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

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 coeﬃcients c f and c m , 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.


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][2][3][4][5] to name just a few. e whole asset management process involves two parties: the investor and the manager. e 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 principalagent conflict. An important part of discussing the asset management problem is finding the investor's optimal incentive mode under the principle agent conflict.
ere are many papers committed to solving principalagent conflict problems. Most of the early literature studies investigate the discrete-time case (we can refer to [6][7][8] or a summary book [9]). e 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][12][13][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. us, there are some contributions in this paper: (i) is 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 attitudes is paper is organized as follows. In Section 2, 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.

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. e investor's return is determined by the terminal investment profit and the manager's salary. e terminal investment profit is related to the manager's investment strategy and effort level, and the incentive mechanism largely determines the manager's strategy. erefore, 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).
is 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 R n t which satisfies where μ ≥ 0, σ ≥ 0, and r > 0 is the risk-free interest rate, W(t) is a Brownian motion on (Ω, F, P), and n t t≥0 is the manager's effort level. Here, for the convenience of calculation, we assume that the drift coefficient of R n t is a linear function of the manager's effort level. In fact, as long as the drift coefficient of R n t has the form of R n t (r + f(n t )) for some function f(n), the same method in this paper can be used after replacing n with f(n). For more general forms of the drift coefficient of R n t , the existence of the time value makes it hard to obtain explicit solutions.
Considering the manager's strategy π � (b π t , n π t ), where b π t represents the wealth that the manager decides to operate at moment t( e manager may not want to operate all the wealth since the cost of the effort will increase with the wealth operated increases. e money left will get a risk-free return.) and n π t represents the manager's effort level at t. By some simple calculations, we can get that the investment income under this strategy satisfies Define the natural filtration produced by W(t) as F W t t≥0 . Now, let us give the definition of both the manager and the investor's admissible strategies. Considering the manager's strategy π � (b π t , n π t ). If b π t and n π t are bounded positive predictable stochastic processes, under the strategy π, (2) has a unique solution.
We call that strategy π � (b π t , n π t ) is admissible. Denote the set of all the manager's admissible strategies by Π.

Remark 1.
Here, we do not consider the case when b � 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 at T and denote it by w(·). If sup π∈Π E[w(X π T )] < ∞, 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. erefore, 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.
en, 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.
erefore, firstly, we fix the investor's incentive strategy w(·) and consider the manager's optimal problem. Suppose that the manager is risk-averse and denote his risk sensitivity coefficient by c m < 0. Referring to [18], we suppose that the manager needs to pay (θn 2 b/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. e 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) Denote the manager's optimal strategy by π w , then the value function is Suppose that the investor is risk-averse too, his risksensitive coefficient is c f < 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. erefore, the investor needs to find a reasonable incentive strategy to maximize his net income, that is, minimize where X w t is the investment income process under strategy π w . us, the investor's value function is Remark 2. e problem discussed above is not a standard stochastic optimal control problem since the form of w(·) 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. en, we can use the dynamic programming principle to solve the problem.

The Manager's Optimization Problem
Define D t � e r(T− t) , β(t, π) � c m D t (θn π2 t /2)b π t , and Γ(t, T, π) � e − T t β(u,π)du . en, J π m (t, x; w) can be denoted by Using the results of Section 3.4 in [21], we know that, under the incentive strategy w(·), the manager's value function V m (t, x; w) satisfies the HJB equation: and the boundary condition Since V m (t, x; w) is a decreasing convex function of x, we can define the Hamiltonian function: x, y, z, c, n, b), where Theorem 1.
is the minimum point of h in (10).
Proof. According to the definition, we know that h 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 h n (n, b; t, x, y, z, c) � − θD t bnc m y + bz, Combining the above two equations, we can obtain the stable point of h: e proof is done. □ Remark 3. In this case, the optimal investment strategy is similar to that without principal-agent relationships. e only difference is that the numerator of the optimal investment strategy is changed from (μ + n * y,z,c t ) into (μ + (n * y,z,c t /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 processes (Z t , Γ t ) to determine the manager's value function and transform the investor's incentive strategy into (Z t , Γ t ). en, the problem in Section 2 can be translated into a classical stochastic optimal control problem.
First, let us give the space of auxiliary stochastic pro-

Mathematical Problems in Engineering 3
Denote the set of all the processes satisfying the above conditions by V(t).
For some (Z, Γ) ∈ V(t) and Y t ≥ 0, define the F W -progressively measurable process Y Z,Γ on the filtration space (Ω, F, P, where X r is the investment income process. Clearly, for fixed Y t , Z, Γ, Y Z,Γ T is only related to the investment income process and is F T measurable, suppose that it is an incentive strategy (we prove it in Corollary 1). In the following, we give the relationship between Y Z,Γ s and the manager's value function. First, we give the following lemma.
Proof. On the one hand, since Z, Γ, Y Z,Γ are all predictable stochastic processes, referring to (12) and (13), we can get that b * Z,Γ and n * Z,Γ are bounded positive predictable stochastic processes. On the other hand, b * y,z,c t and n * y,z,c t are independent of x. Taking b * Z,Γ and n * Z,Γ into (2), we can get the Lipschitz continuity and linear growth of the coefficients in (2) with respect to X t ; then, (2) has a unique solution. e proof is done.
Denote the investment income process under π * Z,Γ by X * Z,Γ . We also have the following theorem.

Theorem 2. Denote the manager's value function with a terminal return (ln
Furthermore, the manager's optimal strategy is π * Z,Γ .
Using Ito's formula, we have It follows from (16) that e − r t β(u,π)du σZ r dW(r) is a martingale. Integrating and taking expectations on both sides of (21), we can get Furthermore, by simple calculations, under π * Z,Γ ∈ Π, we have Using (23) and Ito's formula, we can obtain With similar methods, integrating and taking expectations on both sides of (24), we have is implies that π * Z,Γ is the manager's optimal strategy and Up till now, fixing (Z, Γ) ∈ V(t), 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. at is, finding the optimal (Z, Γ) ∈ V(t) to maximize the investor's net profit.

The Investor's Optimization Problem
Suppose that the investor's wealth is x 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) ∈ V(t) to minimize his value function. Define Referring to eorem 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 Here, R is the minimum pay in order to make sure that the manager takes the job. Section 4.1 gives the verification of the three assumptions.

Proof.
is is the result of eorem 1 and Lemma 1. e Hamiltonian function can be expressed as Here, and we have the following assumption.
□ Assumption 2 (Assumption 4.3 in [19]). F has at least one extreme point n * y,z,b t Proof. On the one hand, the right hand of F is a parabola with an opening up with respect to n; so, the minimum point is attained at the axis of the parabola (z/D t c m θy), that is, n * y,z,b t � (z/D t c m θy). On the other hand, since Z < 0 is predictable, we can get that n * Y Z t ,Z t ,b t � (bZ t /D t c m θbY Z t ) is a positive predictable process. Furthermore, b and n * y,z,b t are independent of x. is 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.
Proof. We can get the result directly from σ > 0, b > 0. □

e Investor's Value Function.
Clearly, as soon as we get v(t, x, y), we can obtain V f (t, x). e following theorem gives the partial differential equation satisfied by v(t, x, y). where at is, v(t, x, y) satisfies (32). e proof is done. Next, we are going to solve (32) and (33). Considering the boundary condition, we guess where E(t) is a function of t which satisfies E(T) � 1.
If the variables in the solution can be separated from each other, (32) can be easily solved. However, (32) contains e c f D t x , which is a cross term of t and x. To cancel the cross term, we introduce z t � D t X * Z,Γ t . Using Ito's formula, we can get dz t � − rD t X * Z,Γ t dt + D t dX * Z,Γ t � D t b * Z,Γ t μ + n * Z,Γ t dt + σdW(t) . (39)

Mathematical Problems in Engineering 5
We can also obtain z T � X * Z,Γ T . Define Obviously, solving v(t, x, y) is equivalent to solving V(t, z, y). Using a similar method as the one in eorem 3, we can get that