An exit contract optimization problem

We study an exit contract design problem, where one provides a universal exit contract to multiple heterogeneous agents, with which each agent chooses an optimal (exit) stopping time. The problem consists in optimizing the universal exit contract w.r.t. some criterion depending on the contract as well as the agents' exit times. Under a technical monotonicity condition, and by using Bank-El Karoui's representation of stochastic processes, we are able to transform the initial contract optimization problem into an optimal control problem. The latter is also equivalent to an optimal multiple stopping problem and the existence of the optimal contract is proved. We next show that the problem in the continuous-time setting can be approximated by a sequence of discrete-time ones, which would induce a natural numerical approximation method. We finally discuss the optimaization problem over the class of all Markovian and/or continuous exit contracts.


Introduction
The contract theory is an important research subject in economics and applied mathematics.A basic problem in the theory is about the design of a contract between the principal(s) and the agent(s), where the principal provides to the agent a contract (reward function) depending on the action and/or the corresponding output of the agent, while the agent takes an optimal action according to the given contract/reward.The problem (of the principal) is then to design a best contract by taking into account the (optimal) action of the agent as well as the corresponding output and reward value paid to the agent.In this paper, we introduce an exit contract optimization problem with n ≥ 1 heterogeneous agents in a stochastic context.Let (Ω, F, F, P) be a filtered probability space, an exit contract is mathematically an adapted reward process Y = (Y t ) t≥0 , with which each agent i = 1, • • • , n solves an optimal stopping problem • In some countries, when a company plans to lay off a number of employees in a progressive way, the manager can not directly fire the chosen employees because of the labour union or law constraint.One needs to provide a time-dependent compensation plan which is identical for everyone, and then let the employees take voluntary leave.
• In some retirement systems, people are allowed to choose their retirement ages in a range.
In particular, everyone has an individual pension basis, and the real retirement pension depends on the basis as well as a parameter.The retirement plan, i.e. how this parameter depends on the age, needs to be universal to all people or all groups of people.
Let us also mention the recent work by Nutz and Zhang [24], which studies an optimal design problem of the exit scheme for a (large) population of agents playing a mean-field game of stopping under the exit scheme.In their context, each agent runs an identical and independent Brownian motion, and the agents are in a symmetric situation with interaction.For our exit contract problem, the main difference is that all agents share the same randomness in the probability space (Ω, F, P) and there is no interaction between the agents.
In the context with only one agent, the exit contract problem could become trivial as the principal can easily manipulate the exit time of the agent, so that it turns to be a so-called first best problem for the optimal stopping part, see e.g.Cvitanić, Wan and Zhang [10].However, in the current context with a universal exit contract for multiple heterogeneous agents, the agents may have different (optimal) exit times as their utility/reward process (f i ) i=1,••• ,n are different.It becomes no more trivial to manipulate directly all exit times of multiple agents as in the one agent case.To the best of our knowledge, this formulation (in both discrete-time and continuous-time settings) has not been studied in the literature.To provide a first approach, we follow the spirit of Sannikov [28] to focus on the value processes of the agents, which allows decoupling the principal's and the agents' problems.In particular, we will apply the remarkable representation theorem of the stochastic processes due to Bank and El Karoui [4] (see also the recent development by Bank and Besslick [3]).By this representation theorem, any optional process Y = (Y t ) t∈[0,T ] satisfying some integrability and regularity in expectation conditions can be represented as an integral of functional of another optional process L = (L t ) t∈[0,T ] .More importantly, the hitting times of the process L at different level provide the (minimal) solutions of a family of optimal stopping problems relying on Y .From this point of view, one can use L to represent the contract Y and at the same time the optimal exit times of the agents.It follows that, at least formally, one can decouple the initial problems and reformulate the exit contract design problem as an optimization problem over a class of processes L.Moreover, the exit design problem can be then reformulated to a multiple optimal stopping problem, where the optimal solution corresponds to the optimal stopping times of the agents with the corresponding optimal exit contract.From this point of view, our approach confirms that, under our technical conditions, our exit contract problem with multiple agents can be transformed as a first best problem (with a well chosen reward function) as in the one agent case (see more discussions in Section 2.3).
In the continuous-time setting, some technical upper-semicontinuous condition is required on the admissible contracts Y to ensure the existence of the optimal exit time of the agents.In the discrete-time setting, such technical condition is not required in the classical optimal stopping theory.We hence provide an analogue of the main results on the exit contract design problem, by developing the same techniques in the discrete-time framework.We can also show the convergence of the discrete-time problems to the continuous-time problem as time step tends to 0. In particular, this could induce natural numerical approximation methods for the initial continuous-time exit contract problem.Finally, by Bank-El Karoui's representation theorem in the discrete-time setting, we can identify the properties of the process L when Y is Markovian and/or continuous w.r.t.some underlying process X, which allows us to study the contract design problem when one is restricted to choose from the class of all Markovian and/or continuous contracts.
The rest of the paper is organized as follows.In Section 2, we formulate our exit contract design problem in a continuous-time framework, and provide an approach to decouple the principal's and agents' problems based on Bank-El Karoui's representation theorem.It is shown that the principal's problem (i.e.exit contract design problem) is equivalent to an optimal control problem or a multiple optimal stopping problem, and the existence of the optimal contract is obtained.Some examples and interpretations are provided in the end.In Section 3, we develop the analogue techniques for the discrete-time problem, and show its convergence to the continuous-time one as time step goes to 0. Finally, in Section 4, we stay in the discrete-time setting and study the problem with Markovian and/or continuous contracts.

Exit contract optimization problem in continuous-time setting
Let (Ω, F, P) be a complete probability space, equipped with a filtration F = (F t ) t∈[0,T ] , for some finite T > 0, satisfying the usual conditions, i.e. the map t −→ F t is right-continuous, and F 0 contains all null sets in F. Let T denote the collection of all F-stopping time taking values in [0, T ].Given τ ∈ T , we also denote

Mathematical formulation of the exit contract problem
We will formulate an exit contract design problem with n agents.An exit contract is a F-optional process Y = (Y t ) t∈[0,T ] , and each agent chooses to quit the contract at time t ∈ [0, T ] to receive a reward value Y t .In the typical case where F is the (augmented) filtration generated by some càdlàg process X = (X t ) t∈[0,T ] , a contract Y is a functional of the underlying process X.The contract design problem consists in choosing an optimal contract Y w.r.t. a criterion depending on Y as well as the agents' exit times.

Let us consider n agents indexed by
which is a F-optional process of class (D) (i.e. the family (Y τ ) τ ∈T is uniformly integrable), the agent i aims at solving the following optimal stopping problem where Namely, the agent i chooses an exit time τ i , before which she/he would receive a reward value with rate f i (t) (w.r.t.µ A ), and at which she/he would receive the compensation Y τ i .We assume that all agents stay in a risk-neutral context under P, so that their optimization problems turn out to be (2.1).
The optimal stopping problem (2.1) can be solved by the classical Snell envelop approach (see the recalling in Theorem A.1). Namely, let us denote and by S i the Snell envelop of G i , so that, for all τ ∈ T , An optional process Y of class (D) is said to be upper-semicontinuous in expectation (USCE) if, for any τ ∈ T and any sequence (τ n ) n≥1 ⊂ T satisfying either By the classical optimal stopping theory (see e.g.Theorem A.1), when the optional process Y is of class (D) and USCE in the sense of Definition 2.1, the smallest optimal stopping time is given by τi := inf t ≥ 0 : We will assume the above technical condition on the admissible contract Y , so that each agent has a unique smallest optimal stopping time.

The exit contract optimization problem
Let ξ be a fixed F T -measurable random variable such that E[|ξ|] < ∞, and define , and satisfies Y T ≥ ξ .
Then for each Y ∈ Y, the optimal stopping problem (2.1) of the agent i has a unique smallest optimal stopping time τi given by (2.8).To make the behaviour of the agents tractable, we fix Y as the set of all admissible contracts, and also assume that the agents will choose to exit the contract at the smallest optimal stopping time among all optimal ones.Next, before the exit time τi of agent i, the principal will receive a (continuous) reward value with rate g i (t) (w.r.t a deterministic atomless finite Borel measure µ P on [0, T ]) due to the work of agent i, where g i : [0, T ] × Ω −→ R is a progressively measurable process.Further, at the exit time τi , the principal pays the agent i the compensation Y τi .We also assume that the principal is risk-neutral, so that the exit contract optimization problem turns out to be The constraint Y T ≥ ξ can be considered as the participation constraint in our exit contract problem.In particular, it ensures that the reward value V A i of agent i satisfies Notice that the optimal stopping times (τ i ) i=1,••• ,n stay unchanged if the principal replaces the contract (Y t ) t∈[0,T ] by (Y t − C) t∈[0,T ] for an arbitrarily big constant C > 0, which would make the reward value V P of the principal in (2.4) to be ∞.The constraint Y T ≥ ξ will prevent the principal to choose the contract in this way.
(ii) Furthermore, the optimal exit times (τ i ) i=1, In the following of the paper, we make the following technical assumptions on ) Remark 2.4.The monotonicity condition (2.5) is a technical condition due to our approach, which would be restrictive in practice.In particular, it implies that the optimal stopping times {τ i } n i=1 are ordered, see also Remark 2.7 below.

Solving the exit contract problem
We will make use of Bank-El Karoui's representation theorem (recalled in Theorem A.4) to solve the principal's problem (2.4).As preparation, let us first interpolate the functionals (f i ) i=1,••• ,n as a functional defined on [0, T ]×Ω×R.By abus of notation, we denote it by f : (2.6) Then under Assumption 2.1, it is clear that, for each fixed ℓ ∈ R, Further, for each (t, ω) ∈ [0, T ] × Ω, the map ℓ −→ f (t, ω, ℓ) is continuous and strictly increasing from −∞ to +∞, and finally, Applying Bank-El Karoui's representation theorem (Theorem A.4), it follows that, for every contract Y ∈ Y, there exists an optional process L Y such that Moreover, the (smallest) optimal stopping time of each agent i = 1, • • • , n can be obtained from the process L Y by τi = inf t ≥ 0 : Notice that in (2.7), there may exist multiple solutions L Y for a given Y , but their running maximum L Y t := sup 0≤s≤t L Y s will be the same, so that the stopping times τi are uniquely defined.Then, at least formally at this stage, one can expect to reformulate the exit contract optimization problem (2.4) as an optimization problem over a class of optional processes L. Let us denote by L the collection of all F-optional processes L : [0, T ] × Ω −→ [0, n], and the subsets .9) Before we provide the solution of the principal's problem based on the representation in (2.7), we show that given any L ∈ L + , one can construct an admissible contract Y L ∈ Y. Lemma 2.5.Let Assumption 2.1 hold true.Then for every L ∈ L + , there exists an optional process Y L such that Moreover, Y L ∈ Y, and it has almost surely right-continuous paths.
Proof (i) Notice that a process L ∈ L + is nondecreasing, so that Therefore, there exists a càdlàg adapted (and hence optional) process Y L satisfying (2.10).
(ii) We next prove that Y L ∈ Y. From (2.10), we first obtain that It is enough to check that Y L is USCE in order to conclude that Y L ∈ Y.It is in fact continuous in expectation.Indeed, let {τ k } ∞ k=1 ⊂ T be such that τ k −→ τ as k → ∞ for some τ ∈ T , then we have by (2.10) and (2.11) where the last limit follows from the fact that and the integrability condition on Theorem 2.2.Let Assumption 2.1 hold true.Then the contract design problem (2.4) is equivalent to Further, their optimal solutions are also related in the following sense: (i) Let L * ∈ L + be an optimal solution to the optimization problem at the r.h.s. of (2.12), then the contract Y L * , defined below, is an optimal solution to the contract design problem (2.4): Moreover, the smallest optimal stopping times (τ Conversely, let Y * be an optimal contract to Problem (2.4), and L Y * be a corresponding optional process in the representation (2.7), let Then L Y * ∈ L + and it is an optimal solution to the optimization problem at the r.h.s. of (2.12).Moreover, the contract Y defined by L Y * through (2.13) is also an optimal contract to (2.4), and it induces the same smallest optimal stopping time τi (defined in (2.8)) for each agent i = 1, • • • , n as those induced by contract Y * .Remark 2.6.There are different ways to interpolate Nevertheless, when we consider the processes in L + 0 , which take values only in {0, 1, • • • , n}, it does not change the problem at the r.h.s. of (2.12).

Proof of Theorem 2.2. (i) Let us first prove that
Given any Y ∈ Y, we denote by L Y an optional process which provides the representation (2.7), and then define

One observes that, for each
It follows that Notice that L Y ∈ L + has almost surely left continuous paths, taking the sum over i = 1, • • • , n, it follows that (2.14) holds.
(ii) We next prove the reverse inequality: For each L ∈ L + , let Y L be the optional process given by Lemma 2.5, so that Y L ∈ Y and L s µ A (dt) F τ , a.s., for all τ ∈ T , then by Bank-El Karoui's representation theorem (Theorem A.4), τi is the smallest optimal stopping time of the i-th agent under the contract Y L .In particular, one has (2.17) It follows that and therefore the inequality in (2.16) holds.
(iii) Notice that for any L ∈ L + , we define L 0 t := [L t ], where [x] denotes the biggest integer less or equal to x.It is easy to verify that So we complete the proof of the second equality in (2.12).
(iv) Given an optimal solution L * ∈ L + to the optimization problem at the r.h.s. of (2.12), together with the equality (2.17), one has where τ * i is the smallest optimal stopping time of agent i with contract Y L * .Hence Y L * is an optimal solution to the exit contract design problem.
Similarly, given an optimal solution Y * , by using (2.15) and the representation (2.7), one can conclude that the corresponding L Y * is an optimal solution to the optimization problem at the r.h.s. of (2.12), L Y * ∈ L + , and the contract Y defined by L Y * through (2.13) is also an optimal contract to (2.4),Moreover, since it follows that Y * and Y induce the same smallest optimal stopping time τi for each agent i = 1, • • • , n.
Remark 2.7.As observed in the above proof, given L ∈ L + and the corresponding contract Y L defined in Lemma 2.5, the optimal stopping time of agent i is given by τi := inf{t ≥ 0 : L t ≥ i}.In particular, they are ordered in the sense that In fact, given an arbitrary contract Y ∈ Y, one can check that the smallest optimal stopping time τi of the agent i in Problem (2.1) satisfies Indeed, under Assumption 2.1 and by the definition of and it follows then from (2.8) that τi ≤ τi+1 , a.s. and hence (2.18) holds.
As noticed in Remark 2.7, the event {L t− ≥ i} at the r.h.s. of (2.12) defines a sequence of ordered stopping times.One can then reformulate the the exit contract design problem as a multiple stopping problem.More importantly, this reformulation allows us to obtain the existence of the optimal contract.Proposition 2.8.Let Assumption 2.1 hold true.
(i) The exit contract design problem (2.4) is equivalent to the following optimal multiple stopping problem, with τ n+1 ≡ T ,

19)
(ii) There exist stopping times τ1 ≤ • • • ≤ τn which solve the optimization problem at the r.h.s of (2.19).Moreover, the process L * t := n i=1 1 {τ i <t} is an optimal solution to the optimization problem at the r.h.s. of (2.12).Consequently, the corresponding contract Y L * defined as in (2.13) is an optimal contract to Problem (2.4).
On the other hand, given L ∈ L + 0 , one can define a sequence of ordered stopping times by 19) from (2.12), together with the equality (ii) By arranging the terms at the r.h.s. of (2. 19), it follows that The stopping time at the r.h.s. of (2. 19) is then equivalent to the optimal multiple stopping problem: (2.20) At the same time, it is easy to check that the continuity (in expectation) of the mapping Then it is enough to apply Theorem A.2 to prove the existence of the optimal stopping times {τ i } n i=1 ⊂ T .
Remark 2.9.The multiple stopping problem has been studied in the classical literature such as in Carmona and Touzi [7], Kobylanski, Quenez and Rouy-Mironescu [21].A main difference of the multiple stopping problem in (2.19) is that the stopping times are required to be ordered.

Further discussions and examples
Our exit contract design problem shares some main features with the classical principal-agent problem (as studied in [28]) since both problems optimize over a class of contracts.However, there would be some structural differences between the two problems: • For a classical principal-agent problem, a basic structure is the following: the agent makes an action a, which induces an output X a , and the contract ξ is a function of the output variable X a .In this setting, there exist two situations: the principal observes both agent's action a and the output X a , or the principal observes only X a .According to the two different situations, the principal's problem would be the so-called first best problem, or the second best problem.
• For our exit contract design problem, the agent makes an action τ , and the contract ξ is a function of (τ, X) with some underlying observable process X.There would be only one situation: the principal observes the agent's action τ so that both can agree with the payoff ξ(τ, X) paid by the principal.
For a standard principal agent problem, where the principal can provide an individual contract to the agent, it is well-known that the principal's problem can be reduced to the so-called first best problem, i.e. the principal controls directly the action of the agent in order to optimize an appropriate reward function.This is also the case for the exit contract design problem in the setting with one agent, see e.g.[10].However, it becomes much less obvious in our setting where the principal needs to provide a universal contract to multiple heterogeneous agents.Our result in Proposition 2.8 shows that, under the monotone condition in Assumption 2.1, one can give an order to different agents and then find an appropriate reward function (as that at the r.h.s. of (2. 19)) so that the initial problem reduces to a first best type optimization problem over the sequences of ordered stopping times.We have found such a reward function thanks to the approach in Theorem 2.2 based on the Bank-El Karoui's representation theorem.
Nevertheless, this seems not to be the feature of our problem without the monotone condition in Assumption 2.1.Indeed, without the monotone condition, there may be different ways to index the agents, and there is no reason that the optimal stopping times of the agents are ordered.Moreover, we show in the following example that, without the monotone condition, the problem is not equivalent to the first best problem if one uses the r.h.s. of (2.19) as the reward function.
Example 2.10.Let us consider a deterministic setting with 2 agents, where T = 3, ξ = 0, and µ A (dt) = µ P (dt) = δ 0 (dt) + δ 1 (dt) + δ 2 (dt), so that the problem can be considered as a discrete-time one on the grid {0, 1, 2, 3}.Let In this deterministic setting, the stopping times become deterministic times taking values in {0, 1, 2, 3}, and as the natural extension of the r.h.s. of (2.19), one can guess that the corresponding first best problem would be By a direct computation, we have , and In fact, by considering all (finitely) possible values of (τ 1 , τ 2 ), one can check that V 1 = V 2 = − 9 2 .Next, let us consider the exit contract optimization problem V P in (2.4).In our deterministic setting, it is enough to consider all possible values of
In the above example, we observe that can not be the corresponding first best problem.At the same time, it seems not clear to us how to formulate an appropriate first best problem for V P in this setting.We next provide an example with explicit solutions to the principal's and agents' problems (under the monotone condition), which could also illustrate the structure of our exit contract design problem.
Example 2.11.Let n = 2, T = 1, ξ = 0, µ A (dt) = µ P (dt) be the Lebesgue measure, f 1 (t) ≡ 1, f 2 (t) ≡ 2, and g 1 , g 2 : [0, T ] −→ R are both deterministic functions.Let us define h 2 : [0, 1] −→ R, and for each and Then the r.h.s. of (2.19) becomes (2.21) In this deterministic setting, given the functions f i and g i , one can easily compute an optimizer (τ 1 , τ2 ) for (2.21).Moreover, by Proposition 2.8, an optimal exit contract for the principal becomes and an optimizer L for (2.19) can be given by Let us provide some interpretations of the functions h 1 , h 2 as well as problem (2.21).First, under the monotone condition that f 1 < f 2 , it is known that the smallest optimal stopping time of Agent 1 will be smaller than that of Agent 2 (see Remark 2.7).Then, for each t ∈ [0, 1], h 2 (t) represents the cumulative reward value that Agent 2 expects to receive from time t if he/she chooses not to stop before T .Thus to encourage Agent 2 to stop at time t < T , the principal should provide at least a compensation value Y t = h 2 (t).Next, depending on the exit time τ2 of Agent 2 and the contract value Y τ2 = h 2 (τ 2 ), the value h 1 (τ 2 , t) denotes the cumulative reward value that Agent 1 expects to receive from time t if he/she chooses not to stop before τ2 .Therefore, to make Agent 1 stop at time t < τ2 , the principal should provide at least a compensation value Y t = h 1 (τ 2 , t).It follows that the principal's optimal exit contract problem turns to be equivalent to (2.21).
Finally, given an optimal solution τ1 ≤ τ2 of (2.21), one can further find an increasing function L : [0, 1] −→ {0, 1, 2} such that the hitting time of L to the level 1 (resp.2) is the time τ1 (resp.τ2 ), which is in fact a solution to the optimization problem at the r.h.s. of (2.19).
Remark 2.12.In view of the above interpretation of h 1 and h 2 , one can in fact provide a direct proof of Proposition 2.8 based on a Snell envelop type argument (in a general stochastic setting).We choose to first prove Theorem 2.2 by the arguments based on Bank-El Karoui's representation theorem, and then to deduce Proposition 2.8, for the following reasons.First, from a numerical point of view, the formulation in (2.12) is a quite standard optimal control problem, which can be solved numerically by the control techniques.We will also develop this point of view and provide a convergence result in Section 3.2.More importantly, the formulation in (2.12) and the corresponding approach are more flexible when one restricts the initial exit contract problem to a subclass of contract, such as the Markovian contract, and/or continuous contracts w.r.t.some underlying processes.We will develop this further in Section 4, see in particular Remark 4.2 for more discussions.
Finally, let us conclude the subsection by another example which highlights the role of the contract Y in incentivizing agents.
Example 2.13.Let us consider a stochastic setting with n = 2 agents, terminal time T < ∞, ξ = 0, and µ A (dt) = µ P (dt) = dt.Let B be a standard Brownian motion, the running rewards f 1 and f 2 of the two agents are given by Notice that f 1 and f 2 clearly satisfy the monotone condition (2.5).
In the case without incentive contract, each agent i = 1, 2 solves the following optimal stopping problem: Notice that both t −→ f 1 (t) and t −→ f 2 (t) are strictly decreasing, then the unique optimal stopping time τi for agent i = 1, 2 will be the first time that f i (t) becomes negative, that is, Next, we analyze the behavior of the agents when the principal provides a contract, which is also optimized w.r.t. the principal's utility functions g 1 and g 2 .By Proposition 2.8, it follows that: Let τ1 , τ2 ∈ T be two stopping times solve respectively the optimal stopping problems sup

.23)
Assume in addition that τ1 ≤ τ2 , a.s.Then the inequality in (2.22) becomes an equality, and (τ 1 , τ2 ) is an optimal solution to V P .Further, by (2.13), it leads to the optimal contract Y * given by (2.24) Moreover, τi is the optimal stopping time of the agents i = 1, 2 with the contract Y * .
In the following, let us consider different examples of the principal's utility function g 1 and g 2 , which leads to different optimal contract Y * as well as different optimal stopping times τ1 and τ2 of the agents.
1. Let g 1 (t) := 2 B t + t, g 2 (t) := 1 + t for all t ∈ [0, T ].Then by (2.23) and (2.24), the optimal contract is and the two optimal stopping problems in (2.23) reduce to so that the optimal stopping times τ1 and τ2 for the agents are given by τ1 = τ2 = T.
One observes that, comparing to the optimal stopping times τ1 and τ2 in the setting without contract, the contract incentivizes the agents to work for a longer period (until the terminal time in fact).

Let
and the optimal stopping problems in (2.23) reduce to so that the optimal stopping times τi of the agent i = 1, 2 are given by τ1 = τ2 = 0.
In this setting, the contract incentivise the agent to work for a shorter period (stop immediately at the beginning in fact).

Let g
The two optimal stopping problems in (2.23) reduce to sup and similarly, the optimal stopping times of the agents are given by Namely, comparing to τ1 and τ2 in the case without contract, the incentive contract makes the agent i = 1 to work for a shorter period, and the agent i = 2 to work for a longer period.

A discrete-time version and its convergence
We now study a discrete-time version of the exit contract design problem, and provide some analogue results to those in the continuous-time setting.In particular, the USCE technical condition is no more required in the discrete-time setting to define the admissible contracts.We next prove its convergence to the continuous-time problem as the time step goes to 0.

A discrete-time version of the exit contract problem
Let us consider a partition π = (t j ) 0≤j≤m of the interval [0, T ], i.e. 0 = t 0 < t 1 < • • • < t m = T , and study the exit contract design problem formulated on the discrete-time grid π.We will in fact embed the discrete-time problem into the continuous-time setting, and reformulate it as a continuous-time problem by considering piecewise constant processes.
We stay in the same probability space setting as in the continuous-time case, i.e. (Ω, F, P) being a complete probability space, equipped with the filtration F = (F t ) t∈[0,T ] satisfying the usual conditions.We next define the filtration F π = (F π t ) t∈[0,T ] by and T π denote the collection of all F π -stopping times taking values in π.Next, let Given a discrete-time contract Y ∈ Y π , the agents' optimal stopping problems are given by whose Snell envelop is also a piecewise constant process, so that the minimum optimal stopping time for the i-th agent is given by and takes only values in π (i.e.τi ∈ T π ), where, as in (2.2), Similarly, let The contract design problem is given by Recall that L, L + and L + 0 are defined above and in (2.9), we further define L π as the set of all F π -optional processes L : [0, T ] × Ω −→ [0, n] which is constant on each interval [t j , t j+1 ), and In this context, one still has the extension of the Bank-El Karoui's representation theorem (see Theorem A.4) without the USCE condition on Y .It follows that, for any Y ∈ Y π , there exists an F π -optional process L which is piecewise constant on each interval [t j , t j+1 ) such that where sup s∈[τ,t) L s will be replaced by L t− if L ∈ L π,+ , and the hitting time τ i := inf t ≥ 0 : L t ≥ i is the minimum solution to the i-th agent's optimal stopping problem (3.3).
On the other hand, for any L ∈ L π,+ , the process satisfies clearly that One can follow almost the same arguments in Theorem 2.2 and Proposition 2.8, but use the discrete-time version of the optimal stopping theory and Bank-El Karoui's representation theorem (see Theorems A.1 and A.4), to obtain the analogue solution to the discrete-time exit contract problem.Let us just state the results and omit the proof.
Then one has the following equivalence for the contract design problem: Moreover, there exists an optimal contract for Problem (3.4).
Remark 3.1.As in Theorem 2.2 and Proposition 2.8, one can also construct an optimal solution to (3.4) from a solution to (3.8), and vice-versa.We nevertheless skip this for simplicity.

Convergence of the discrete-time value function to the continuous-time one
As illustrated in Section 5 of Bank and Föllmer [5], the discrete-time version of the representation theorem could provide a numerical algorithm for the continuous-time problem.Here we consider the exit contract design problem, and provide a convergence result of the discrete-time problems to the continuous-time problem.
Let us consider a sequence (π m ) m≥1 of partitions of [0, T ], with π m = (t m j ) 0≤j≤m , and such that We also fix µ A,πm and µ P,πm by µ A,πm (dt) := m j=1 c A,m δ t m j (dt), and µ P,πm (dt) := ) and c P,m := µ P ((t m j−1 , t m j ]).Recall also that the corresponding value function V P,πm is defined by (3.4) with partition π m .
Theorem 3.3.Let Assumptions 2.1 and 3.2 hold true.Then Proof (i) For any fixed L ∈ L + 0 , and m ≥ 1, let us define so that L m ∈ L πm,+ 0 .Notice that L m , L are nondereasing and take only values in {0, 1, holds except for at most n numbers of time t m j .Therefore, for each It follows that Further, for L ∈ L + 0 , L t− remains constant on (t m j−1 , t m j ] except for at most n numbers of time t m j .It follows that, By (3.9) and dominated convergence theorem, we have for i = 0, 1, and it follows that Then by (3.12) and (3.13), one obtains that Similarly, one can obtain that This implies that Further, by the arbitrariness of L, this leads to the inequality (ii) To prove the reverse inequality, we notice that L ∈ L + 0 for any L ∈ L πm,+ 0 .Then, for each i = 1, • • • , n, one has the estimation where ε m is independent of L, and satisfies Similarly, one has, for any Then it follows that V P,πm ≤ V P + ε m + ε ′ m , which concludes the proof.4 The problem with Markovian and/or continuous contract In this section, we will investigate a version of the exit contract design problem, where the admissible contracts are required to be Markovian and/or continuous w.r.t.some underlying process X.We will restrict ourself in the following discrete-time setting with a partition π = m+1) be the canonical space with canonical process X = (X t j ) 1≤j≤m , i.e.X t j (ω) := ω j for all ω = (ω 0 , • • • , ω m ) ∈ Ω.We further extend it to be a continuous-time process by setting X t = X t j , for all t ∈ [t j , t j+1 ), j = 0, • • • , m − 1.By abus of notation, we still denote it by X = (X t ) t∈[0,T ] .Let F := F T , with F t := σ(X s : s ∈ [0, t]) for every t ∈ [0, T ], so that the filtration F = (F t ) t∈[0,T ] is right-continuous and F π = F by (3.1).We next assume that P is a probability measure on (Ω, F), under which (X t j ) j=0,••• ,m is a Markovian process, i.e.
Then there exists also a family of probability measures {P j x : j = 0, • • • , m, x ∈ R d } such that (P j x ) x∈R d consists of a family of conditional probability distribution of (X t j , • • • , X tm ) known X t j .Since F is generated by X, an admissible contract Y ∈ Y π (see its definition in (3.2)) is a functional of process X.We will further define a class of Markovian contracts in the sense that Y t j = y j (X t j ) for some function y j , and a class of continuous Markovian contracts by assuming y to be continuous.Let us denote by B(R d ) the collection of all Borel measurable functions defined on R d , and by C b (R d ) the collection of all bounded continuous functions defined on R d .Let We then obtain two variations of the contract design problem: V P,π m,c := sup where τi is the minimum optimal stopping time of the agent i (with the corresponding given contract Y ) in (3.3).
We can naturally expect to solve the above contract design problem as in Theorem 3.1, but consider a subset of L π .Let us define and with L π , L π,+ and L π,+ 0 defined above and in (3.5), we let We next formulate some additional technical conditions on the coefficient functions ξ, f and probability kernels (P Recall that, in (2.6), the functionals Notice also that g i (t, •) is only F t -measurable, it could depends on the whole past path of X t∧• .To emphasize this point, we will write f i (t, X t ), f (t, X t , ℓ), g(t, X t∧• ), ξ(X T ) in place of f i (t, ω), f (t, ω, ℓ), g(t, ω), ξ(ω) .x is continuous under the weak convergence topology.
Example 4.1.Let X be a diffusion process defined by the stochastic differential equation with a Brownian motion W in the filtered probability space (Ω, F, F, P) and the Lipschitz coefficient functions b and σ.Let X t = X t j for all t ∈ [t j , t j+1 ) and j = 0, • • • , m − 1.Then it satisfies Assumption 4.2.
Theorem 4.3.Let Assumption 4.1 hold true.Then Remark 4.2.Similarly to Theorem 2.2 and Proposition 2.8, it is equivalent to take the supremum over L π,+ m in place of L π,+ m,0 at the r.h.s. of (4.3) (reps.(4.4)).Moreover, one can construct the corresponding optimal solutions from the initial contract design problem in (4.1) (resp.(4.2)), and vice versa.Nevertheless, we do not have now the equivalent optimal stopping formulation for (4.3) and (4.4) as in Proposition 2.8.Intuitively, given a solution τ1 ≤ • • • τn to the optimal stopping problem as at the r.h.s. of 2.8, the corresponding L * t := n i=1 1 {τ i <t} is a priori measurable w.r.t.F t = σ(X s , s ≤ t), but not measurable w.r.t.σ(X t ).Therefore, at time t ∈ [0, T ], the contract value Y L * t defined in (2.13) has no reason to be a measurable function of X t .
As preparation of the above theorem, let us provide some technical lemmas.m and j = 0, • • • , m, the random variable Z ℓ t j defined below is σ(X t j )-measurable (up to the complementation of the σfield): Proof Recall that µ A,π is sum of Dirac measures on π, then by the dynamic programming principle, one has Z ℓ tm = Y tm a.s., and It is then enough to apply the induction argument.First, Z ℓ tm = Y tm is σ(X tm )-measurable as t m = T .Next, assume that that Z ℓ t j+1 is σ(X t j+1 ) measurable, then by the Markov property of X, one has and given L ∈ L π,+ m , we define This is enough to prove that L Y t j is σ(X t j )-measurable as Y t j and Z ℓ t j (for all ℓ ∈ R) are all σ(X t j )-measurable.Therefore, L Y ∈ L π .
(ii) Given L ∈ L π,+ m , it is easy to check that Y L is piecewise constant on each interval [t j , t j+1 ).Moreover, for t j = s < t ≤ t j+1 , one has sup v∈[0,t) L v = L t j .Then it is enough to apply the Markovian property of X to prove that Y L t j is σ(X t j )-measurable.
Then ĥ(x, ℓ) is also bounded continuous in x, and Lipschitz in ℓ uniformly in x.
where L Y t j is defined by (4.6), so that L Y ∈ L π m,c .
(ii) For any L ∈ L π,+ m,c , there exist mappings y where Y L is defined by (4.7), so that Y L ∈ Y π m,c .
. By definition, there exist bounded continuous mappings y j (•) : R d −→ R, such that Y (t j ) = y j (X t j ) for all j = 0, • • • , m. Recall that Z ℓ t j is defined in (4.5) and satisfies Then by Lemma 4.5, together with a backward induction argument, there exist bounded continuous functions z ℓ j : R d −→ R such that To conclude the proof of (i), it is enough to prove that Notice that Lemma 4.4 implies that L Y ∈ L π m , then it is easy to verify that L Y ∈ L π,+ m .Finally, taking the sum over i = 1, • • • , n, we prove the inequality (4.9).
(ii) We next prove the reverse inequality: (4.10)For each L ∈ L π,+ m , let Y L be defined by (4.7).Then Lemma 4.4 gives that Y L ∈ Y π m .So we can derive as in Theorem 2.2 similarly the inequality and therefore the reverse inequality (4.10) holds.
(iii) For any L ∈ L π,+ m , let us define L 0 So we complete the proof of (i) in the statement.
(iv) The second part can be proved in the same way as the first part using Lemma 4.6 instead of Lemma 4.4.

Conclusion
We have introduced an exit contract design problem in this work, where the principal provides a universal exit contract to multiple heterogeneous agents.Under a technical monotone condition, we have developed a systematic technique to transform the exit contract design problem into an optimal control problem, which can be formulated equivalently as a multiple optimal stopping problem, and it can be considered as a first best problem.This general method can be easily adapted to the discrete-time setting, and for some variations of the problem where the contract is required to be a Markovian and/or continuous functional of the underlying process.
An interesting future topic would be the mean-field extension of the exit contract design problem, where the agents interact between each other and the number of agents turns to infinite.To achieve this, one should rely on a mean-field extension of the Bank-El Karoui's representation theorem.Another interesting topic would be the application of our general approach to study more concrete economic problems, such as the optimal subsidy/tax policy problem for the government, where one looks for a universal subsidy/tax policy for different electricity companies in order to encourage them to replace the traditional fossil-fuel electricity generators by green energy ones.
stopping time τ ℓ is the smallest solution to (A.6).In particular, the stopping time τ ℓ does not depend on the choice of L.
(iii) Although the process L giving the representation (A.4) is not unique, it has a unique maximum solution in the following sense (see Theorem 2.16 of [3]).For all τ ∈ T and σ ∈ T τ , let us denote by l τ,σ the unique F τ -measurable random variable satisfying Then ( L(τ )) τ ∈T can be aggregated into an optional process L, and it is the maximum solution to (A.4) the sense that for any optional process L satisfying the representation (A.4), one has L τ ≤ L τ for all τ ∈ T .
Proof of Theorem A.4.We will prove the theorem separately for the continuous setting (i.e.µ is atomless and Y is USCE) and the discrete-time setting (i.e.µ is sum of Dirac measures).L s µ(dt) F τ .
As µ is atomless, one can change the interval of integration from [τ, T ) to (τ, T ].Moreover, for the term sup s∈[τ,t] L s , the fact that t → sup s∈[τ,t] L s is increasing implies that sup s∈[τ,t] L s = sup s∈[τ,t) L s , µ(dt)-a.s.We hence obtain (A.4).
(i.b)In the discrete-time setting, the representation (A.4) can be proved easily by a backward induction argument.However, to unify the proof of Item (ii) in both continuous-time and discrete-time setting, we provide the proof of (A.4) as in the continuous-time setting.When µ is sum of Dirac measures, we denote π := (t j ) 0≤j≤m of [0, T ] and write µ π (dt) := m j=1 c j δ t j (dt) instead of µ to remind the difference of the context.Let Y be an optional process such that Y T = ξ.We define the optional process Z ℓ such that and further a family of stopping times τ ℓ as well as a process L: τ ℓ t := min{s ≥ t : Z ℓ s = Y s }, L t := sup{l ∈ R : Z ℓ t = Y t }.
We will then prove that L is the required optional process by a backward induction argument.
For a fixed j ∈ {0, 1, • • • , m − 1}, we assume that L satisfies We will prove that Y t j can also be represented by L as above.
(ii) The optimal stopping theory implies that it is sufficient to prove Z ℓ τ ℓ = Y τ ℓ a.s. and for any 0 ≤ t < τ ℓ , Z A,ℓ t > Y t a.s., where Z ℓ is defined as (A.7).We observe that, for each ℓ ∈ R, one has While Z ℓ τ ℓ ≥ Y τ ℓ a.s. is trivial, to prove the reverse inequality, we have that L v µ(ds) F t = Y t , a.s.
We hence conclude the proof.
(i.a)In the continuous-time setting where µ is atomless and Y is USCE, one can assume Y T = 0 by considering the process(Y t − E[Y T |F t ]) t∈[0,T ] in place of Y .Then by[3], there exists an optional process L :[0, T ] × Ω −→ R, such that, for all τ ∈ T , E [τ,T ) h t, sup v∈[τ,t] L(v) µ(dt) < ∞, and Y τ = E [τ,T ) h t, sup s∈[τ,t] ••• ,n of the agents will not change if the principal replaces Y by Y − M for some martingale M .Therefore, the principal will always choose a contract Y such that Y T = ξ.Otherwise, he/she can replace Y by Y − M with M t := E[Y T − ξ|F t ] for t ∈ [0, T ] to have a better reward value.For this reason, one can assume, additionally and w.l.o.g., that Y T = ξ for all admissible contracts in Y.