# The Principle-Agent Conflict Problem in a Continuous-Time Delegated Asset Management Model

Authors: Yanan Li; Chuanzheng Li
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper considers the principle-agent conflict problem in a continuous-time delegated asset management model when the investor and the fund manager are all risk-averse with risk sensitivity coefficientsγfandγm, respectively. Suppose that the investor entrusts his money to the fund manager. The return of the investment is determined by the manager’s effort level and incentive strategy, but the benefit belongs to the investor. In order to encourage the manager to work hard, the investor will determine the manager’s salary according to the terminal income. This is a stochastic differential game problem, and the distribution of income between the manager and the investor is a key point to be solved in the custody model. The uncertain form of the incentive strategy implies that the problem is different from the classical stochastic optimal control problem. In this paper, we first express the investor’s incentive strategy in term of two auxiliary processes and turn this problem into a classical one. Then, we employ the dynamic programming principle to solve the problem.


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## Body

## 1. Introduction

Since professional asset management institutions can make efficient investment decisions, save investors’ time and effort, and simplify the investment process, more and more investors now entrust their money to fund managers, securities firms, and other asset management organizations. Nowadays, scholars pay more and more attention to asset management problems. We can refer to [1–5] to name just a few.The whole asset management process involves two parties: the investor and the manager. The return of the investment is closely related to the manager’s effort level and investment strategy, but the interests belong to the investor. So, the investor and manager’s relation poses a principal-agent conflict. An important part of discussing the asset management problem is finding the investor’s optimal incentive mode under the principle agent conflict.There are many papers committed to solving principal-agent conflict problems. Most of the early literature studies investigate the discrete-time case (we can refer to [6–8] or a summary book [9]). The problem in continuous-time models is discussed for the first time in [10]. It points out that the investor’s optimal incentive mode is linear. See references [11–14] for further work. In recent years, the maximum principle or the martingale representation theorem is often used to solve this problem in continuous-time models. For the literature using the maximum principle, we can refer to [15, 16], and for the literature of using the martingale representation theorem, we can refer to [17, 18]. However, since this problem often needs to solve a backward stochastic differential equation (BSDE) that rarely has explicit solutions, there are few articles which give analytical solutions to this problem. In order to get explicit solutions of principal-agent conflict problems, the authors of [19] express the investor’s incentive strategy in terms of two auxiliary processes and turn the principle agent problem into a classical stochastic differential game problem.Although there are many papers committed to solving principal-agent conflict problems in continuous-time models, the delegated asset management problems are usually investigated in discrete-time models for the sake of simplicity. Thus, there are some contributions in this paper:(i)
This paper considers the delegated asset management problem in a continuous-time model(ii)
Learning from [19], this paper gives explicit value functions and the optimal strategies of both sides by expressing the investor’s incentive strategy in terms of two auxiliary processes and turning the problem into a classical stochastic differential game problem(iii)
In order to make the model more realistic, this paper brings in risk sensitivity coefficients to represent the subjects’ risk aversion attitudesThis paper is organized as follows. In Section2, we establish a continuous-time model of the fund management problem. In Section 3, we discuss the manager’s optimization problem under fixed investor’s incentive strategy. By substituting the manager’s optimal strategy into the investor’s optimal problem, both the investor and the manager’s optimal strategies are obtained in Section 4.


## 2. The Principal-Agent Conflict Model

Similar to the model in [20], let us assume that the investor employs a professional fund controller (manager) to invest and the investor will get a profit and pay the manager at the terminal moment T. Since the manager’s effort level cannot be observed, the investor will determine the manager’s salary according to the terminal profit of the investment. The investor’s return is determined by the terminal investment profit and the manager’s salary. The terminal investment profit is related to the manager’s investment strategy and effort level, and the incentive mechanism largely determines the manager’s strategy. Therefore, the investor needs to find the optimal incentive mechanism (the manager’s salary) to maximize his terminal net income. Meanwhile, according to the investor’s incentive mechanism, the manager shall decide his investment strategy and the best effort level to maximize his net salary (terminal salary minus effort cost). This is a non-cooperative game problem. Next, let us build a mathematical model of this problem in probability space Ω,ℱ,P.Similar to the model in [18], we suppose that the manager’s effort will affect the fund income Rtn which satisfies(1)dRtn=Rtnr+μ+ntdt+σdWt,where μ≥0, σ≥0, and r>0 is the risk-free interest rate, Wt is a Brownian motion on Ω,ℱ,P, and ntt≥0 is the manager’s effort level. Here, for the convenience of calculation, we assume that the drift coefficient of Rtn is a linear function of the manager’s effort level. In fact, as long as the drift coefficient of Rtn has the form of Rtnr+fnt for some function fn, the same method in this paper can be used after replacing n with fn. For more general forms of the drift coefficient of Rtn, the existence of the time value makes it hard to obtain explicit solutions.Considering the manager’s strategyπ=btπ,ntπ, where btπ represents the wealth that the manager decides to operate at moment t(The manager may not want to operate all the wealth since the cost of the effort will increase with the wealth operated increases. The money left will get a risk-free return.) and ntπ represents the manager’s effort level at t. By some simple calculations, we can get that the investment income under this strategy satisfies(2)dXtπ=rXtπ+btπμ+ntπdt+btπσdWt.Define the natural filtration produced byWt as ℱtWt≥0. Now, let us give the definition of both the manager and the investor’s admissible strategies. Considering the manager’s strategy π=btπ,ntπ. If btπandntπ are bounded positive predictable stochastic processes, under the strategy π, (2) has a unique solution.We call that strategyπ=btπ,ntπ is admissible. Denote the set of all the manager’s admissible strategies by Π.Remark 1.
Here, we do not consider the case whenb=0 or n=0 since in that case, the model is meaningless.Suppose that the investor’s incentive strategy is a function of the investment income atT and denote it by w·. If supπ∈ΠEwXTπ<∞, the manager’s value function under w· is a decreasing convex function with respect to the initial wealth, we say that w· is the investor’s admissible strategy. Denote the set of all the investor’s admissible strategies by Π^.Now, let us analyze the whole game process. Referring to [15], we know that investors play a leading role in the game. Managers need to decide their effort level and investment strategy according to the investors’ incentive strategy. Therefore, first, we need to fix w· and investigate the manager’s optimal problem. We can get the manager’s optimal effort and investment strategy in terms of w· as a byproduct. Then, by substituting the manager’s optimal strategy into the wealth process, we can solve the investor’s optimal problem by using the dynamic programming principle.Therefore, firstly, we fix the investor’s incentive strategyw· and consider the manager’s optimal problem. Suppose that the manager is risk-averse and denote his risk sensitivity coefficient by γm<0. Referring to [18], we suppose that the manager needs to pay θn2b/2 to manage b units of capital in unit time under the effort level n. Here, θ>0 is a constant which represents the effort cost parameter. The objective of the manager is to find the optimal effort level and investment strategy to maximize his net income (salary minus effort cost), which is equivalent to minimize(3)Jmπt,x;w=EeγmwXTπ−∫tTerT−tθntπ2/2btπdt|Xtπ=x.Denote the manager’s optimal strategy byπw, then the value function is(4)Vmt,x;w=infπ∈ΠJmπt,x;w=Jmπwt,x;w.Suppose that the investor is risk-averse too, his risk-sensitive coefficient isγf<0. Next, we consider the investor’s optimal problem.If the manager’s salary is too high, the investor’s income will be reduced. If the manager’s salary is too low, the manager’s enthusiasm wanes, which also deduces the investor’s terminal income. Therefore, the investor needs to find a reasonable incentive strategy to maximize his net income, that is, minimize(5)Jfwt,x=EeγfXTw−wXTw|Xtw=x,where Xtw is the investment income process under strategy πw. Thus, the investor’s value function is(6)Vft,x=infw∈Π^Jfwt,x.Remark 2.
The problem discussed above is not a standard stochastic optimal control problem since the form ofw· is uncertain, and we cannot solve it directly by using standard stochastic optimal methods. In Section 3, we give another form of the incentive strategy and transform the game problem into a classical one. Then, we can use the dynamic programming principle to solve the problem.


## 3. The Manager’s Optimization Problem

DefineDt=erT−t, βt,π=γmDtθntπ2/2btπ, and Γt,T,π=e−∫tTβu,πdu. Then, Jmπt,x;w can be denoted by(7)Jmπt,x;w=EΓt,T,πeγmwXTπ|Xtπ=x.Using the results of Section 3.4 in [21], we know that, under the incentive strategy w·, the manager’s value function Vmt,x;w satisfies the HJB equation:(8)−Vmtt,x;w=infπ∈Π−βt,πVmt,x;w+rx+btπμ+ntπVmxt,x;w+btπ2σ22Vmxxt,x;wand the boundary condition(9)VmT,x;w=eγmwx.SinceVmt,x;w is a decreasing convex function of x, for ∀t,x,y,z,γ∈0,T×ℝ×0,∞×−∞,0×0,∞, we can define the Hamiltonian function:(10)Ht,x,y,z,γ=infn>0,b>0ht,x,y,z,γ,n,b,where(11)ht,x,y,z,γ,n,b=−Dtγmθn2b2y+rx+bμ+nz+b2σ22γ.Theorem 1.
(12)nt∗y,z,γ=zθγmyDt,(13)bt∗y,z,γ=−μ+nt∗y,z,γ/2σ2zγ,is the minimum point of h in (10).Proof.
According to the definition, we know thath is a convex function of n,b. So, the minimum point of h in (10) is the stable point under constraint conditions n>0,b>0. By some simple calculations, we have(14)hnn,b;t,x,y,z,γ=−θDtbnγmy+bz,hbn,b;t,x,y,z,γ=σ2γb+μ+nz−Dtθnγmy2.
Combining the above two equations, we can obtain the stable point ofh:(15)nt∗y,z,γ=zθγmyDt>0,bt∗y,z,γ=−μ+nt∗y,z,γ/2σ2zγ>0.
The proof is done.Remark 3.
In this case, the optimal investment strategy is similar to that without principal-agent relationships. The only difference is that the numerator of the optimal investment strategy is changed fromμ+nt∗y,z,γ into μ+nt∗y,z,γ/2. Clearly, this is due to the existence of the agency relationship.Apparently, the investor’s incentive strategy and the manager’s value function are one-to-one. In the following, we will use auxiliary stochastic processesZt,Γt to determine the manager’s value function and transform the investor’s incentive strategy into Zt,Γt. Then, the problem in Section 2 can be translated into a classical stochastic optimal control problem.First, let us give the space of auxiliary stochastic processesZ,Γ. Fix t∈0,T, let Z:t,T×Ω⟶−∞,0,Γ:t,T×Ω⟶0,∞ be ℱW-predicable processes which satisfy(16)E∫tTZs2σs2+Γsσs2ds<+∞.Denote the set of all the processes satisfying the above conditions byVt.For someZ,Γ∈Vt and Yt≥0, define the ℱW-progressively measurable process YZ,Γ on the filtration space Ω,ℱ,P,ℱtWt≥0 by(17)YsZ,Γ=Yt−∫tsHr,Xr,YrZ,Γ,Zr,Γrdr+∫tsZrdXr+12∫tsΓrdXr,s∈t,T,where Xr is the investment income process. Clearly, for fixed Yt,Z,Γ, YTZ,Γ is only related to the investment income process and is ℱT measurable, suppose that it is an incentive strategy (we prove it in Corollary 1). In the following, we give the relationship between YsZ,Γ and the manager’s value function. First, we give the following lemma.Lemma 1.
Define(18)π∗Z,Γ=b∗Z,Γ,n∗Z,Γ=bt∗YtZ,Γ,Zt,Γtt≥0,nt∗YtZ,Γ,Zt,Γtt≥0,and then we have π∗Z,Γ∈Π.Proof.
On the one hand, sinceZ,Γ,YZ,Γ are all predictable stochastic processes, referring to (12) and (13), we can get that b∗Z,Γandn∗Z,Γ are bounded positive predictable stochastic processes. On the other hand, bt∗y,z,γandnt∗y,z,γ are independent of x. Taking b∗Z,Γandn∗Z,Γ into (2), we can get the Lipschitz continuity and linear growth of the coefficients in (2) with respect to Xt; then, (2) has a unique solution. The proof is done.
Denote the investment income process underπ∗Z,Γ by X∗Z,Γ. We also have the following theorem.Theorem 2.
Denote the manager’s value function with a terminal returnlnYTZ,Γ/γm by Vmt,x;YTZ,Γ. We can obtain that(19)Yt=Vmt,x;YTZ,Γ.Furthermore, the manager’s optimal strategy isπ∗Z,Γ.Proof.
∀π∈Π,s∈t,T, we have(20)YsZ,Γ=Yt−∫tsHr,Xrπ,YrZ,Γ,Zr,Γrdr+∫tsZrdXrπ+12∫tsΓrdXπr.
Using Ito’s formula, we have(21)de−∫trβu,πduYrZ,Γ=e−∫trβu,πdu−Hr,Xrπ,YrZ,Γ,Zr,Γr+rXrπ+brπμ+nrπZr+brπ2σ22Γr−βr,πdr+e−∫trβu,πduσZrdWr.
It follows from (16) that e−∫trβu,πduσZrdWr is a martingale. Integrating and taking expectations on both sides of (21), we can get(22)Yt≥Ee−∫tTβu,πduYTZ,Γ|Xtπ=x=Jmπt,x;YTZ,Γ.
Furthermore, by simple calculations, underπ∗Z,Γ∈Π, we have(23)dYtZ,Γ=βt,π∗Z,ΓYtZ,Γdt+bt∗YtZ,Γ,Zt,ΓtZtσdWt.
Using (23) and Ito’s formula, we can obtain(24)de−∫trβu,π∗Z,ΓduYrZ,Γ=e−∫trβu,π∗Z,Γdubt∗YtZ,Γ,Zt,ΓtZtσdWt.
With similar methods, integrating and taking expectations on both sides of (24), we have(25)Yt=Ee−∫tTβu,π∗Z,ΓduYTZ,Γ|Xt∗Z,Γ=x=Jmπ∗Z,Γt,x;YTZ,Γ≥Jmπt,x;YTZ,Γ.
This implies thatπ∗Z,Γ is the manager’s optimal strategy and(26)Yt=Vmt,x;YTZ,Γ.
Up till now, fixingZ,Γ∈Vt, we can get the manager’s optimal strategy and represent the manager’s value function. In Section 4, we begin to consider the investor’s optimization problem. That is, finding the optimal Z,Γ∈Vt to maximize the investor’s net profit.


## 4. The Investor’s Optimization Problem

Suppose that the investor’s wealth isx at t. Apparently, the investor’s value function is uniquely determined by the wealth process and the manager’s value function. So, the objective of the investor is to find the optimal Z,Γ∈Vt to minimize his value function. Define(27)vt,x,y=infZ,Γ∈VtEeγfXT∗Z,Γ−lnYTZ,Γ/γm|Xtπ=x,YtZ,Γ=y.Referring to Theorem 4.1 in [19], we know that if Assumption 3.2, Assumption 4.3, and Assumption 4.4 in [19] hold, the investor’s value function satisfies(28)Vft,x=infy∈0,eγmRvt,x,y.Here,R is the minimum pay in order to make sure that the manager takes the job.Section4.1 gives the verification of the three assumptions.


### 4.1. The Verification of Assumptions

Assumption 1 (Assumption 3.2 in [19]).
H has at least one extreme point bt∗y,z,γ,nt∗y,z,γ. For any t∈0,T,Z,Γ∈Vt, we have π∗Z,Γ∈Π.Proof.
This is the result of Theorem1 and Lemma 1.
The Hamiltonian function can be expressed as(29)Ht,x,y,z,γ=infb>0Ft,x,y,z,b+b2σ22γ.
Here,(30)Ft,x,y,z,b=infn>0−Dtγmθn2b2y+rx+bμ+nz.
Define(31)YsZ=Yt−∫tsFr,Xr,YrZ,Zrdr+∫tsZrdXr,s∈t,T,and we have the following assumption.Assumption 2 (Assumption 4.3 in [19]).
F has at least one extreme point nt∗y,z,b; furthermore, b,n∗YZ,Z,b∈Π.Proof.
On the one hand, the right hand ofF is a parabola with an opening up with respect to n; so, the minimum point is attained at the axis of the parabola z/Dtγmθy, that is, nt∗y,z,b=z/Dtγmθy. On the other hand, since Z<0 is predictable, we can get that nt∗YtZ,Zt,b=bZt/DtγmθbYtZ is a positive predictable process. Furthermore, bandnt∗y,z,b are independent of x. This implies the Lipschitz continuity and linear growth of the coefficients in (2) with respect to the investment income process; then, (2) has a unique solution.Assumption 3 (Assumption 4.4 in [19]).
∀b>0, 1/b2σ2 is bounded.Proof.
We can get the result directly fromσ>0,b>0.


### 4.2. The Investor’s Value Function

Clearly, as soon as we getvt,x,y, we can obtain Vft,x. The following theorem gives the partial differential equation satisfied by vt,x,y.Theorem 3.
vt,x,y is the viscosity solution of(32)−vtt,x,y=infZ,Γ∈VtGt,x,y,Z,Γ,(33)vT,x,y=eγfxy−γf/γm,where(34)Gt,x,y,Z,Γ=rx+bt∗Z,Γμ+nt∗Z,Γvx+σ2bt∗Z,Γ22vxx+Dtγmθnt∗Z,Γ22bt∗Z,Γyvy+σ2bt∗Z,Γ22Z2vyy+σ2bt∗Z,Γ2Zvxy.Proof.
By the definition ofvt,x,y, we can obtain that it satisfies (33). Furthermore, according to the dynamic programming principle, we have(35)vt,x,y=infZ,Γ∈Vtvt+h,Xt+h∗Z,Γ,Yt+hZ,Γ.
By using Ito’s formula with respect tovs,Xs∗Z,Γ,YsZ,Γ from t to t+h, we have(36)vt+h,Xt+h∗Z,Γ,Yt+hZ,Γ=vt,x,y+∫tt+hvts,Xs∗Z,Γ,YsZ,Γ+Gs,Xs∗Z,Γ,YsZ,Γ,Zs,Γsds.
Combining with the above two equations, we can get(37)vtt,x,y+infZ,Γ∈VtGt,x,y,Z,Γ=0.
That is,vt,x,y satisfies (32). The proof is done.
Next, we are going to solve (32) and (33). Considering the boundary condition, we guess(38)vt,x,y=eγfDtxy−γf/γmEt,where Et is a function of t which satisfies ET=1.
If the variables in the solution can be separated from each other, (32) can be easily solved. However, (32) contains eγfDtx, which is a cross term of t and x. To cancel the cross term, we introduce zt=DtXt∗Z,Γ. Using Ito’s formula, we can get(39)dzt=−rDtXt∗Z,Γdt+DtdXt∗Z,Γ=Dtbt∗Z,Γμ+nt∗Z,Γdt+σdWt.
We can also obtainzT=XT∗Z,Γ. Define(40)Vt,z,y=infZ,Γ∈VtEeγfzT−lnYTZ,Γ/γm|zt=z=infZ,Γ∈VtEeγfXT∗Z,Γ−lnYTZ,Γ/γm|Xt∗Z,Γ=zDt=vt,zDt,y.
Obviously, solvingvt,x,y is equivalent to solving Vt,z,y. Using a similar method as the one in Theorem 3, we can get that(41)−Vt=infZ,Γ∈Vt−μ+nt∗Z,Γ/2μ+nt∗Z,Γσ2DtZΓVz−γmθnt∗Z,Γ2μ+nt∗Z,Γ/22σ2yDtZΓVy+μ+nt∗Z,Γ/222σ2Dt2Z2Γ2Vzz+nt∗Z,Γ2μ+nt∗Z,Γ/222σ2γmθyDt2Z2Γ2Vyy+nt∗Z,Γμ+nt∗Z,Γ/22σ2γmθyDt2Z2Γ2Vzy,(42)VT,z,y=eγfzy−γf/γm.
The first step in solving (41) is to find its minimum point. Define MZ,Γ=Z/Γ, it is shown in Section 3 that Z,Γ and MZ,Γ,n∗Z,Γ are one-to-one. Then, (41) is transformed into(43)−Vt=infn,M∈ℝ+×ℝ−−μ+n/2μ+nσ2DtMVz+μ+n/222σ2Dt2M2Vzz−γmθn2μ+n/22σ2yDtMVy+n2μ+n/222σ2γmθyDt2M2Vyy+nμ+n/22σ2γmθyDt2M2Vzy.
Now, the problem of finding the minimum point in (41) is changed into a problem of finding the minimum point in (43).
According to (38), we suppose that Vt,z,y=Eteγfzy−γf/γm. By some simple calculations, we can get that(44)Vzt,z,y=γfVt,z,y,Vzzt,z,y=γf2Vt,z,y,yVyt,z,y=−γfγmVt,z,y,y2Vyyt,z,y=γfγf+γmγm2Vt,z,y,yVzyt,z,y=−γf2γmVt,z,y.
Taking them into (43), we have(45)−E′tVt,z,y=infn,M∈ℝ+×ℝ−−μ+n/2μ+nσ2DtMγf+μ+n/222σ2Dt2M2γf2+γfθn2μ+n/22σ2DtM+n2μ+n/222σ2θ2Dt2M2γfγf+γm−nμ+n/22σ2γf2θDt2M2EtVt,z,y.
Since the right hand of (45) is continuous, the minimum point can only be attained at the stable points or the boundary points, which depends on the parameter values. Denote the minimum point of (45) by nt∗,Mt∗, and denote the corresponding minimum point of (41) by Zt∗,Γt∗. It is shown from the Appendix that nt∗ and DtMt∗ are constants concerning μ,θ,γf, and γm. Let nt∗=n∗.Remark 4.
On the one hand, the exponential form of the objective function implies thatbt∗ is independent of Xt∗. On the other hand, the benefit and the cost brought by the manager’s effort are only related to bt∗, so nt∗ is independent of Xt∗. Furthermore, in this paper, we consider the discounted benefit and cost brought by the manager’s effort; so, nt∗ is independent of t.Remark 5.
It is shown from figures in the Appendix thatn∗ decreases with an increase in μ(the drift coefficient of the fund wealth process), θ(the effort cost coefficient), and γm(the manager’s risk aversion level). It increases with an increase in γf(the investor’s risk aversion level).Remark 6.
DefineYt∗=Vmt,x;YTZ∗,Γ∗, considering (12) and (13), we can get that Zt∗/Yt∗=θγmDtn∗ and Dtbt∗=−μ+n∗/2/σ2DtMt∗ are constants.Taking the minimum point into (45) and solving it, we can get(46)Vt,z,y=eBT−teγfzy−γf/γm.Here,(47)B=−μ+n∗/2μ+n∗σ2DtMt∗γf+μ+n∗/222σ2Dt2Mt∗2γf2+γfθn∗2μ+n∗/22σ2DtMt∗+n∗2μ+n∗/222σ2θ2Dt2Mt∗2γfγf+γm−n∗μ+n∗/22σ2γf2θDt2Mt∗2is a constant. As a consequence, we can also get the following results.(48)vt,x,y=eBT−teγfDtxy−γf/γm,Vft,x=vt,x,eγmR=eBT−teγfDtx−R.


### 4.3. The Investor’s Excitation Mechanism

In this section, let us analyze the investor’s excitation mechanism. DenoteYt∗=YtZ∗,Γ∗. From the above analysis, we know that(49)dYt∗=Dtγmθn∗2bt∗Yt∗2dt+bt∗Zt∗σdWt,Yt∗=eγmR.Using Ito’s formula, we have(50)dlnYt∗=Dtγmθn∗2bt∗2dt+bt∗Zt∗Yt∗σdWt−bt∗2σ2Zt∗22Yt∗2dt.Furthermore, we can get that the investment income undern∗andbt∗ satisfies(51)dXt∗=rXt∗+bt∗n∗+μdt+bt∗σdWt,which implies(52)dlnYt∗=Dtγmθn∗2bt∗2−bt∗2σ2Zt∗22Yt∗2dt+Zt∗Yt∗dXt∗−rXt∗+bt∗n∗+μdt=γmDtbt∗θn∗22−12Dt2bt∗2θ2γmn∗2σ2+Dtbt∗θn∗n∗+μdt+γmθn∗DtdXt∗−rXt∗dt.Define constant(53)A=Dtbt∗θn∗22−12Dt2bt∗2θ2γmn∗2σ2+Dtbt∗θn∗n∗+μ>0,and then we can obtain(54)dlnYt∗=γmAdt+γmθn∗DtdXt∗−rXt∗dt=γmAdt+θn∗dDtXt∗.So,(55)lnYT∗−γmR=γmAT−t+n∗θXT∗−Dtxcan be deduced immediately. Since lnYT=γmwXT, we can get the strategy(56)wXT=AT−t+n∗θXT∗−Dtx+R.It is a linear function ofXT∗−Dtx, which is the discounted profit of the investment.Remark 7.
It follows from the above results that the manager’s wages increase with the increase of the cost coefficient, the effort level and the discounted profit of the investment. Furthermore, the longer the work, the higher the salary. It is consistent with reality.We can also get the following corollary.Corollary 1.
(57)Ys∗=Vms,Xs∗;YT∗=eγmR+AT−t+n∗θXs∗−Dtx.This implies thatVms,Xs∗;YT∗ is a decreasing convex function of Xs∗. Thus, the assumption in Section 2 that YT∗ is an incentive strategy is proved.


## 4.1. The Verification of Assumptions

Assumption 1 (Assumption 3.2 in [19]).
H has at least one extreme point bt∗y,z,γ,nt∗y,z,γ. For any t∈0,T,Z,Γ∈Vt, we have π∗Z,Γ∈Π.Proof.
This is the result of Theorem1 and Lemma 1.
The Hamiltonian function can be expressed as(29)Ht,x,y,z,γ=infb>0Ft,x,y,z,b+b2σ22γ.
Here,(30)Ft,x,y,z,b=infn>0−Dtγmθn2b2y+rx+bμ+nz.
Define(31)YsZ=Yt−∫tsFr,Xr,YrZ,Zrdr+∫tsZrdXr,s∈t,T,and we have the following assumption.Assumption 2 (Assumption 4.3 in [19]).
F has at least one extreme point nt∗y,z,b; furthermore, b,n∗YZ,Z,b∈Π.Proof.
On the one hand, the right hand ofF is a parabola with an opening up with respect to n; so, the minimum point is attained at the axis of the parabola z/Dtγmθy, that is, nt∗y,z,b=z/Dtγmθy. On the other hand, since Z<0 is predictable, we can get that nt∗YtZ,Zt,b=bZt/DtγmθbYtZ is a positive predictable process. Furthermore, bandnt∗y,z,b are independent of x. This implies the Lipschitz continuity and linear growth of the coefficients in (2) with respect to the investment income process; then, (2) has a unique solution.Assumption 3 (Assumption 4.4 in [19]).
∀b>0, 1/b2σ2 is bounded.Proof.
We can get the result directly fromσ>0,b>0.


## 4.2. The Investor’s Value Function

Clearly, as soon as we getvt,x,y, we can obtain Vft,x. The following theorem gives the partial differential equation satisfied by vt,x,y.Theorem 3.
vt,x,y is the viscosity solution of(32)−vtt,x,y=infZ,Γ∈VtGt,x,y,Z,Γ,(33)vT,x,y=eγfxy−γf/γm,where(34)Gt,x,y,Z,Γ=rx+bt∗Z,Γμ+nt∗Z,Γvx+σ2bt∗Z,Γ22vxx+Dtγmθnt∗Z,Γ22bt∗Z,Γyvy+σ2bt∗Z,Γ22Z2vyy+σ2bt∗Z,Γ2Zvxy.Proof.
By the definition ofvt,x,y, we can obtain that it satisfies (33). Furthermore, according to the dynamic programming principle, we have(35)vt,x,y=infZ,Γ∈Vtvt+h,Xt+h∗Z,Γ,Yt+hZ,Γ.
By using Ito’s formula with respect tovs,Xs∗Z,Γ,YsZ,Γ from t to t+h, we have(36)vt+h,Xt+h∗Z,Γ,Yt+hZ,Γ=vt,x,y+∫tt+hvts,Xs∗Z,Γ,YsZ,Γ+Gs,Xs∗Z,Γ,YsZ,Γ,Zs,Γsds.
Combining with the above two equations, we can get(37)vtt,x,y+infZ,Γ∈VtGt,x,y,Z,Γ=0.
That is,vt,x,y satisfies (32). The proof is done.
Next, we are going to solve (32) and (33). Considering the boundary condition, we guess(38)vt,x,y=eγfDtxy−γf/γmEt,where Et is a function of t which satisfies ET=1.
If the variables in the solution can be separated from each other, (32) can be easily solved. However, (32) contains eγfDtx, which is a cross term of t and x. To cancel the cross term, we introduce zt=DtXt∗Z,Γ. Using Ito’s formula, we can get(39)dzt=−rDtXt∗Z,Γdt+DtdXt∗Z,Γ=Dtbt∗Z,Γμ+nt∗Z,Γdt+σdWt.
We can also obtainzT=XT∗Z,Γ. Define(40)Vt,z,y=infZ,Γ∈VtEeγfzT−lnYTZ,Γ/γm|zt=z=infZ,Γ∈VtEeγfXT∗Z,Γ−lnYTZ,Γ/γm|Xt∗Z,Γ=zDt=vt,zDt,y.
Obviously, solvingvt,x,y is equivalent to solving Vt,z,y. Using a similar method as the one in Theorem 3, we can get that(41)−Vt=infZ,Γ∈Vt−μ+nt∗Z,Γ/2μ+nt∗Z,Γσ2DtZΓVz−γmθnt∗Z,Γ2μ+nt∗Z,Γ/22σ2yDtZΓVy+μ+nt∗Z,Γ/222σ2Dt2Z2Γ2Vzz+nt∗Z,Γ2μ+nt∗Z,Γ/222σ2γmθyDt2Z2Γ2Vyy+nt∗Z,Γμ+nt∗Z,Γ/22σ2γmθyDt2Z2Γ2Vzy,(42)VT,z,y=eγfzy−γf/γm.
The first step in solving (41) is to find its minimum point. Define MZ,Γ=Z/Γ, it is shown in Section 3 that Z,Γ and MZ,Γ,n∗Z,Γ are one-to-one. Then, (41) is transformed into(43)−Vt=infn,M∈ℝ+×ℝ−−μ+n/2μ+nσ2DtMVz+μ+n/222σ2Dt2M2Vzz−γmθn2μ+n/22σ2yDtMVy+n2μ+n/222σ2γmθyDt2M2Vyy+nμ+n/22σ2γmθyDt2M2Vzy.
Now, the problem of finding the minimum point in (41) is changed into a problem of finding the minimum point in (43).
According to (38), we suppose that Vt,z,y=Eteγfzy−γf/γm. By some simple calculations, we can get that(44)Vzt,z,y=γfVt,z,y,Vzzt,z,y=γf2Vt,z,y,yVyt,z,y=−γfγmVt,z,y,y2Vyyt,z,y=γfγf+γmγm2Vt,z,y,yVzyt,z,y=−γf2γmVt,z,y.
Taking them into (43), we have(45)−E′tVt,z,y=infn,M∈ℝ+×ℝ−−μ+n/2μ+nσ2DtMγf+μ+n/222σ2Dt2M2γf2+γfθn2μ+n/22σ2DtM+n2μ+n/222σ2θ2Dt2M2γfγf+γm−nμ+n/22σ2γf2θDt2M2EtVt,z,y.
Since the right hand of (45) is continuous, the minimum point can only be attained at the stable points or the boundary points, which depends on the parameter values. Denote the minimum point of (45) by nt∗,Mt∗, and denote the corresponding minimum point of (41) by Zt∗,Γt∗. It is shown from the Appendix that nt∗ and DtMt∗ are constants concerning μ,θ,γf, and γm. Let nt∗=n∗.Remark 4.
On the one hand, the exponential form of the objective function implies thatbt∗ is independent of Xt∗. On the other hand, the benefit and the cost brought by the manager’s effort are only related to bt∗, so nt∗ is independent of Xt∗. Furthermore, in this paper, we consider the discounted benefit and cost brought by the manager’s effort; so, nt∗ is independent of t.Remark 5.
It is shown from figures in the Appendix thatn∗ decreases with an increase in μ(the drift coefficient of the fund wealth process), θ(the effort cost coefficient), and γm(the manager’s risk aversion level). It increases with an increase in γf(the investor’s risk aversion level).Remark 6.
DefineYt∗=Vmt,x;YTZ∗,Γ∗, considering (12) and (13), we can get that Zt∗/Yt∗=θγmDtn∗ and Dtbt∗=−μ+n∗/2/σ2DtMt∗ are constants.Taking the minimum point into (45) and solving it, we can get(46)Vt,z,y=eBT−teγfzy−γf/γm.Here,(47)B=−μ+n∗/2μ+n∗σ2DtMt∗γf+μ+n∗/222σ2Dt2Mt∗2γf2+γfθn∗2μ+n∗/22σ2DtMt∗+n∗2μ+n∗/222σ2θ2Dt2Mt∗2γfγf+γm−n∗μ+n∗/22σ2γf2θDt2Mt∗2is a constant. As a consequence, we can also get the following results.(48)vt,x,y=eBT−teγfDtxy−γf/γm,Vft,x=vt,x,eγmR=eBT−teγfDtx−R.


## 4.3. The Investor’s Excitation Mechanism

In this section, let us analyze the investor’s excitation mechanism. DenoteYt∗=YtZ∗,Γ∗. From the above analysis, we know that(49)dYt∗=Dtγmθn∗2bt∗Yt∗2dt+bt∗Zt∗σdWt,Yt∗=eγmR.Using Ito’s formula, we have(50)dlnYt∗=Dtγmθn∗2bt∗2dt+bt∗Zt∗Yt∗σdWt−bt∗2σ2Zt∗22Yt∗2dt.Furthermore, we can get that the investment income undern∗andbt∗ satisfies(51)dXt∗=rXt∗+bt∗n∗+μdt+bt∗σdWt,which implies(52)dlnYt∗=Dtγmθn∗2bt∗2−bt∗2σ2Zt∗22Yt∗2dt+Zt∗Yt∗dXt∗−rXt∗+bt∗n∗+μdt=γmDtbt∗θn∗22−12Dt2bt∗2θ2γmn∗2σ2+Dtbt∗θn∗n∗+μdt+γmθn∗DtdXt∗−rXt∗dt.Define constant(53)A=Dtbt∗θn∗22−12Dt2bt∗2θ2γmn∗2σ2+Dtbt∗θn∗n∗+μ>0,and then we can obtain(54)dlnYt∗=γmAdt+γmθn∗DtdXt∗−rXt∗dt=γmAdt+θn∗dDtXt∗.So,(55)lnYT∗−γmR=γmAT−t+n∗θXT∗−Dtxcan be deduced immediately. Since lnYT=γmwXT, we can get the strategy(56)wXT=AT−t+n∗θXT∗−Dtx+R.It is a linear function ofXT∗−Dtx, which is the discounted profit of the investment.Remark 7.
It follows from the above results that the manager’s wages increase with the increase of the cost coefficient, the effort level and the discounted profit of the investment. Furthermore, the longer the work, the higher the salary. It is consistent with reality.We can also get the following corollary.Corollary 1.
(57)Ys∗=Vms,Xs∗;YT∗=eγmR+AT−t+n∗θXs∗−Dtx.This implies thatVms,Xs∗;YT∗ is a decreasing convex function of Xs∗. Thus, the assumption in Section 2 that YT∗ is an incentive strategy is proved.



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