionux satisfies u0=0,u∞=∞, and(28)ux≤C11+xk,x≥0,for some constants C1>0 and 0<k<1. Since ux is continuous, increasing, and concave on 0,∞, then we have u′0=∞ and u′∞=0. Moreover, the dual function of ux is defined by(29)u^y=supx≥0ux−yx,for y≥0, then u^y is a continuous, decreasing, and convex function on 0,∞, and u^0=∞,u^∞=0,u^′0=−∞,u^′∞=0, and moreover, according to (28) and (29), it is easy to obtain that u^y satisfies(30)u^y≤C21+yp,y>0,for some constants C2>0 and p=k/k−1<0.


### 4.1. Existence and Dual Representation of Solutions

In order to use the dual theory to solve the optimal control problem (9) with general utility functions, we need first to construct a dual minimization problem. For all t∈0,T and any process Xt satisfying (25), according to Bian and Zheng [14], a dual process Yt can be selected as a nonnegative supermartingale process which makes XtYt be a supermartingale. The set of such process Yt is denoted as(31)Ωy=Yt:Yt≥0,Y0=y,XtYtisasuper−martingale, for allt∈0,T.Then the dual minimum problem can be characterized byinfY∈ΩyEu^YT.Lemma 1.
The dual problem of the primal problem (9) can be expressed as(32)infY∈Ω˜yEu^YT,where for t∈0,T,∀γt≥0 and(33)Ω¯y=Yt:Y0=yanddYt=Yt−rdt−b−rσdW1t−γt+λμXησXdW0t.Proof.
According to Doob-Mayer decomposition theorem, a non-negative super-martingaleYt can be decomposed into(34)Yt=yε¯−αtW1t−βtW0tDt,where ε¯ is Doleans-Dade coefficient, Dt is a positive non-increasing process, and D0=1. Since u^ is a decreasing function and Dt≤1, the dual problem can be characterized as infY^Eu^YT, where Y^ satisfies the following SDE:(35)dYt=Yt−αtdW1t−βtdW0t,Y0=y.
According to (25), let(36)dYt=Yt−rdt−αtdW1t−βtdW0t,Y0=y.
ByIto^ formula, we get(37)dXtYt=Ytπtb−r−σαt+λqtλμXη−σXβtdt+YtσπtdW1t+YtλσXqtdW0t+XtYt−αtdW1t−βtdW0t.
SinceXY is a super-martingale, for all t, it means that(38)πtb−r−σαt+λqtλμXη−σXβt≤0.
Sinceπt∈R, we consider a sufficient condition, i.e., b−r−σαt=0 and ∀γt≥0 for all t∈0,T, which can ensure (38) holds. Denote(39)αt=b−rσ,βt=λμXησX+γtσX,
and then we can see that (38) holds.
LetJt,y be the value function of this dual problem for y≥0,t∈0,T, and then we have(40)Jt,y=infy∈Ω˜yEu^YT|Yt=y,y≥0,t∈0,T.
The dual HJB equation is given by(41)Jt−ryJy+infγt≥012b−rσ2+12λμXησX+γtσX2y2Jyy=0,with the boundary condition(42)JT,y=u^y.
It is easy to verify thatJt,⋅ is convex for fixed t∈0,T. The dual HJB (41) has the unique minimizer when γt=0, and therefore (41) can be rewritten as(43)Jt−ryJy+ϑ22y2Jyy=0,where(44)ϑ=b−rσ2+λμXησX2.
Equation (43) is a linear PDE whose solution can be expressed by(45)Jt,y=∫−∞+∞Kt,lny,T,ξu^eξdξ,where(46)Kt,z,s,ξ=14πμt,sexp−14μt,s−rs−t−μt,s+z−ξ2.
Andμt,s=ϑ2/2s−t for 0≤t<s≤T and z,ξ∈R.Lemma 2.
Jt,y is continuous on 0,T×0,∞, and is a classical solution to (43), satisfying(47)0≤Jt,y≤C31+yp,
for some constantC3 depending on T. Furthermore, if ϑ2−2r≥0, for all fixed t∈0,T,Jt,y is strictly decreasing, is strictly convex with respect to y, and has the following limits properties:(48)Jt,0=∞,Jt,∞=0,Jyt,0=−∞,Jyt,∞=0.Proof.
Firstly, from (45), we can rewrite Jt,y as(49)Jt,y=∫−∞+∞Kt,ω,T,0u^ye−ωdω.
Notice that∫−∞+∞Kt,ω,T,0dω=1 and according to (30), we get u^ye−ω≤C21+ype−pω, and thus(50)Jt,y≤∫−∞+∞Kt,ω,T,0C21+ype−pωdω=C2+ype−p−rT−t.
LetC3=maxC2,e−p−rT−t, and we have 0≤Jt,y≤C31+yp for y>0,t∈0,T.
Secondly, sinceu^ is a continuous, decreasing, and convex function and K is a fundamental solution to (43), Jt,y is continuous on 0,T×0,∞, and decreasing and convex in y for fixed t∈0,T, that is Jyt,y≤0 and Jyyt,y≥0. Let ft,y=Jyyt,y. Differentiating y twice with respect to both sides of the dual HJB equation (43), it is not difficult to see that ft,y is a non-negative classical solution to the following equation when ϑ2−2r≥0,(51)ft+ϑ2−2rf+2ϑ2−ryfy+ϑ22y2fyy=0,y>0,t∈0,T.
Ifft0,y0=0 for some point t0,y0,t0<T, then t0,y0 is a minimum point of ft,y. The strong maximum principle leads to ft,y=Jyyt,y=0 for all t,y∈t0,T×0,∞, which means that Jt,y is linear in y for fixed t∈t0,T. In particular, u^y is linear, which is a contradiction with the given conditions. Thus we have Jyyt,y>0 for fixed t∈t0,T. Similarly, we deduce that Jyt,y<0 for fixed t∈0,T.
Finally, we will prove the limiting properties (48). To show Jt,0=∞, notice that u^ye−ω is non-negative and increasing as y⟶0. Applying the Monotone Convergence Theorem (MCT) yields(52)Jt,0=limy⟶0Jt,y=limy⟶0∫−∞+∞Kt,ω,T,0u^ye−ωdω=∫−∞+∞Kt,ω,T,0limy⟶0u^ye−ωdω=∫−∞+∞Kt,ω,T,0u^0dω=u^0=∞.
To proveJt,∞=0. when y>1 we have yp<1, and then u^ye−ω≤C21+ype−pω≤C21+e−pω. Based on the Dominated Convergence Theorem (DCT), we have(53)Jt,∞=limy⟶∞Jt,y=limy⟶∞∫−∞+∞Kt,ω,T,0u^ye−ωdω=∫−∞+∞Kt,ω,T,0limy⟶∞u^ye−ωdω=∫−∞+∞Kt,ω,T,0u^∞dω=0.
To proveJyt,0=−∞, let gω,y,h=u^ye−ω+he−ω−u^ye−ω/h, and we get(54)Jt,y+h−Jt,yh=∫−∞+∞Kt,ω,T,0gω,y,hdω.
Sinceu^ is decreasing and convex, we have gω,y,h is decreasing as h⟶0, and for all h≥0, we have gω,y,h≤0. Employing the MCT arrives at(55)Jyt,y=limh⟶0Jt,y+h−Jt,yh=∫−∞+∞Kt,ω,T,0limh⟶0gω,y,hdω=∫−∞+∞Kt,ω,T,0e−ωu^′ye−ωdω.
Sinceu^ is convex, we have u^′ is increasing and u^′ye−ω≤u^′∞. When y converges monotonically decreasing to 0, the function u^′ye−ω is decreasing. Using the MCT achieves(56)Jyt,0=limy⟶0Jyt,y=∫−∞+∞Kt,ω,T,0e−ωlimy⟶0u^′ye−ωdω=e−rT−tu^′0=−∞.
Notice that, fory≥1,(57)0≤u^ye−ωy≤C21+e−pωy≤C21+e−pω.
Applying the DCT yields(58)Jyt,∞=limy⟶∞Jyt,y=limy⟶∞Jt,yy=∫−∞+∞Kt,ω,T,0limy⟶∞u^ye−ωydω=0.Theorem 3.
Assume that the utility functionux satisfies (28). Then there exists a function Ht,x∈C00,T×0,∞ which is a classical solution to the HJB equation (26) and Ht,x can be expressed by(59)Ht,x=Jt,yt,x+xyt,x,where yt,x∈C1,∞0,T×0,∞ satisfies(60)Jyt,yt,x+x=0.Furthermore, for fixedt∈0,T,Ht,⋅ is strictly concave and strictly increasing and satisfies HT,x=ux and0≤Ht,x≤C41+xk for constants C4>0 and 0<k<1. Furthermore, the maximum of the HJB (26) is arrived at(61)q∗t,x=−ημXσX2Hxt,xHxxt,x,π∗t,x=−b−rσ2Hxt,xHxxt,x.Proof.
Fort,x∈0,T×0,∞ define(62)Ht,x=infy>0Jt,y+xy.
Ifx>0, then the minimum is reached at a point y satisfying (60). Let yt,⋅ be an inverse function of −Jyt,y, and then we get(63)−Jyt,yt,x=x,yt,−Jyt,y=y,for fixed t∈0,T. Since J∈C1,∞0,T×0,∞ and Jyyt,y>0,x<−Jyt,0=∞, according to the implicit function theorem, we have yt,x∈C1,∞0,T×0,∞ and Ht,x=Jt,yt,x+xyt,x. Hence,(64)Hxxt,x=−1Jyyt,yt,x<0,
which meansHt,⋅ is strictly convex and strictly increasing in x for fixed t∈0,T.
In what follows, we will prove thatH is the classical solution to the HJB (26). Because Jt,y is a classical solution of dual HJB (43), by the relationship between J and H, we have from (43)(65)Ht+rxHx−ϑ22Hx2Hxx=0.
For a given numbera and t∈0,T, let(66)Fq,π=12λσX2q2+aλμXηq+12σ2π2+ab−rπ,over q,π∈0,∞×0,∞. Both ∂F/∂q=0 and ∂F/∂π=0 yield(67)q^=−ημXσX2a,π^=−b−rσ2a.
The Hessian matrix ofFq,π can be written as(68)∂2F∂q2∂2F∂q∂π∂2F∂π∂q∂2F∂π2|q^,π^=λσX200σ2,
which is a positive definite matrix. This implies thatq^,π^ is the unique minimizer of the function F and we get(69)Fminq^,π^=−ϑ22a2.
Takea=Hx/Hxx<0 in (69); then it is easy to obtain that(70)minq,π∈∏b−rπ+λμXηqHxHxx+12σ2π2+λσX2q2=−ϑ22Hx2Hxx2.
According to (64), since Hxx<0, the above equation is equivalent to(71)supq,π∈∏b−rπ+λμXηqHx+12σ2π2+λσX2q2Hxx=−ϑ22Hx2Hxx.
Based on (65) and (71), comparing with (26), we can know that H(t, x) is a classical solution to the primal HJB (26). Moreover, according to (67), we can see that the maximum of the HJB equation (26) is achieved at q∗t,x and π∗t,x given by (61).
Furthermore, according to (62) and Lemma 2, we have(72)Ht,x≤infy>0C31+yp+xy.
It is easy to see that the minimum value is achieved aty=x1−k/kC3k−1, and thus we arrive at(73)Ht,x≤C3+1−kkC3k−11kxk.
LetC4=maxC3,1/k1−k/kC3k−1, and we obtain 0≤Ht,x≤C41+xk.Theorem 4.
LetHt,x be given by Theorem 3, and then Vt,x≤Ht,x on 0,T×0,∞. Furthermore, if SDE (25) admits a unique nonnegative strong solution X¯ with an optimal strategy q∗,π∗ given by (61) and q∗,π∗ satisfies the following conditions:(74)Eexp12∫0Tσ2π∗v,X¯v2dv<∞andEexp12∫0TλσX2q∗v,X¯v2dv<∞,then Vt,x=Ht,x and q∗,π∗ is the indeed optimal reinsurance-investment strategy for problem (9) with the general utility function ux when θ=η.Proof.Step 1.
Ifθ=η, by Itô’s formula on Ht,x for the wealth process Xt in (25) corresponding to an arbitrary admissible strategy qt,πt∈Π, we have(75)HT,XT=Ht,x+∫tTΑq,πHv,xdv+∫tTHxv,XvσπvdW1v+λσXqvdW0v,where(76)Aq,πHt,x=Htt,Xt+Hxt,XtrXt+b−rπt+λμXηqt+12Hxxt,Xtσ2π2t+λ2σX2q2t.
SinceHt,x is a classical solution of the HJB (26), we have supq,π∈ΠAq,πHt,x=0, which means that Aq,πHt,x≤0, and then(77)HT,XT≤Ht,x+∫tTHxv,XvσπvdW1v+λσXqvdW0v.
The last two terms are square-integrable martingales with zero expectation; taking conditional expectation givent,x on both sides of (77), then we get(78)EHT,XT|Xt=x≤Ht,x.
Furthermore, taking the supremum, we obtain(79)supq,π∈ΠEHT,XT|Xt=x≤Ht,x.
It implies thatVt,x≤Ht,x.Step 2.
For alls∈t,T, any stopping time τ∈t,∞,s∧τ=mins,τ, and optimal strategy q∗,π∗ given by (61) satisfying (74), let(80)τn=infs≥t:∫tsHxv,X¯v2σ2π∗v,X¯v2+λσX2q∗v,X¯v2dv≥n.
Whenn⟶∞ we have τn⟶∞ and τn∧T⟶T.
Employing Itô’s formula forHt,x on 0,τn∧T, we have(81)Hτn∧T,X¯τn∧T=H0,x0+∫0τn∧TAq∗,π∗Hv,X¯vdv+∫0τn∧THxv,X¯vσπ∗v,X¯vdW1v+λσXq∗v,X¯vdW0v.
Since the optimal strategyq∗,π∗ given by (61) satisfies Aq∗,π∗Ht,Xt=0, and the last two terms in (81) are square-integrable martingales with zero expectation, taking conditional expectation given t,x on both sides of (81) yields(82)EHτn∧T,X¯τn∧T=H0,x0<∞.Step 3.
For the optimal control strategyq∗,π∗ given by (61), applying Itôts formula for Ht,x on t,τn∧T leads to(83)Hτn∧T,X¯τn∧T=Ht,x+∫tτn∧TAq∗,π∗Hv,X¯vdv+∫tτn∧THxv,X¯vσπ∗v,X¯vdW1v+λσXq∗v,X¯vdW0v.
Similarly, taking conditional expectation leads to(84)EHτn∧T,X¯τn∧T=Ht,x.
Letn⟶∞, and we get(85)Ht,x=EHT,X¯T≤Vt,x.
Therefore, it means thatHt,x=Vt,x and q∗,π∗ given by (61) is indeed the optimal reinsurance-investment strategy for problem (9). Theorem 4 is proved.
In what follows, we give two examples to illustrate Theorems3 and 4.Example 1.
Assume thatux=xk/k,0<k<1. We can derive the dual function of ux which is given by u^y=−yp/p for y > 0, where p=−k/1−k. According to (49), the solution to the dual HJB equation (43) is given by(86)Jt,y=u^yexpϑ22pp−1−prT−t,where ϑ is given by (44). From (60), we get(87)−yp−1expϑ22pp−1−prT−t+x=0,
which can be rewritten as(88)yt,x=x1/p−1exp−ϑ22p+pp−1rT−t.
According to (59), we obtain a smooth solution to the HJB equation (26) expressed by(89)Ht,x=xkkexp−ϑ22kk−1+krT−t.
Then from (61), the maximum of the primal HJB (26) is achieved at(90)q∗t,x=−ημXσX2Hxt,xHxxt,x=ημXσX2x1−k,π∗t,x=−b−rσ2Hxt,xHxxt,x=b−rσ2x1−k.Remark 2.
Compared with Theorem1, we can see that (90) is consistent with (14), (15), and (110) when θ=η. To solve the HJB equation (26), a standard method in the literature such as that used in Theorem 1 is to guess the solution having the form from (18) according to the form of the power utility (12) and then solve an ordinary differential equation to get the expression of gt and h(t). However, for some complex utility function, the solution form cannot be easily guessed; as a result, the standard method may be ineffective. With the help of Theorem 3, we do not need to guess the solution form and we can find the solution directly with our dual control method here. Particularly, in Example 2 which would be difficult with the standard method, one can feel the usefulness of the dual method in finding a solution to the HJB equation.Example 2.
Define(91)ux=13w−3x+w−1x+xwx,for x > 0, where(92)wx=2−1+1+4x1/2.
By a simple calculus, it is easy to derive that(93)u′x=wx>0,u″x=w′x=−2−1+1+4x−3/21+4x−1/2<0,
which meansux is strictly increasing and strictly concave. Moreover, ux satisfies the following properties:(a)
limx⟶0ux=0; (we may define u0=0), u∞=∞,u′∞=0,u′0=∞. It means that ux is a utility function.(b)
ux satisfies condition (28). On the one hand, since(94)limx⟶∞wxx−1/4=1,there exists a constantN > 1 such that 1/2x−1/4≤wx≤2x−1/4 when N>1. According to w′x<0, we have wx≥2/−1+1+4N1/2 for x≤N. Then we get from x1/4≤x3/4 with x > N > 1 that(95)w−3x≤8x3/4,x>N,−1+1+4N23/2,x≤N,and(96)w−1x≤2x3/4,x>N,−1+1+4N21/2,x≤N.
Sincewx>0,w′x<0, and(97)wxxw′x=−wxxw′x=41+1/1+4x>1,x>0,
it means that(98)xwx′=wx+xw′x=wx−xw′x>0,so xwx is increasing for x > 0, and then we get(99)xwx≤2x3/4,x>N,N2−1+1+4N1/2,x≤N.
Therefore,(100)ux≤N¯+203x3/4,where(101)N¯=13−1+1+4N23/2+−1+1+4N21/2+N2−1+1+4N1/2.
LetC0=maxN¯,20/3, and then we obtain ux≤C01+xk,x>0, where k=3/4, which satisfies (28).
The dual function ofux is defined by u^y, see (29). Since the optimal point x satisfies the equation u′x−y=wx−y=0, we have x=y−2+y−4 and(102)u^y=13y−3+y−1.
Since the dual value functionJt,y is the solution to the linear PDE (43), applying Poisson’s formula, then we obtain from (49) that(103)Jt,y=13y−3e6ϑ2+3rT−t+y−1eϑ2+rT−t.
Furthermore, sincey is the solution to (60), we need to solve the equation(104)−y−4exp6ϑ2+3rT−t−y−2expϑ2+rT−t+x=0,
and then(105)y2=12xeϑ2+rT−t+e2ϑ2+rT−t+4xe32ϑ2+rT−t.
Therefore, according to Theorem3 and the expression of Jt,y, we can derive the expression of the optimal reinsurance-investment strategy from (61) and (64) as follows:(106)q∗t,x=ημXσX2yJyyt,y=ημXσX2y−24y−2e6ϑ2+3rT−t+2eϑ2+rT−t,π∗t,x=b−rσ2yJyyt,y=b−rσ2y−24y−2e6ϑ2+3rT−t+2eϑ2+rT−t,where y is given by (105).
Furthermore, according to Theorem3, we can derive a classical solution to the primal HJB (26) by computing Vt,x=infy>0Jt,y+xy, which leads to Jyt,y+x=0 and(107)Vt,x=Jt,yt,x+xyt,x=y3y−4e6ϑ2+3rT−t+3y−2eϑ2+rT−t+3x=y3−y−2eϑ2+rT−t+x+3y−2eϑ2+rT−t+3x=23y−1eϑ2+rT−t+2xy,where y>0 is given by (105). We have found a closed-form classical solution to the primal HJB equation for a highly complicated utility function ux in (91).


### 4.2. Asymptotic Behavior of the Optimal Control Strategy

In this section we discuss the asymptotic behavior of the optimal strategy given by Theorem3 by a new and simple proof with the duality method. We continue to write Vt,x instead of V¯τ,x and t instead of τ=T−t in this section and the next with the understanding that t is a time-to-horizon variable, which means that t=0 is the horizon time T and t⟶∞ equals T⟶∞.Next we can see from (27) that Vt,x is a classical solution to the following HJB equation:(108)−Vt+rxVx−ϑ22Vx2Vxx=0,t,x∈R+×R+,withV0,x=ux,x∈R+, and Vt,⋅ is strictly increasing and strictly concave for fixed t > 0, and ϑ is given by (44).From Theorem4, the optimal reinsurance strategy q∗t,x and the optimal investment strategy π∗t,x under the general utility function ux are given by(109)q∗t,x=−ημXσX2Vxt,xVxxt,x,π∗t,x=−b−rσ2Vxt,xVxxt,x,t>0.Moreover, Theorem1 says that the optimal strategy qpow∗,πpow∗ with the power utility (12) when θ=η can be expressed by(110)qpow∗=ημXσX21−kx,πpow∗=b−rσ21−kx.Recall that the dual function ofux is u^y for y≥0, and u^y is a nonnegative, continuous, convex, and decreasing function on 0,∞. From (43), we can see that the dual function Jt,y of Vt,x satisfies(111)Jt+ryJy−ϑ22y2Jyy=0,t,y∈R+×R+,with J0,y=u^y for y∈R+ and J∈C1,∞. Then (109) can be rewritten as(112)q∗t,x=ημXσX2yJyyt,y,π∗t,x=b−rσ2yJyyt,y,t>0,where y satisfies Jyt,y+x=0, or y=Vxt,x.We are interested in estimating the difference of optimal strategyq∗t,x,π∗t,x for the general utility function and the optimal strategy qpow∗,πpow∗ for the power utility 1/kxk0<k<1 when t⟶∞. In what follows, we will prove that the limit of q∗t,x,π∗t,x with general utility functions u(x) satisfying (28) converges to qpow∗,πpow∗ with the power utility function (12) when t⟶∞. We therefore need to estimate(113)q∗t,x−qpow∗=ημXσX2yJyyt,y+1−pJyt,y,(114)π∗t,x−πpow∗=b−rσ2yJyyt,y+1−pJyt,y,where p=k/k−1 and y=Vxt,x.Remark 3.
Equations (113) and (114) are two key relations and useful to prove Theorem 5 which leads to the asymptotic behavior below.
We first give the two following lemmas which will be used for estimating (113) and (114).Lemma 3.
Letωt,x∈C1,2R+×R+ be a solution to the following heat equation:(115)∂ω∂t−α2ωxx=0,ω0,x=ψxα>0.DenoteΦx=eβxψx with a constant β>0. Assume that ψx∈C1R and(116)limx⟶−∞Φ′xep¯x=−1,Φ′x≤L1ep¯x,x≤0,L1,x≥0,for constantsLi≥1,i=1,2, and p¯<1 (note that if p¯<0, then p¯=p=k/k−1 holds). Then we get(117)eβxωt,xx≤L1⋅L0t,eβxωt,xxx≤L2⋅p¯+β+1απtL0tforx≥0, where L0t=eβ2α2t+eβ−p¯2α2t, and(118)limx⟶−∞eβxωt,xxep¯−β2α2t+p¯x=−1,limx⟶−∞eβxωt,xxxep¯−β2α2t+p¯x=−p¯.Proof.
Poisson formula yields(119)ωt,x=12απt∫−∞+∞e−ξ−x2/4α2tψξdξ.
Letξ=x+αtη, and we get(120)eβxωt,x=12π∫−∞+∞e−η2/4−αβtηΦx+αtηdη.
An easy calculus arrives at(121)eβxωt,xx=12π∫−∞+∞e−η2/4−αβtηΦ′x+αtηdη,(122)eβxωt,xxx=12π∫−∞+∞e−η2/4−αβtηη2αt+βΦ′x+αtηdη.
Forx≥0, from (116), we get(123)Φ′x+αtη≤L1⋅1+ep¯αtη,∀η∈R,
and then there holds(124)eβxωt,xx≤L12π∫−∞+∞e−η2/4−αβtη1+ep¯αtηdη=L1⋅L0t,where L0t=eα2β2t+ep¯−β2α2t. Similarly, we derive from (122) that(125)eβxωt,xxx≤12π∫−∞+∞e−η2/4−αβtηη2αt+βΦ′x+αtηdη≤L2⋅p¯+β+1απtL0t,
which implies that (117) holds.
Next we prove (118). Using (121), we have(126)eβxωt,xxep¯−β2α2t+p¯x+1=12π∫−∞+∞e−η−2p¯−βαt2/4Φ′x+αtηep¯x+αtη+1dη,
and forx≤0, we get(127)Φ′x+αtηep¯x+αtη≤L1⋅1+e−p¯αtη,∀η∈R.
Applying the control convergence theorem with (116) arrives at(128)limx⟶−∞eβxωt,xxep¯−β2α2t+p¯x+1≤12π∫−∞+∞e−η−2p¯−βαt2/4limx⟶−∞Φ′x+αtηep¯x+αtη+1dη=0,
which means the first limit in (116) is proved. Similarly, using (122), we obtain(129)eβxωt,xxxep¯−β2α2t+p¯x+p¯=12π∫−∞+∞e−η−2p¯−βαt2/4η2αt+βΦ′x+αtηep¯x+αtη+1dη,
and applying the control convergence theorem yields(130)limx⟶−∞eβxωt,xxxep¯−β2α2t+p¯x+p¯≤12π∫−∞+∞e−η−2p¯−βαt2/4η2αt+βlimx⟶−∞Φ′x+αtηep¯x+αtη+1dη=0.
Thus we have proved the second limit in (116).
Next an estimate is given for the dual value functionu^.Lemma 4.
Suppose thatu^∈C1R+ satisfies(131)limy⟶0u^′yyp¯−1=−1,yu^′y≤L1yp¯,y≤1,L1,y≥1,
for constantsL1≥1 and p¯<1 (note that if p¯<0, then p¯=p=k/k−1 holds). Then for y≥1 we have(132)yJyt,y≤L1eγtL0t,y2Jyyt,y≤L2⋅eγt1+p¯+β+1απtL0t,and(133)limy⟶0Jyt,yeρtyp¯−1=−1,limy⟶0Jyyt,yeρtyp¯−2=1−p¯,where β=1/2+r/ϑ2,α=1/2ϑ,γ=−α2β2,ρ=1/2ϑ2p¯p¯−1−rp¯, and ϑ is given by (44).Proof.
Letωt,x=e−βx+γtJt,ex, and it can be proved that ωt,x is a solution to equation (115) with the initial condition ω0,x=ψx=e−βxJ0,ex=e−βxu^ex.
Notice thatΦx=eβxψx=u^ex, let y=ex, and Lemma 3 leads to(134)limx⟶−∞Φ′xep¯x=limy⟶0u^′yyp¯−1=−1,Φ′x=exu^′ex=yu^′y≤L1yp¯,y≤1,x≤0,L1,y≥1,x≥0,
and fory≥1,x≥0,(135)eβxωt,xx=e−γtyJyt,y≤L1⋅L0t,eβxωt,xxx=e−γty2Jyyt,y+yJyt,y≤L2⋅1+p¯+β+1απtL0t.
Sinceβ−p¯2α2=ρ−γ, Lemma 3 yields(136)limx⟶−∞eβxωt,xxep¯−β2α2t+p¯x=limy⟶0Jyt,yeρtyp¯−1=−1,(137)limx⟶−∞eβxωt,xxxep¯−β2α2t+p¯x=limy⟶0yJyyt,y+Jyt,yeρtyp¯−1=−p¯,
and then we can obtain from (136) and (137) that(138)limy⟶0Jyyt,yeρtyp¯−2=1−p¯.
Thus we complete the proof of Lemma4.
In what follows, the asymptotic behavior of the optimal reinsurance-investment strategy with general utility functions will be given in the theorem below, which is the main result of this section.Theorem 5.
Assume thatu^∈C1R+ satisfies (131). Then, for x > 0, we get(139)limt⟶∞q∗t,x=qpow∗,limt⟶∞π∗t,x=πpow∗,where q∗t,x,π∗t,x,qpow∗ and πpow∗ are given by (109) and (110), respectively.Proof.
Recall thatp=k/k−1<0,0<k<1, and according to (133), we have(140)limy⟶0Jyt0,y+yp−1eρt0yp−1=0,limy⟶∞Jyt0,y+yp−1eρt0=0,
for any fixedt0>0. Therefore, for any fixed ε>0, there exists δ=δε>0, such that(141)Jyt0,y+yp−1eρt0<εyp−1+δ,∀y≥0.
For∀t,y∈t0,t1×R+ with any t1>t0, we define(142)zt,y=±Jyt,y+yp−1eρt+εyp−1eρt−t0+1+δe−rt−t0.
Notice that we obtain the following equation by (111):(143)Jty+r−ϑ2yJyy−ϑ22y2Jyyy+rJy=0.
Based on (143), we can prove that zt,x satisfies the following equation and boundary conditions:(144)∂z∂t+r−ϑ2yzy−ϑ22y2zyy+rz=rε,t,y∈t0,t1×R+,zt0,y>0,y∈R+,liminfy⟶0zt,y>0,liminfy⟶∞zt,y>0,t∈t0,t1.
According to the maximum theorem, we conclude thatzt,y>0 for all t,y∈t0,t1×R+. This means that(145)Jyt,y+yp−1eρt≤εyp−1eρt−t0+1+δe−rt−t0.
Notice thaty=Vxt,x,Jyt,y=−x for any x>0. Equation (145) yields(146)Vxt,xp−1eρt≤x+εVxt,xp−1eρt−t0+1+δe−rt−t0.
Takingε=1/2eρt0, we conclude that(147)Vxt,xp−1eρt≤Lx.for t≥t0, where Lx=2x+eρt0+2δ1/2eρt0.
Similarly, equations (132) and (133) yield(148)limy⟶0yJyyt0,y+1−pJyt0,yyp−1=0,limy⟶∞yJyyt0,y+1−pJyt0,y=0.
For any fixedε>0, there exists δ=δε>0, such that(149)yJyyt0,y+1−pJyt0,y<εyp−1+δ,∀y≥0.
For∀t,y∈t0,t1×R+ with any t1>t0, we define(150)z¯t,y=±yJyyt,y+1−pJyt,y+εyp−1eρt−t0+1+δe−rt−t0.
Then we can prove thatz¯t,y satisfies (144). Similarly, the maximum theorem leads to(151)yJyyt,y+1−pJyt,y≤εyp−1eρt−t0+1+δe−rt−t0,for all t,y∈t0,t1×R+.
For fixedx>0, by (113), (147), and (151), we get(152)q∗t,x−qpow∗=ημXσX2yJyyt,y+1−pJyt,y=ημXσX2Vxt,xJyyt,Vxt,x+1−pJyt,Vxt,x≤ημXσX2εVxt,xp−1eρt−t0+1+δe−rt−t0≤ημXσX2εLxe−ρt0+1+δe−rt−t0,for t≥t0. Let t⟶∞ and then let ε⟶0 (or according to the definition of the limit) in (152); we obtain limt⟶∞q∗t,x=qpow∗.
Similarly, from (114), we can prove that limt⟶∞π∗t,x=πpow∗ holds. This completes the proof of Theorem 5.
Next, we will verify that the optimal investment-reinsurance strategy given by (106) with a general utility function (91) converges to that with the power utility function 1/kxkk=3/4 at any level of the wealth, which is consistent with Theorem 5.Example 3.
For Example2, according to the expression of u^y given by (102), it is easy to check that (131) holds with p¯=p=k/k−1=−3k=3/4 and L1=2. Based on (110) with k=3/4 and (112), we get from (113) that(153)q∗t,x−qpow∗=ημXσX2yJyyt,y+1−pJyt,y=ημXσX24x1+1+4xe4ϑ2+rt.
Note that, compared with (106), τ=T−t in (106) is written as t in (153). Thus, we can obtain from (153) that(154)limt⟶∞q∗t,x−qpow∗=ημXσX2limt⟶∞4x1+1+4xe4ϑ2+rt=0,
which implieslimt⟶∞q∗t,x=qpow∗. Similarly, we achieve limt⟶∞π∗t,x=πpow∗.
In addition, unless otherwise stated, the basic parameters in Example2 are given by(155)r=0.03,b=0.05,λ=1,μX=0.5,σX2=0.5,η=0.2,σ=0.16,
andk=3/4 which has been stated in Example 2. We then draw the optimal control strategies given by (106) and (110) in Figure 1, Figure 2, and Figure 3 respectively (notice that we here write t instead of T−t in (106)). These figures show that, at any level of the wealth, the optimal investment-reinsurance strategy with the general utility function converges to that with the power utility function, which means that the asymptotic behavior (139) in Theorem 5 holds.Figure 1
Impacts of (t) on optimal strategies with (x) = 5.Figure 2
Impacts of (x) on optimal reinsurance strategies with different values of (t).Figure 3
Impacts of (x) on optimal investment strategies with different values of (t).


## 4.1. Existence and Dual Representation of Solutions

In order to use the dual theory to solve the optimal control problem (9) with general utility functions, we need first to construct a dual minimization problem. For all t∈0,T and any process Xt satisfying (25), according to Bian and Zheng [14], a dual process Yt can be selected as a nonnegative supermartingale process which makes XtYt be a supermartingale. The set of such process Yt is denoted as(31)Ωy=Yt:Yt≥0,Y0=y,XtYtisasuper−martingale, for allt∈0,T.Then the dual minimum problem can be characterized byinfY∈ΩyEu^YT.Lemma 1.
The dual problem of the primal problem (9) can be expressed as(32)infY∈Ω˜yEu^YT,where for t∈0,T,∀γt≥0 and(33)Ω¯y=Yt:Y0=yanddYt=Yt−rdt−b−rσdW1t−γt+λμXησXdW0t.Proof.
According to Doob-Mayer decomposition theorem, a non-negative super-martingaleYt can be decomposed into(34)Yt=yε¯−αtW1t−βtW0tDt,where ε¯ is Doleans-Dade coefficient, Dt is a positive non-increasing process, and D0=1. Since u^ is a decreasing function and Dt≤1, the dual problem can be characterized as infY^Eu^YT, where Y^ satisfies the following SDE:(35)dYt=Yt−αtdW1t−βtdW0t,Y0=y.
According to (25), let(36)dYt=Yt−rdt−αtdW1t−βtdW0t,Y0=y.
ByIto^ formula, we get(37)dXtYt=Ytπtb−r−σαt+λqtλμXη−σXβtdt+YtσπtdW1t+YtλσXqtdW0t+XtYt−αtdW1t−βtdW0t.
SinceXY is a super-martingale, for all t, it means that(38)πtb−r−σαt+λqtλμXη−σXβt≤0.
Sinceπt∈R, we consider a sufficient condition, i.e., b−r−σαt=0 and ∀γt≥0 for all t∈0,T, which can ensure (38) holds. Denote(39)αt=b−rσ,βt=λμXησX+γtσX,
and then we can see that (38) holds.
LetJt,y be the value function of this dual problem for y≥0,t∈0,T, and then we have(40)Jt,y=infy∈Ω˜yEu^YT|Yt=y,y≥0,t∈0,T.
The dual HJB equation is given by(41)Jt−ryJy+infγt≥012b−rσ2+12λμXησX+γtσX2y2Jyy=0,with the boundary condition(42)JT,y=u^y.
It is easy to verify thatJt,⋅ is convex for fixed t∈0,T. The dual HJB (41) has the unique minimizer when γt=0, and therefore (41) can be rewritten as(43)Jt−ryJy+ϑ22y2Jyy=0,where(44)ϑ=b−rσ2+λμXησX2.
Equation (43) is a linear PDE whose solution can be expressed by(45)Jt,y=∫−∞+∞Kt,lny,T,ξu^eξdξ,where(46)Kt,z,s,ξ=14πμt,sexp−14μt,s−rs−t−μt,s+z−ξ2.
Andμt,s=ϑ2/2s−t for 0≤t<s≤T and z,ξ∈R.Lemma 2.
Jt,y is continuous on 0,T×0,∞, and is a classical solution to (43), satisfying(47)0≤Jt,y≤C31+yp,
for some constantC3 depending on T. Furthermore, if ϑ2−2r≥0, for all fixed t∈0,T,Jt,y is strictly decreasing, is strictly convex with respect to y, and has the following limits properties:(48)Jt,0=∞,Jt,∞=0,Jyt,0=−∞,Jyt,∞=0.Proof.
Firstly, from (45), we can rewrite Jt,y as(49)Jt,y=∫−∞+∞Kt,ω,T,0u^ye−ωdω.
Notice that∫−∞+∞Kt,ω,T,0dω=1 and according to (30), we get u^ye−ω≤C21+ype−pω, and thus(50)Jt,y≤∫−∞+∞Kt,ω,T,0C21+ype−pωdω=C2+ype−p−rT−t.
LetC3=maxC2,e−p−rT−t, and we have 0≤Jt,y≤C31+yp for y>0,t∈0,T.
Secondly, sinceu^ is a continuous, decreasing, and convex function and K is a fundamental solution to (43), Jt,y is continuous on 0,T×0,∞, and decreasing and convex in y for fixed t∈0,T, that is Jyt,y≤0 and Jyyt,y≥0. Let ft,y=Jyyt,y. Differentiating y twice with respect to both sides of the dual HJB equation (43), it is not difficult to see that ft,y is a non-negative classical solution to the following equation when ϑ2−2r≥0,(51)ft+ϑ2−2rf+2ϑ2−ryfy+ϑ22y2fyy=0,y>0,t∈0,T.
Ifft0,y0=0 for some point t0,y0,t0<T, then t0,y0 is a minimum point of ft,y. The strong maximum principle leads to ft,y=Jyyt,y=0 for all t,y∈t0,T×0,∞, which means that Jt,y is linear in y for fixed t∈t0,T. In particular, u^y is linear, which is a contradiction with the given conditions. Thus we have Jyyt,y>0 for fixed t∈t0,T. Similarly, we deduce that Jyt,y<0 for fixed t∈0,T.
Finally, we will prove the limiting properties (48). To show Jt,0=∞, notice that u^ye−ω is non-negative and increasing as y⟶0. Applying the Monotone Convergence Theorem (MCT) yields(52)Jt,0=limy⟶0Jt,y=limy⟶0∫−∞+∞Kt,ω,T,0u^ye−ωdω=∫−∞+∞Kt,ω,T,0limy⟶0u^ye−ωdω=∫−∞+∞Kt,ω,T,0u^0dω=u^0=∞.
To proveJt,∞=0. when y>1 we have yp<1, and then u^ye−ω≤C21+ype−pω≤C21+e−pω. Based on the Dominated Convergence Theorem (DCT), we have(53)Jt,∞=limy⟶∞Jt,y=limy⟶∞∫−∞+∞Kt,ω,T,0u^ye−ωdω=∫−∞+∞Kt,ω,T,0limy⟶∞u^ye−ωdω=∫−∞+∞Kt,ω,T,0u^∞dω=0.
To proveJyt,0=−∞, let gω,y,h=u^ye−ω+he−ω−u^ye−ω/h, and we get(54)Jt,y+h−Jt,yh=∫−∞+∞Kt,ω,T,0gω,y,hdω.
Sinceu^ is decreasing and convex, we have gω,y,h is decreasing as h⟶0, and for all h≥0, we have gω,y,h≤0. Employing the MCT arrives at(55)Jyt,y=limh⟶0Jt,y+h−Jt,yh=∫−∞+∞Kt,ω,T,0limh⟶0gω,y,hdω=∫−∞+∞Kt,ω,T,0e−ωu^′ye−ωdω.
Sinceu^ is convex, we have u^′ is increasing and u^′ye−ω≤u^′∞. When y converges monotonically decreasing to 0, the function u^′ye−ω is decreasing. Using the MCT achieves(56)Jyt,0=limy⟶0Jyt,y=∫−∞+∞Kt,ω,T,0e−ωlimy⟶0u^′ye−ωdω=e−rT−tu^′0=−∞.
Notice that, fory≥1,(57)0≤u^ye−ωy≤C21+e−pωy≤C21+e−pω.
Applying the DCT yields(58)Jyt,∞=limy⟶∞Jyt,y=limy⟶∞Jt,yy=∫−∞+∞Kt,ω,T,0limy⟶∞u^ye−ωydω=0.Theorem 3.
Assume that the utility functionux satisfies (28). Then there exists a function Ht,x∈C00,T×0,∞ which is a classical solution to the HJB equation (26) and Ht,x can be expressed by(59)Ht,x=Jt,yt,x+xyt,x,where yt,x∈C1,∞0,T×0,∞ satisfies(60)Jyt,yt,x+x=0.Furthermore, for fixedt∈0,T,Ht,⋅ is strictly concave and strictly increasing and satisfies HT,x=ux and0≤Ht,x≤C41+xk for constants C4>0 and 0<k<1. Furthermore, the maximum of the HJB (26) is arrived at(61)q∗t,x=−ημXσX2Hxt,xHxxt,x,π∗t,x=−b−rσ2Hxt,xHxxt,x.Proof.
Fort,x∈0,T×0,∞ define(62)Ht,x=infy>0Jt,y+xy.
Ifx>0, then the minimum is reached at a point y satisfying (60). Let yt,⋅ be an inverse function of −Jyt,y, and then we get(63)−Jyt,yt,x=x,yt,−Jyt,y=y,for fixed t∈0,T. Since J∈C1,∞0,T×0,∞ and Jyyt,y>0,x<−Jyt,0=∞, according to the implicit function theorem, we have yt,x∈C1,∞0,T×0,∞ and Ht,x=Jt,yt,x+xyt,x. Hence,(64)Hxxt,x=−1Jyyt,yt,x<0,
which meansHt,⋅ is strictly convex and strictly increasing in x for fixed t∈0,T.
In what follows, we will prove thatH is the classical solution to the HJB (26). Because Jt,y is a classical solution of dual HJB (43), by the relationship between J and H, we have from (43)(65)Ht+rxHx−ϑ22Hx2Hxx=0.
For a given numbera and t∈0,T, let(66)Fq,π=12λσX2q2+aλμXηq+12σ2π2+ab−rπ,over q,π∈0,∞×0,∞. Both ∂F/∂q=0 and ∂F/∂π=0 yield(67)q^=−ημXσX2a,π^=−b−rσ2a.
The Hessian matrix ofFq,π can be written as(68)∂2F∂q2∂2F∂q∂π∂2F∂π∂q∂2F∂π2|q^,π^=λσX200σ2,
which is a positive definite matrix. This implies thatq^,π^ is the unique minimizer of the function F and we get(69)Fminq^,π^=−ϑ22a2.
Takea=Hx/Hxx<0 in (69); then it is easy to obtain that(70)minq,π∈∏b−rπ+λμXηqHxHxx+12σ2π2+λσX2q2=−ϑ22Hx2Hxx2.
According to (64), since Hxx<0, the above equation is equivalent to(71)supq,π∈∏b−rπ+λμXηqHx+12σ2π2+λσX2q2Hxx=−ϑ22Hx2Hxx.
Based on (65) and (71), comparing with (26), we can know that H(t, x) is a classical solution to the primal HJB (26). Moreover, according to (67), we can see that the maximum of the HJB equation (26) is achieved at q∗t,x and π∗t,x given by (61).
Furthermore, according to (62) and Lemma 2, we have(72)Ht,x≤infy>0C31+yp+xy.
It is easy to see that the minimum value is achieved aty=x1−k/kC3k−1, and thus we arrive at(73)Ht,x≤C3+1−kkC3k−11kxk.
LetC4=maxC3,1/k1−k/kC3k−1, and we obtain 0≤Ht,x≤C41+xk.Theorem 4.
LetHt,x be given by Theorem 3, and then Vt,x≤Ht,x on 0,T×0,∞. Furthermore, if SDE (25) admits a unique nonnegative strong solution X¯ with an optimal strategy q∗,π∗ given by (61) and q∗,π∗ satisfies the following conditions:(74)Eexp12∫0Tσ2π∗v,X¯v2dv<∞andEexp12∫0TλσX2q∗v,X¯v2dv<∞,then Vt,x=Ht,x and q∗,π∗ is the indeed optimal reinsurance-investment strategy for problem (9) with the general utility function ux when θ=η.Proof.Step 1.
Ifθ=η, by Itô’s formula on Ht,x for the wealth process Xt in (25) corresponding to an arbitrary admissible strategy qt,πt∈Π, we have(75)HT,XT=Ht,x+∫tTΑq,πHv,xdv+∫tTHxv,XvσπvdW1v+λσXqvdW0v,where(76)Aq,πHt,x=Htt,Xt+Hxt,XtrXt+b−rπt+λμXηqt+12Hxxt,Xtσ2π2t+λ2σX2q2t.
SinceHt,x is a classical solution of the HJB (26), we have supq,π∈ΠAq,πHt,x=0, which means that Aq,πHt,x≤0, and then(77)HT,XT≤Ht,x+∫tTHxv,XvσπvdW1v+λσXqvdW0v.
The last two terms are square-integrable martingales with zero expectation; taking conditional expectation givent,x on both sides of (77), then we get(78)EHT,XT|Xt=x≤Ht,x.
Furthermore, taking the supremum, we obtain(79)supq,π∈ΠEHT,XT|Xt=x≤Ht,x.
It implies thatVt,x≤Ht,x.Step 2.
For alls∈t,T, any stopping time τ∈t,∞,s∧τ=mins,τ, and optimal strategy q∗,π∗ given by (61) satisfying (74), let(80)τn=infs≥t:∫tsHxv,X¯v2σ2π∗v,X¯v2+λσX2q∗v,X¯v2dv≥n.
Whenn⟶∞ we have τn⟶∞ and τn∧T⟶T.
Employing Itô’s formula forHt,x on 0,τn∧T, we have(81)Hτn∧T,X¯τn∧T=H0,x0+∫0τn∧TAq∗,π∗Hv,X¯vdv+∫0τn∧THxv,X¯vσπ∗v,X¯vdW1v+λσXq∗v,X¯vdW0v.
Since the optimal strategyq∗,π∗ given by (61) satisfies Aq∗,π∗Ht,Xt=0, and the last two terms in (81) are square-integrable martingales with zero expectation, taking conditional expectation given t,x on both sides of (81) yields(82)EHτn∧T,X¯τn∧T=H0,x0<∞.Step 3.
For the optimal control strategyq∗,π∗ given by (61), applying Itôts formula for Ht,x on t,τn∧T leads to(83)Hτn∧T,X¯τn∧T=Ht,x+∫tτn∧TAq∗,π∗Hv,X¯vdv+∫tτn∧THxv,X¯vσπ∗v,X¯vdW1v+λσXq∗v,X¯vdW0v.
Similarly, taking conditional expectation leads to(84)EHτn∧T,X¯τn∧T=Ht,x.
Letn⟶∞, and we get(85)Ht,x=EHT,X¯T≤Vt,x.
Therefore, it means thatHt,x=Vt,x and q∗,π∗ given by (61) is indeed the optimal reinsurance-investment strategy for problem (9). Theorem 4 is proved.
In what follows, we give two examples to illustrate Theorems3 and 4.Example 1.
Assume thatux=xk/k,0<k<1. We can derive the dual function of ux which is given by u^y=−yp/p for y > 0, where p=−k/1−k. According to (49), the solution to the dual HJB equation (43) is given by(86)Jt,y=u^yexpϑ22pp−1−prT−t,where ϑ is given by (44). From (60), we get(87)−yp−1expϑ22pp−1−prT−t+x=0,
which can be rewritten as(88)yt,x=x1/p−1exp−ϑ22p+pp−1rT−t.
According to (59), we obtain a smooth solution to the HJB equation (26) expressed by(89)Ht,x=xkkexp−ϑ22kk−1+krT−t.
Then from (61), the maximum of the primal HJB (26) is achieved at(90)q∗t,x=−ημXσX2Hxt,xHxxt,x=ημXσX2x1−k,π∗t,x=−b−rσ2Hxt,xHxxt,x=b−rσ2x1−k.Remark 2.
Compared with Theorem1, we can see that (90) is consistent with (14), (15), and (110) when θ=η. To solve the HJB equation (26), a standard method in the literature such as that used in Theorem 1 is to guess the solution having the form from (18) according to the form of the power utility (12) and then solve an ordinary differential equation to get the expression of gt and h(t). However, for some complex utility function, the solution form cannot be easily guessed; as a result, the standard method may be ineffective. With the help of Theorem 3, we do not need to guess the solution form and we can find the solution directly with our dual control method here. Particularly, in Example 2 which would be difficult with the standard method, one can feel the usefulness of the dual method in finding a solution to the HJB equation.Example 2.
Define(91)ux=13w−3x+w−1x+xwx,for x > 0, where(92)wx=2−1+1+4x1/2.
By a simple calculus, it is easy to derive that(93)u′x=wx>0,u″x=w′x=−2−1+1+4x−3/21+4x−1/2<0,
which meansux is strictly increasing and strictly concave. Moreover, ux satisfies the following properties:(a)
limx⟶0ux=0; (we may define u0=0), u∞=∞,u′∞=0,u′0=∞. It means that ux is a utility function.(b)
ux satisfies condition (28). On the one hand, since(94)limx⟶∞wxx−1/4=1,there exists a constantN > 1 such that 1/2x−1/4≤wx≤2x−1/4 when N>1. According to w′x<0, we have wx≥2/−1+1+4N1/2 for x≤N. Then we get from x1/4≤x3/4 with x > N > 1 that(95)w−3x≤8x3/4,x>N,−1+1+4N23/2,x≤N,and(96)w−1x≤2x3/4,x>N,−1+1+4N21/2,x≤N.
Sincewx>0,w′x<0, and(97)wxxw′x=−wxxw′x=41+1/1+4x>1,x>0,
it means that(98)xwx′=wx+xw′x=wx−xw′x>0,so xwx is increasing for x > 0, and then we get(99)xwx≤2x3/4,x>N,N2−1+1+4N1/2,x≤N.
Therefore,(100)ux≤N¯+203x3/4,where(101)N¯=13−1+1+4N23/2+−1+1+4N21/2+N2−1+1+4N1/2.
LetC0=maxN¯,20/3, and then we obtain ux≤C01+xk,x>0, where k=3/4, which satisfies (28).
The dual function ofux is defined by u^y, see (29). Since the optimal point x satisfies the equation u′x−y=wx−y=0, we have x=y−2+y−4 and(102)u^y=13y−3+y−1.
Since the dual value functionJt,y is the solution to the linear PDE (43), applying Poisson’s formula, then we obtain from (49) that(103)Jt,y=13y−3e6ϑ2+3rT−t+y−1eϑ2+rT−t.
Furthermore, sincey is the solution to (60), we need to solve the equation(104)−y−4exp6ϑ2+3rT−t−y−2expϑ2+rT−t+x=0,
and then(105)y2=12xeϑ2+rT−t+e2ϑ2+rT−t+4xe32ϑ2+rT−t.
Therefore, according to Theorem3 and the expression of Jt,y, we can derive the expression of the optimal reinsurance-investment strategy from (61) and (64) as follows:(106)q∗t,x=ημXσX2yJyyt,y=ημXσX2y−24y−2e6ϑ2+3rT−t+2eϑ2+rT−t,π∗t,x=b−rσ2yJyyt,y=b−rσ2y−24y−2e6ϑ2+3rT−t+2eϑ2+rT−t,where y is given by (105).
Furthermore, according to Theorem3, we can derive a classical solution to the primal HJB (26) by computing Vt,x=infy>0Jt,y+xy, which leads to Jyt,y+x=0 and(107)Vt,x=Jt,yt,x+xyt,x=y3y−4e6ϑ2+3rT−t+3y−2eϑ2+rT−t+3x=y3−y−2eϑ2+rT−t+x+3y−2eϑ2+rT−t+3x=23y−1eϑ2+rT−t+2xy,where y>0 is given by (105). We have found a closed-form classical solution to the primal HJB equation for a highly complicated utility function ux in (91).


## 4.2. Asymptotic Behavior of the Optimal Control Strategy

In this section we discuss the asymptotic behavior of the optimal strategy given by Theorem3 by a new and simple proof with the duality method. We continue to write Vt,x instead of V¯τ,x and t instead of τ=T−t in this section and the next with the understanding that t is a time-to-horizon variable, which means that t=0 is the horizon time T and t⟶∞ equals T⟶∞.Next we can see from (27) that Vt,x is a classical solution to the following HJB equation:(108)−Vt+rxVx−ϑ22Vx2Vxx=0,t,x∈R+×R+,withV0,x=ux,x∈R+, and Vt,⋅ is strictly increasing and strictly concave for fixed t > 0, and ϑ is given by (44).From Theorem4, the optimal reinsurance strategy q∗t,x and the optimal investment strategy π∗t,x under the general utility function ux are given by(109)q∗t,x=−ημXσX2Vxt,xVxxt,x,π∗t,x=−b−rσ2Vxt,xVxxt,x,t>0.Moreover, Theorem1 says that the optimal strategy qpow∗,πpow∗ with the power utility (12) when θ=η can be expressed by(110)qpow∗=ημXσX21−kx,πpow∗=b−rσ21−kx.Recall that the dual function ofux is u^y for y≥0, and u^y is a nonnegative, continuous, convex, and decreasing function on 0,∞. From (43), we can see that the dual function Jt,y of Vt,x satisfies(111)Jt+ryJy−ϑ22y2Jyy=0,t,y∈R+×R+,with J0,y=u^y for y∈R+ and J∈C1,∞. Then (109) can be rewritten as(112)q∗t,x=ημXσX2yJyyt,y,π∗t,x=b−rσ2yJyyt,y,t>0,where y satisfies Jyt,y+x=0, or y=Vxt,x.We are interested in estimating the difference of optimal strategyq∗t,x,π∗t,x for the general utility function and the optimal strategy qpow∗,πpow∗ for the power utility 1/kxk0<k<1 when t⟶∞. In what follows, we will prove that the limit of q∗t,x,π∗t,x with general utility functions u(x) satisfying (28) converges to qpow∗,πpow∗ with the power utility function (12) when t⟶∞. We therefore need to estimate(113)q∗t,x−qpow∗=ημXσX2yJyyt,y+1−pJyt,y,(114)π∗t,x−πpow∗=b−rσ2yJyyt,y+1−pJyt,y,where p=k/k−1 and y=Vxt,x.Remark 3.
Equations (113) and (114) are two key relations and useful to prove Theorem 5 which leads to the asymptotic behavior below.
We first give the two following lemmas which will be used for estimating (113) and (114).Lemma 3.
Letωt,x∈C1,2R+×R+ be a solution to the following heat equation:(115)∂ω∂t−α2ωxx=0,ω0,x=ψxα>0.DenoteΦx=eβxψx with a constant β>0. Assume that ψx∈C1R and(116)limx⟶−∞Φ′xep¯x=−1,Φ′x≤L1ep¯x,x≤0,L1,x≥0,for constantsLi≥1,i=1,2, and p¯<1 (note that if p¯<0, then p¯=p=k/k−1 holds). Then we get(117)eβxωt,xx≤L1⋅L0t,eβxωt,xxx≤L2⋅p¯+β+1απtL0tforx≥0, where L0t=eβ2α2t+eβ−p¯2α2t, and(118)limx⟶−∞eβxωt,xxep¯−β2α2t+p¯x=−1,limx⟶−∞eβxωt,xxxep¯−β2α2t+p¯x=−p¯.Proof.
Poisson formula yields(119)ωt,x=12απt∫−∞+∞e−ξ−x2/4α2tψξdξ.
Letξ=x+αtη, and we get(120)eβxωt,x=12π∫−∞+∞e−η2/4−αβtηΦx+αtηdη.
An easy calculus arrives at(121)eβxωt,xx=12π∫−∞+∞e−η2/4−αβtηΦ′x+αtηdη,(122)eβxωt,xxx=12π∫−∞+∞e−η2/4−αβtηη2αt+βΦ′x+αtηdη.
Forx≥0, from (116), we get(123)Φ′x+αtη≤L1⋅1+ep¯αtη,∀η∈R,
and then there holds(124)eβxωt,xx≤L12π∫−∞+∞e−η2/4−αβtη1+ep¯αtηdη=L1⋅L0t,where L0t=eα2β2t+ep¯−β2α2t. Similarly, we derive from (122) that(125)eβxωt,xxx≤12π∫−∞+∞e−η2/4−αβtηη2αt+βΦ′x+αtηdη≤L2⋅p¯+β+1απtL0t,
which implies that (117) holds.
Next we prove (118). Using (121), we have(126)eβxωt,xxep¯−β2α2t+p¯x+1=12π∫−∞+∞e−η−2p¯−βαt2/4Φ′x+αtηep¯x+αtη+1dη,
and forx≤0, we get(127)Φ′x+αtηep¯x+αtη≤L1⋅1+e−p¯αtη,∀η∈R.
Applying the control convergence theorem with (116) arrives at(128)limx⟶−∞eβxωt,xxep¯−β2α2t+p¯x+1≤12π∫−∞+∞e−η−2p¯−βαt2/4limx⟶−∞Φ′x+αtηep¯x+αtη+1dη=0,
which means the first limit in (116) is proved. Similarly, using (122), we obtain(129)eβxωt,xxxep¯−β2α2t+p¯x+p¯=12π∫−∞+∞e−η−2p¯−βαt2/4η2αt+βΦ′x+αtηep¯x+αtη+1dη,
and applying the control convergence theorem yields(130)limx⟶−∞eβxωt,xxxep¯−β2α2t+p¯x+p¯≤12π∫−∞+∞e−η−2p¯−βαt2/4η2αt+βlimx⟶−∞Φ′x+αtηep¯x+αtη+1dη=0.
Thus we have proved the second limit in (116).
Next an estimate is given for the dual value functionu^.Lemma 4.
Suppose thatu^∈C1R+ satisfies(131)limy⟶0u^′yyp¯−1=−1,yu^′y≤L1yp¯,y≤1,L1,y≥1,
for constantsL1≥1 and p¯<1 (note that if p¯<0, then p¯=p=k/k−1 holds). Then for y≥1 we have(132)yJyt,y≤L1eγtL0t,y2Jyyt,y≤L2⋅eγt1+p¯+β+1απtL0t,and(133)limy⟶0Jyt,yeρtyp¯−1=−1,limy⟶0Jyyt,yeρtyp¯−2=1−p¯,where β=1/2+r/ϑ2,α=1/2ϑ,γ=−α2β2,ρ=1/2ϑ2p¯p¯−1−rp¯, and ϑ is given by (44).Proof.
Letωt,x=e−βx+γtJt,ex, and it can be proved that ωt,x is a solution to equation (115) with the initial condition ω0,x=ψx=e−βxJ0,ex=e−βxu^ex.
Notice thatΦx=eβxψx=u^ex, let y=ex, and Lemma 3 leads to(134)limx⟶−∞Φ′xep¯x=limy⟶0u^′yyp¯−1=−1,Φ′x=exu^′ex=yu^′y≤L1yp¯,y≤1,x≤0,L1,y≥1,x≥0,
and fory≥1,x≥0,(135)eβxωt,xx=e−γtyJyt,y≤L1⋅L0t,eβxωt,xxx=e−γty2Jyyt,y+yJyt,y≤L2⋅1+p¯+β+1απtL0t.
Sinceβ−p¯2α2=ρ−γ, Lemma 3 yields(136)limx⟶−∞eβxωt,xxep¯−β2α2t+p¯x=limy⟶0Jyt,yeρtyp¯−1=−1,(137)limx⟶−∞eβxωt,xxxep¯−β2α2t+p¯x=limy⟶0yJyyt,y+Jyt,yeρtyp¯−1=−p¯,
and then we can obtain from (136) and (137) that(138)limy⟶0Jyyt,yeρtyp¯−2=1−p¯.
Thus we complete the proof of Lemma4.
In what follows, the asymptotic behavior of the optimal reinsurance-investment strategy with general utility functions will be given in the theorem below, which is the main result of this section.Theorem 5.
Assume thatu^∈C1R+ satisfies (131). Then, for x > 0, we get(139)limt⟶∞q∗t,x=qpow∗,limt⟶∞π∗t,x=πpow∗,where q∗t,x,π∗t,x,qpow∗ and πpow∗ are given by (109) and (110), respectively.Proof.
Recall thatp=k/k−1<0,0<k<1, and according to (133), we have(140)limy⟶0Jyt0,y+yp−1eρt0yp−1=0,limy⟶∞Jyt0,y+yp−1eρt0=0,
for any fixedt0>0. Therefore, for any fixed ε>0, there exists δ=δε>0, such that(141)Jyt0,y+yp−1eρt0<εyp−1+δ,∀y≥0.
For∀t,y∈t0,t1×R+ with any t1>t0, we define(142)zt,y=±Jyt,y+yp−1eρt+εyp−1eρt−t0+1+δe−rt−t0.
Notice that we obtain the following equation by (111):(143)Jty+r−ϑ2yJyy−ϑ22y2Jyyy+rJy=0.
Based on (143), we can prove that zt,x satisfies the following equation and boundary conditions:(144)∂z∂t+r−ϑ2yzy−ϑ22y2zyy+rz=rε,t,y∈t0,t1×R+,zt0,y>0,y∈R+,liminfy⟶0zt,y>0,liminfy⟶∞zt,y>0,t∈t0,t1.
According to the maximum theorem, we conclude thatzt,y>0 for all t,y∈t0,t1×R+. This means that(145)Jyt,y+yp−1eρt≤εyp−1eρt−t0+1+δe−rt−t0.
Notice thaty=Vxt,x,Jyt,y=−x for any x>0. Equation (145) yields(146)Vxt,xp−1eρt≤x+εVxt,xp−1eρt−t0+1+δe−rt−t0.
Takingε=1/2eρt0, we conclude that(147)Vxt,xp−1eρt≤Lx.for t≥t0, where Lx=2x+eρt0+2δ1/2eρt0.
Similarly, equations (132) and (133) yield(148)limy⟶0yJyyt0,y+1−pJyt0,yyp−1=0,limy⟶∞yJyyt0,y+1−pJyt0,y=0.
For any fixedε>0, there exists δ=δε>0, such that(149)yJyyt0,y+1−pJyt0,y<εyp−1+δ,∀y≥0.
For∀t,y∈t0,t1×R+ with any t1>t0, we define(150)z¯t,y=±yJyyt,y+1−pJyt,y+εyp−1eρt−t0+1+δe−rt−t0.
Then we can prove thatz¯t,y satisfies (144). Similarly, the maximum theorem leads to(151)yJyyt,y+1−pJyt,y≤εyp−1eρt−t0+1+δe−rt−t0,for all t,y∈t0,t1×R+.
For fixedx>0, by (113), (147), and (151), we get(152)q∗t,x−qpow∗=ημXσX2yJyyt,y+1−pJyt,y=ημXσX2Vxt,xJyyt,Vxt,x+1−pJyt,Vxt,x≤ημXσX2εVxt,xp−1eρt−t0+1+δe−rt−t0≤ημXσX2εLxe−ρt0+1+δe−rt−t0,for t≥t0. Let t⟶∞ and then let ε⟶0 (or according to the definition of the limit) in (152); we obtain limt⟶∞q∗t,x=qpow∗.
Similarly, from (114), we can prove that limt⟶∞π∗t,x=πpow∗ holds. This completes the proof of Theorem 5.
Next, we will verify that the optimal investment-reinsurance strategy given by (106) with a general utility function (91) converges to that with the power utility function 1/kxkk=3/4 at any level of the wealth, which is consistent with Theorem 5.Example 3.
For Example2, according to the expression of u^y given by (102), it is easy to check that (131) holds with p¯=p=k/k−1=−3k=3/4 and L1=2. Based on (110) with k=3/4 and (112), we get from (113) that(153)q∗t,x−qpow∗=ημXσX2yJyyt,y+1−pJyt,y=ημXσX24x1+1+4xe4ϑ2+rt.
Note that, compared with (106), τ=T−t in (106) is written as t in (153). Thus, we can obtain from (153) that(154)limt⟶∞q∗t,x−qpow∗=ημXσX2limt⟶∞4x1+1+4xe4ϑ2+rt=0,
which implieslimt⟶∞q∗t,x=qpow∗. Similarly, we achieve limt⟶∞π∗t,x=πpow∗.
In addition, unless otherwise stated, the basic parameters in Example2 are given by(155)r=0.03,b=0.05,λ=1,μX=0.5,σX2=0.5,η=0.2,σ=0.16,
andk=3/4 which has been stated in Example 2. We then draw the optimal control strategies given by (106) and (110) in Figure 1, Figure 2, and Figure 3 respectively (notice that we here write t instead of T−t in (106)). These figures show that, at any level of the wealth, the optimal investment-reinsurance strategy with the general utility function converges to that with the power utility function, which means that the asymptotic behavior (139) in Theorem 5 holds.Figure 1
Impacts of (t) on optimal strategies with (x) = 5.Figure 2
Impacts of (x) on optimal reinsurance strategies with different values of (t).Figure 3
Impacts of (x) on optimal investment strategies with different values of (t).


## 5. Conclusion

Since the complexity of general utility functions leads to deriving an analytical solution difficultly for some optimal control problems, there are few articles considering the optimal reinsurance-investment problems with general utility functions, while this paper provides a dual control method to investigate the optimal investment-reinsurance problem with general utility functions. We show the existence of the classical solution to the HJB equation and find the representation of the solution in terms of the dual function. Based on our dual control method, we achieve the expression of the optimal investment-reinsurance strategy with general form utility functions and give some examples to illustrate the usefulness of our dual method; moreover, the asymptotic behavior of the optimal control strategy is verified.As for future research, there are still some possible extensions of this paper. For example, this paper assumes that the risky asset price process follows geometric Brownian motion; one can consider if the risky asset price process is described by CEV model, Heston’s SV model, and other more general stochastic processes, how to derive the optimal control strategy, and whether the asymptotic behavior still holds under general utility functions. These open problems are interesting and challenging and require further research and investigation.



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