Optimal Insurance with Limited Commitment in a Finite Horizon
aa r X i v : . [ ec on . T H ] J a n Optimal Insurance with Limited Commitment in aFinite Horizon ∗ Junkee Jeon † Hyeng Keun Koo ‡ Kyunghyun Park § January 14, 2019
Abstract
We study a finite horizon optimal contracting problem with limited commitment.A risk-neutral principal enters into an insurance contract with a risk-averse agent whoreceives a stochastic income stream and is unable to make any commitment. This probleminvolves an infinite number of constraints at all times and at each state of the world.Miao and Zhang (2015) have developed a dual approach to the problem by consideringa Lagrangian and derived a Hamilton-Jacobi-Bellman equation in an infinite horizon.We consider a similar Lagrangian in a finite horizon, but transform the dual probleminto an infinite series of optimal stopping problems. For each optimal stopping problemwe provide an analytic solution by providing an integral equation representation for thefree boundary. We provide a verification theorem that the value function of the originalprincipal’s problem is the Legender-Fenchel transform of the integral of the value functionsof the optimal stopping problems. We also provide numerical simulations results of theoptimal contracting strategies.
Keywords : Optimal contract, Limited commitment, Principal-Agent problem, Opti-mal stopping problem, Variational inequality, Singular control problem ∗ Junkee Jeon gratefully acknowledges the support of the National Research Foundation of Korea (NRF)grant funded by the Korea government (Grant No. NRF-2017R1C1B1001811). Hyeng Kuen Koo gratefullyacknowledges the support of the National Research Foundation of Korea (NRF) grant funded by the Koreagovernment (MSIP) (Grant No. NRF-2016R1A2B4008240). Kyunghyun Park is supported by NRF GlobalPh.D Fellowship (2016H1A2A1908911). † E-mail: [email protected]
Department of Applied Mathematics, Kyung Hee University, Korea. ‡ E-mail: [email protected]
Department of Financial Engineering, Ajou University, Korea. § E-mail: [email protected]
Department of Mathematical Sciences, Seoul National University, Ko-rea. Introduction
In this paper we investigate a contracting problem with limited commitment in continuoustime. A risk-neutral principal enters into an insurance contract with a risk-averse agentwho receives a stream of random income. The principal, typically a large institutionalagent such as a government agency or a financial institution, is able to make firm commit-ment to keep the contract for reputational or for other reasons. However, the agent, whomay be an individual or a small country, is not able to make such commitment. If bothparties were able to make firm commitment, the contract would result in a classical out-come of full insurance, where the principal absorbs all the risk in the agent’s income andprovides a constant stream of income to the agent. In the limited commitment case, thefull insurance outcome is not attainable and the principal provides only partial insuranceto the agent.There has been intensive economic research on optimal contract s with limited commit-ment (see e.g., Eaton and Gersovitz (1981), Thomas and Worrall (1988), Kehoe and Levine(1993)). Typically, the authors in the literature show that credit limits are used as amechanism to enforce the contract and try to characterize the credit limits. They alsoshow qualitative features of the contract. Recently, Grochulski and Zhang (2011) andMiao and Zhang (2015) proposed formulating the problem in continuous time and ob-taining a closed form characterization of the optimal contract. The closed-form outcomeexhibits the following feature of the optimal contract: the optimal contract starts witha payment much lower than in the full commitment case and ratchets up whenever theincome process hits a new high level. They also show that the contract outcome can be en-forced by providing unlimited insurance through futures contracts and optimal imposingcredit limits.We consider the continuous-time contracting problem with a finite contracting period.The infinite horizon models studied by Grochulski and Zhang (2011) and Miao and Zhang(2015) do not capture an important feature of real-world contracts: contracts mostly havefinite maturity dates. This is the motivation of our paper.We use the dual approach which allows us to use the Lagrangian method to solve theoptimization problem. To apply th is approach, we need to transform an infinite numberof constraints into one constraint. We utilize a method proposed by He and Pagés (1993)or Miao and Zhang (2015). We construct the Lagrangian and define the dual problem byusing the state variable , i.e., which is the agent’s income process, and a costate variable,i.e., the cumulative Lagrange multiplier process arising from the dynamic participationconstraints. We solve the dual problem by transforming the problem into a series ofoptimal stopping problems. The optimal stopping problems are, in a formal sense, equiv-alent to those of early exercise of American options. Then, we can apply well-developedtechniques to the latter contracting problem. In particular, we apply the integral repre-sentation of an American option value (see e.g. Kim (1990), Jacksa (1991), Carr et al.(1992), Detemple (2005)) and derive the optimal contract in analytic form. To obtain concrete solution, we apply the recursive integration method proposed by Huang et al.(1996) to solve numerically the integral equation. Contributions.
Our contributions are as follows. First, we provide an analytic solutionto the optimal contracting problem with limited commitment with a finite contractingperiod. Second, we make a technical contribution, providing a connection between thecontracting problem and the irreversible investment problem involving real options stud-ied by Dixit and Pindyck (1994). Miao and Zhang (2015) establish the duality theoremand provide a dynamic programming characterization of the dual problem. However, itis difficult to apply their method to the problem with a finite period, since one has toconsider a Hamilton-Jacobi-Bellman(HJB) equation with a gradient constraint involvingthree variables : time, income, and the agent’s continuation value. Moreover, it is noteasy to find a solution to the HJB equation. We overcome th is difficulty by considering atransformed problem, which is similar in its formal structure to an irreversible incremen-tal problem and equivalent to an infinite series of optimal stopping problems, so that itessentially becomes a single optimal stopping problem in our model.
Related literature.
In addition to the research mentioned above, there is extensive litera-ture on the contracting problem with limited commitment. Kehoe and Levine (1993) andAlvarez and Jermann (2000) investigate asset pricing based on the model with limitedcommitment. Zhang (2013) provides a solution to a long-term contracting problem in dis-crete time using a stopping-time approach. Bolton et al. (2016) shows that there is equiva-lence between the household problem and the contracting problem. Ai and Hartman-Glaser(2014), Ai and Li (2015), Ai et al. (2016), and Bolton et al. (2016) study optimal con-tracting between investors and an entrepreneur in a continuous-time model. Our paper,however, is different from the papers in the literature in the following aspects. First,the contract maturity is finite in our continuous-time model, whereas most papers con-sider either discrete-time long-horizon or continuous-time infinite-horizon models. Second,most authors use dynamic programming, and Miao and Zhang (2015) use in particularthe dual approach, whereas we also use the dual approach as well as transform ing theoriginal problem into optimal stopping problems. The duality approach has been usedto study continuous-time portfolio selection problems (see e.g., Cox and Huang (1989),Karatzas et al. (1987)). Jeon et al. (2018) also apply the dual approach and the trans-formation to study an optimal consumption and portfolio selection problem in which aneconomic agent does not tolerate a decline in standard of living.
Organization.
The rest of paper is organized as follows. Section 2 explains the model ofthe optimal contracting problem with limited commitment in a finite-horizon framework.In Section 3, we state our optimization problem. By constructing the Lagrangian of theoptimization problem, we define the dual problem. In Section 4, we analyse the variationalinequality arising from the dual problem. Section 5 establishes the duality theorem and rovides the analytic representation of the optimal strategies. In Section 6, we providenumerical simulation results. In Section 7 we draw conclusions. We extend a contracting model with limited commitment in a continuous-time frameworkstudied in Grochulski and Zhang (2011), and Miao and Zhang (2015). The crucial differ-ence between the existing models and ours is that our model is set up in the finite horizonto study how the horizon affect s the contract.The agent receives an income stream Y t which is an F t -adapted process. We considerthe following geometric Brownian motion for the income process: dY t = µY t dt + σY t dB t , Y = y, where µ > , σ > is constant and { B t } Tt =0 is a standard Brownian motion on the under-lying probability space (Ω , F , P ) and {F t } Tt =0 is the augmentation under P of the naturalfiltration generated by the standard Brownian motion { B t } Tt =0 . The agent’s income pro-cess Y is publicly observable by both the principal and the agent.The agent is risk-averse with subject discount rate ρ > . The contract horizon is [0 , T ] .At time , the principal offers a contract ( C, T ) to the agent. Then the instantaneouspayment or consumption of the agent C = { C t } Tt =0 is a non-negative F t -adapted processsatisfying E "Z T e − rt C t dt < ∞ . Note that the contract is dependent on the entire history of the Y t process in our model.We assume that the agent has no access to the financial market. The agent’s utilityat time is defined by U a ( C ) ≡ E "Z T e − ρt u ( C t ) dt , where u ( · ) is a continuously differentiable, strictly concave, and strictly increasing func-tion. Thus, the agent’s continuation utility at date t is given by U at ( C ) ≡ E t "Z Tt e − ρ ( s − t ) u ( C s ) ds , where E t [ · ] = E [ · | F t ] is the conditional expectation at time t on the filtration F t .We assume that the principal can freely access the financial market with constantrisk-free rate r > and can derive utility according to U p ( y, C ) ≡ E "Z T e − rt ( Y t − C t ) dt . Without loss of generality, we assume < r ≤ ρ . This also means that the principal ismore patient than or equally patient with the agent. o obtain explicit solutions, we assume that the agent uses the the constant relativerisk aversion (CRRA) utility function as a representative utility function. Assumption 1. u ( c ) ≡ c − γ − γ , γ > , γ = 1 , (1) where γ is the agent’s coefficient of relative risk aversion. The agent has a limited commitment. He/she can walk away from the contract andtake an outside value at any time after signing the contract. The outside value (or autarkyvalue) U d at time t is given by U d ( t, Y t ) ≡ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds = ( Y t ) − γ − γ − e − ˆ ρ ( T − t ) ˆ ρ , where ˆ ρ is defined by ˆ ρ ≡ ρ − (1 − γ ) µ + 12 γ (1 − γ ) σ . We make the following assumption to ensure a finite outside value for sufficiently large T . Assumption 2. ˆ ρ > . To ensure that the agent does not walk away, we impose the following dynamic par-ticipation constraint: U at ( C ) ≥ U d ( t, Y t ) , ∀ t ∈ [0 , T ] . (2)We also impose the promise keeping constraint (or individual rationality constraint): w = U a ( C ) , (3)where w is an initial promised value to the agent. Assumption 3.
We assume that the initial promised value w satisfies the followinginequality: w ≥ U d (0 , y ) . Lastly, the continuation utility of the principal at date t is defined by U pt ( Y, C ) ≡ E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds . We call a consumption plan { C s } Ts = t enforceable at time t ∈ [0 , T ] if the following condi-tions hold.(i) (Integrability condition) E t "Z Tt e − r ( s − t ) C s ds < ∞ . (4) ii) (Participation constraint) U as ( C ) ≥ U d ( s, Y s ) , ∀ s ∈ [ t, T ] . (5)Let Γ t ( y, w ) be the set of all enforceable consumption plans at time t .For some technical aspects, we make the following assumption: Assumption 4. ˆ r ≡ r − µ > , K ≡ r + ρ − rγ > , and µ > σ . (6) We first consider the first-best allocation contracting problem without participation con-straint (2) as the first-best benchmark. We use the agent’s continuation value w t ≡ U at ( C ) as a state variable. Problem 1 (First-Best Problem) . For given w t = w, Y t = y , the principal’s problem is to maximize V F B ( t, y, w ) = sup Γ ′ t ( y,w ) U pt ( y, C ) , where Γ ′ t ( y, w ) is the set of all consumption plans { C s } Ts = t satisfying the integrabilitycondition (4) . Note that the actual principal’s problem for first-best case is to find V F B (0 , y, w ) at t = 0 . However, the value function of Problem 1 is well-defined for any t > by thedynamic programming principle (Bellman (1954)) since the principal fully commits to thecontract until T .For Lagrangian multiplier λ ∗ > , let us consider the following Lagrangian for thefirst-best case: L F B ≡ E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds + λ ∗ E t "Z Tt e − ρ ( s − t ) u ( C s ) ds − w ! . Define the convex dual function of u as follows ˜ u ( z ) ≡ max c> { zu ( c ) − c } , for z > . (7)In the case of the CRRA utility function, the dual function is derived as follows: ˜ u ( z ) = γ − γ z γ . (8)Here, the first-best consumption is given by C F Bs = ( u ′ ) − (cid:16) e ( ρ − r )( s − t ) /λ ∗ (cid:17) , s ∈ [ t, T ] . (9) rom Lagrangian L F B , we can define the dual value function ˜ V F B ( t, λ ∗ , y ) as follows: ˜ V F B ( t, λ ∗ , y ) = E t "Z Tt e − r ( s − t ) ˜ u (cid:16) e − ( ρ − r )( s − t ) λ ∗ (cid:17) ds + E t "Z Tt e − r ( s − t ) Y s ds . Then, we can deduce the following duality-relationship: V F B ( t, y, w ) = inf λ ∗ > (cid:16) ˜ V F B ( t, λ ∗ , y ) − wλ ∗ (cid:17) . By a first-order condition, we can obtain w = 1 − e − K ( T − t ) K − γ ( λ ∗ ) γ − . From (9), C F Bs = e − ( ρ − r ) γ ( s − t ) (cid:18) K (1 − γ ) w − e − K ( T − t ) (cid:19) − γ , s ∈ [ t, T ] . (10)For the first-best consumption C F B , the following equality holds: U a ( C F B ) = w. This means that the risk-neutral principal bears all uncertainty and fully insures therisk-averse agents.Since < r ≤ ρ , we can easily confirm that the first-best consumption process C F B is non-increasing function over time.
We now write down the optimal contract problem with limited commitment between theprincipal and the agent.
Problem 2 (Primal problem) . Given w t = w and Y t = y , we consider the followingmaximization problem: V ( t, y, w ) ≡ sup C ∈ Γ t ( y,w ) U pt ( y, C ) . To obtain the solution of Problem 2, we will construct a Lagrangian for Problem2. The key is to write the part of the Lagrangian corresponding to the participationconstraint (5), which should hold at every s ∈ [ t, T ] and thus is comprised of infiniteindividual constraints. By utilizing the similar method proposed by He and Pagés (1993),Miao and Zhang (2015) wrote an infinite number of constraints as an integral of theconstraints. We write down the Lagrangian as follows: L ≡ E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds + λ E t "Z Tt e − ρ ( s − t ) u ( C s ) ds − w ! + E t "Z Tt e − r ( s − t ) η s Z Ts e − ρ ( ξ − s ) ( u ( C ξ ) − u ( Y ξ )) dξ ! ds , (11) here λ > is the Lagrange multiplier associated with the promise-keeping constraint(3) at each time t ≥ and e − r ( s − t ) η s ≥ is the Lagrange multiplier associated with theparticipation constraint (2) at each time s ∈ [ t, T ] .Using integration by parts, the third term of right-hand side in (11) can be given asfollows: E t "Z Tt e − r ( s − t ) η s Z Ts e − ρ ( ξ − s ) ( u ( C ξ ) − u ( Y ξ )) dξ ! ds = E t "Z Tt e − ρ ( s − t ) · e ( ρ − r )( s − t ) η s Z Ts e − ρ ( ξ − s ) ( u ( C ξ ) − u ( Y ξ )) dξ ! ds = E t "Z Tt d (cid:18)Z st e ( ρ − r )( ξ − t ) η ξ dξ (cid:19) Z Ts e − ρ ( ξ − t ) ( u ( C ξ ) − u ( Y ξ )) dξ ! ds = E t "Z Tt (cid:18)Z st e ( ρ − r )( ξ − t ) η ξ dξ (cid:19) e − ρ ( s − t ) ( u ( C s ) − u ( Y s )) ds . (12)Plugging th e equation (12) into the Lagrangian (11), we obtain L = E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds + E t "Z Tt (cid:18)Z st e ( ρ − r )( ξ − t ) η ξ dξ + λ (cid:19) e − ρ ( s − t ) ( u ( C s ) − u ( Y s )) ds − λw. (13)We define a costate process { X s } Ts = t as the cumulative amounts of the Lagrangian multi-pliers, X s ≡ Z st e ( ρ − r )( ξ − t ) η ξ dξ + λ, X t = λ. (14)This process is non-decreasing, continuous, and satisfies dX s = e ( ρ − r )( s − t ) η s ds, s ∈ [ t, T ] . Then we can write down L = E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds + E t "Z Tt X s e − ρ ( s − t ) ( u ( C s ) − u ( Y s )) ds − λw. (15)For every enforceable consumption plan { C s } Ts = t , we can deduce that E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds ≤ sup { C t } L . (16)To derive the dual problem, we first choose the consumption at each time to obtain themaximum of the Lagrangian (15), which takes the following form: L ( X ) = sup { C t } ( E t "Z Tt e − r ( s − t ) (cid:16) e − ( ρ − r )( s − t ) X s u ( C s ) − C s (cid:17) ds + E t "Z Tt e − r ( s − t ) Y s − e − ρ ( s − t ) X s u ( Y s ) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds − λw ) . (17) y the first-order condition for a consumption plan { C s } Ts = t in (17), the optimal con-sumption { C ∗ s } Ts = t is given as: C ∗ s = ( u ′ ) − (cid:18) e ( ρ − r )( s − t ) X s (cid:19) = ( e − ( ρ − r )( s − t ) X s ) γ , s ∈ [ t, T ] . (18)Then L ( X ) can be given by L ( X ) = E t "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) X s ) + Y s − e − ( ρ − r )( s − t ) X s u ( Y s ) (cid:17) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds − λw. (19)To ensure that the Lagrangian (19) is finite, we assume the following integrability condi-tions: E t "Z Tt e − ρ ( s − t ) | u ( Y s ) | X s ds < ∞ , E t "Z Tt e − r ( s − t ) | ˜ u ( X s e − ( ρ − r )( s − t ) ) | ds < ∞ . (20)Let us define J ( t, λ, y, X ) ≡ E t "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) X s ) + Y s − e − ( ρ − r )( s − t ) X s u ( Y s ) (cid:17) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds . (21)By the definition of the Lagrangian L ( X ) and the function J defined in (21), E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds ≤ J ( t, λ, y, X ) − λw. Then, it is clear that E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds ≤ inf λ> ,X ∈N D ( λ ) [ J ( t, λ, y, X ) − λw ] , where N D ( λ ) denotes the set of all positive non-decreasing, adaptable with respect to F ,right-continuous processes X with left-limits (RCLL) and starting at X t = λ , satisfying(20).Hence, the value function V ( t, y, w ) , the maximized utility value of the principal,satisfies the following inequality: V ( t, y, w ) = sup C ∈ Γ t ( y,w ) E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds ≤ inf λ> ,X ∈N D ( λ ) [ J ( t, λ, y, X ) − λw ] . (22)We will show in Theorem 5.1 that the maximized value is indeed equal to the right-handside of the inequality in (22) with the infimum being replaced by the minimum, i.e., V ( t, y, w ) = min λ> ,X ∈N D ( λ ) [ J ( t, λ, y, X ) − λw ] = min λ> (cid:20) min X ∈N D ( λ ) J ( t, λ, y, X ) − λw (cid:21) . (23) hat is, we should choose the process X to minimize L ( X ) : inf X ∈N D ( λ ) L ( X ) . (24)We now study the minimization problem inside the bracket of the right-hand side of thelast equality in (23), which we will call as the dual problem of Problem 2. Problem 3 (Dual problem) . Given λ > , consider the minimization problem: J ( t, λ, y ) = inf X ∈N D ( λ ) J ( t, λ, y, X ) . (25)To obtain the optimal costate process { X ∗ s } Ts = t , we transform Problem 3 to the optimalstopping problem . This transformation is due to the existence of one-to-one correspon-dence between the set of all costate processes { X s } Ts = t ∈ N D ( λ ) and the set of all infiniteseries of F -stopping times { τ ( x ) } x ≥ λ taking values in [ t, T ] which is non-decreasing andleft-continuous with right limits as a function of x .The correspondence is given by τ ( x ) = inf { s ≥ t | X s ≥ x } ∧ T. The problem of choosing a non-decreasing process { X s } Ts = t is similar to an irreversibleincremental investment problem studied by Pindyck (1988) and Dixit and Pindyck (1994).They consider the capacity expansion decision of a firm as a series of optimal stoppingproblems: for each level of capacity, there is a corresponding stopping problem wherethe firm chooses the optimal time to expand its capacity to an appropriate level. Basedon this idea, we transform Problem 3 into a series of optimal stopping problems in thefollowing lemma. Lemma 3.1.
We can write the dual value function as follows: J ( t, λ, y ) = − y − γ Z ∞ λ sup τ ( x ) ∈ [ t,T ] E Q t h e − ˆ ρ ( τ ( x ) − t ) h ( τ ( x ) , x H τ ( x ) ) i! dx + J ( t, λ, y ) , (26) where H s ≡ e − ( ρ − r )( s − t ) Y − γs ,h ( t, z ) ≡ − γ (cid:18) − e − ˆ ρ ( T − t ) ˆ ρ − − e − K ( T − t ) K z γ − (cid:19) ,J ( t, λ, y ) ≡ γ − γ − e − K ( T − t ) K λ γ + 1 − e − ˆ r ( T − t ) ˆ r y. (27) and the measure Q is defined in the proof. E Q [ · ] is the expectation with respect to measure Q . The corresponding standard Brownian motion B Q s is defined as B Q s ≡ B s − (1 − γ ) σs, s ∈ [ t, T ] . Proof.
Define a function f as follows f ( x ) = e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) x ) + Y s − e − ( ρ − r )( s − t ) x · u ( Y s ) (cid:17) . y assigning (1) and (8) to f , f ( x ) = e − r ( s − t ) (cid:18) γ − γ ( e − ( ρ − r )( s − t ) x ) γ + Y s − e − ( ρ − r )( s − t ) x − γ Y − γs (cid:19) (28)and f ′ ( x ) = e − ρ ( s − t ) − γ (cid:16) ( e − ( ρ − r )( s − t ) x ) γ − − Y − γs (cid:17) . (29)Then, J ( t, λ, y )= inf X s ∈N D ( λ ) E t "Z Tt f ( X s ) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds = inf X s ∈N D ( λ ) E t "Z Tt Z X s X t f ′ ( x ) dx + f ( X t ) ! ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds = inf X s ∈N D ( λ ) E t "Z Tt (cid:18)Z ∞ λ f ′ ( x ) · { x ≤ X s } dx (cid:19) ds + E t "Z Tt f ( λ ) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds = inf X s ∈N D ( λ ) E t "Z Tt (cid:18)Z ∞ λ f ′ ( x ) · { x ≤ X s } dx (cid:19) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds + E t "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) λ ) + Y s − e − ( ρ − r )( s − t ) λ · u ( Y s ) (cid:17) ds = inf X s ∈N D ( λ ) E t "Z Tt (cid:18)Z ∞ λ f ′ ( x ) · { w ≤ X s } dx (cid:19) ds + E t "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) λ ) + Y s (cid:17) ds = inf τ ( x ) ∈ [ t,T ] E t "Z ∞ λ Z Tt f ′ ( x ) · { s>τ ( x ) } ds ! dx + E t "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) λ ) + Y s (cid:17) ds = Z ∞ λ inf τ ( x ) ∈ [ t,T ] E t "Z Tτ ( x ) f ′ ( x ) ds dx + E t "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) λ ) + Y s (cid:17) ds . (30)where a stopping time τ ( x ) is defined by τ ( x ) = inf { s ≥ t | X s ≥ x } ∧ T, ∀ x ≥ λ. (31)Note that Fubini’s theorem implies that inf X s ∈N D ( λ ) E t "Z Tt (cid:18)Z ∞ λ f ′ ( x ) · { x ≤ X s } dx (cid:19) ds = inf τ ( x ) ∈ [ t,T ] E t "Z ∞ λ Z Tt f ′ ( x ) · { s>τ ( x ) } ds ! dx . Define a exponential martingale process Z ts and corresponding probability measure Q foreach s ∈ [ t, T ] , Z ts = exp (cid:26) −
12 (1 − γ ) σ ( s − t ) + (1 − γ ) σ ( B s − B t ) (cid:27) , and d Q d P = Z ts , respectively.Girsanov’s theorem implies that dB Q s = dB s − (1 − γ ) σds, s ∈ [ t, T ] s a standard Brownian motion under the measure Q .Since Y s = Y t e ( µ − σ )( s − t )+ σ ( B s − B t ) and Y t = y , let us define Y ts ≡ e ( µ − σ )( s − t )+ σ ( B s − B t ) .By using (29), the first term of the last equation in (30) can be derived as follows. inf τ ( x ) ∈ [ t,T ] E t "Z Tτ ( x ) f ′ ( x ) ds = inf τ ( x ) ∈ [ t,T ] E t "Z Tτ ( x ) e − ρ ( s − t ) − γ (cid:16) ( e − ( ρ − r )( s − t ) x ) γ − − Y − γs (cid:17) ds = inf τ ( x ) ∈ [ t,T ] − γ E t " e − ρ ( τ ( x ) − t ) E τ ( x ) "Z Tτ ( x ) e − ρ ( s − τ ( x )) (cid:16) ( e − ( ρ − r )( s − t ) x ) γ − − Y − γs (cid:17) ds = inf τ ( x ) ∈ [ t,T ] y − γ − γ E t e − ρ ( τ ( x ) − t ) ( Y tτ ( x ) ) − γ E τ ( x ) Z Tτ ( x ) e − ρ ( s − τ ( x )) ( Y τ ( x ) s ) − γ (cid:18) e − ( ρ − r )( s − t ) x ( Y s ) γ (cid:19) γ − − ds = inf τ ( x ) ∈ [ t,T ] y − γ − γ E t e − ˆ ρ ( τ ( x ) − t ) Z tτ ( x ) E τ ( x ) Z Tτ ( x ) e − ˆ ρ ( s − τ ( x )) Z τ ( x ) s (cid:18) e − ( ρ − r )( s − t ) x ( Y s ) γ (cid:19) γ − − ds = inf τ ( x ) ∈ [ t,T ] y − γ − γ E Q t e − ˆ ρ ( τ ( x ) − t ) E Q τ ( x ) Z Tτ ( x ) e − ˆ ρ ( s − τ ( x )) (cid:18) e − ( ρ − r )( s − t ) x ( Y s ) γ (cid:19) γ − − ds = inf τ ( x ) ∈ [ t,T ] y − γ − γ E Q t " e − ˆ ρ ( τ ( x ) − t ) E Q τ ( x ) "Z Tτ ( x ) (cid:18) e − ˆ ρ ( s − τ ( x )) ( x H s ) γ − · e − ( ρ − r )( s − τ ( x )) (cid:16) γ − (cid:17) ( Y τ ( x ) s ) γ − − e − ˆ ρ ( s − τ ( x )) (cid:17) ds ii = inf τ ( x ) ∈ [ t,T ] y − γ − γ E Q t (cid:20) e − ˆ ρ ( τ ( x ) − t ) (cid:18) − e − K ( T − τ ( x )) K ( x H τ ( x ) ) γ − − − e − ˆ ρ ( T − τ ( x )) ˆ ρ (cid:19)(cid:21) . (32) Remark 3.1.
Under the measure Q , dY s = ( µ + (1 − γ ) σ ) Y s ds + σY s dB Q s ,d H s = (ˆ r − ˆ ρ + σ γ ) H s ds − γσ H s dB Q s . (33) The process H s can be easily derived by Itô’s lemma. We will use this process to provevariational inequality(VI) later. The other term of last equation in (30) can be directly derived. E t "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) λ ) + Y s (cid:17) ds = γ − γ − e − K ( T − t ) K λ γ + 1 − e − ˆ r ( T − t ) ˆ r y. (34)By (32) and (34), J ( t, λ, y )= Z ∞ λ (cid:18) inf τ ( x ) ∈ [ t,T ] y − γ − γ E Q t (cid:20) e − ˆ ρ ( τ ( x ) − t ) (cid:18) − e − K ( T − τ ( x )) K ( x H τ ( x ) ) γ − − − e − ˆ ρ ( T − τ ( x )) ˆ ρ (cid:19)(cid:21)(cid:19) dx + γ − γ (cid:18) − e − K ( T − t ) K λ γ + 1 − e − ˆ r ( T − t ) ˆ r y (cid:19) = − y − γ Z ∞ λ sup τ ( x ) ∈ [ t,T ] E Q t (cid:20) e − ˆ ρ ( τ ( x ) − t ) − γ (cid:18) − e − ˆ ρ ( T − τ ( x )) ˆ ρ − − e − K ( T − τ ( x )) K ( x H τ ( x ) ) γ − (cid:19)(cid:21)! dx + γ − γ (cid:18) − e − K ( T − t ) K λ γ + 1 − e − ˆ r ( T − t ) ˆ r y (cid:19) . (35) his ends the proof of Lemma 3.1. (cid:3) In Lemma 3.1, the dual problem can be solved by converting it to the infinite series ofoptimal stopping time problems. Since the underlying process ( H s ) Ts = t has an exponen-tial form, however, the multiplicative factor x can be absorbed into the initial conditionproblem. Thus, the problems can be combined into a single problem, as shown below. Problem 4 (Optimal stopping problem) . We consider the following optimal stoppingproblem: g ( t, z ) = sup τ ∈S ( t,T ) E Q h e − ˆ ρ ( τ − t ) h ( τ, H τ ) (cid:12)(cid:12) H t = z i , where S ( t, T ) denotes the set of all stopping times of the filtration F taking values in [ t, T ] , and h ( t, z ) = 11 − γ (cid:18) − e − ˆ ρ ( T − t ) ˆ ρ − − e − K ( T − t ) K z γ − (cid:19) and ( H s ) s ≥ t satisfies the following stochastic diffusion process: d H s = (ˆ r − ˆ ρ + σ γ ) H s ds − γσ H s dB Q s . Notice that Problem 4 is equivalent to that of finding the optimal exercise time of anAmerican option written on the underlying process {H s } Ts = t with payoff equal to h ( τ, H τ ) at the time of exercise time τ . The exercise time is characterized as the first time forthe underlying process to hit the early exercise boundary(or free boundary), and thusthe problem is to derive the early exercise boundary. Kim (1990), Carr et al. (1992), andDetemple (2005) provide an integral equation representation of the American option valuefrom which one can derive a functional equation for the early exercise boundary.According to the standard technique for the optimal stopping problem, g ( T, z ) can bederived from the following variational inequality(VI). (See Chapter 2 of Karatzas and Shreve(1998) or Yang and Koo (2018)): − ∂ t g − L g = 0 , if g ( t, z ) > h ( t, z ) and ( t, z ) ∈ M T , − ∂ t g − L g ≥ , if g ( t, z ) = h ( t, z ) and ( t, z ) ∈ M T ,g ( T, z ) = h ( T, z ) , ∀ z ∈ (0 , + ∞ ) , (36)where M T = [0 , T ) × (0 , + ∞ ) and the operator L is generated by the process H s : L = γ σ z ∂ zz + (ˆ r − ˆ ρ + σ γ ) z∂ z − ˆ ρ. In this section we provide a complete self-contained derivation of the solution to VI (36)arising from Problem 4 by borrowing the ideas and proofs in Yang and Koo (2018). or convenience of proof, we substitute the above VI (36) for the following to makethe lower obstacle 0. Q ( t, z ) = g ( t, z ) − h ( t, z ) . Then, the (36) can be converted to − ∂ t Q − L Q = 11 − γ (cid:16) z γ − − (cid:17) , if Q ( t, z ) > and ( t, z ) ∈ M T , − ∂ t Q − L Q ≥ − γ (cid:16) z γ − − (cid:17) , if Q ( t, z ) = 0 and ( t, z ) ∈ M T ,Q ( T, z ) = 0 , ∀ z ∈ (0 , + ∞ ) . (37)Now, we will prove the existence and uniqueness of W , p,loc solution ( p ≥ to VI (37) anddescribe properties of the solution. Lemma 4.1.
VI (37) has a unique strong solution Q satisfying the following properties:1. Q ∈ W , p,loc ( M T ) ∩ C ( g M T ) for any p ≥ and ∂ z Q ∈ C ( g M T ) , where f M T = [0 , T ] × (0 , + ∞ ) .2. ∂ z Q ≥ a.e. in g M T and ∂ t Q ≤ a.e. in g M T . Proof.
1. To use the Theorem of Friedman (1975), replace the PDE operator L with anon-degenerate parabolic equation.Define ζ = log z, ¯ Q ( t, ζ ) = Q ( t, z ) . Then ¯ Q ( t, ζ ) satisfies − ∂ t ¯ Q − ¯ L ¯ Q = 11 − γ (cid:16) e ζ ( γ − − (cid:17) , if ¯ Q ( t, ζ ) > and ( t, e ζ ) ∈ M T − ∂ t ¯ Q − ¯ L ¯ Q ≥ − γ (cid:16) e ζ ( γ − − (cid:17) , if ¯ Q ( t, ζ ) = 0 and ( t, e ζ ) ∈ M T ¯ Q ( T, ζ ) = 0 , ∀ ζ ∈ (0 , + ∞ ) . (38)where ¯ L = γ σ ∂ ζζ + (ˆ r − ˆ ρ + σ γ ∂ ζ − ˆ ρ. Since the inhomogeneous term ‘ − γ (cid:16) e ζ ( γ − − (cid:17) ’, the lower obstacle ‘0’ and the ter-minal value ‘0’ are all smooth functions, we can easily show that VI(38) has a uniquesolution satisfying ¯ Q ∈ W , p,loc ( M T ) ∩ C ( g M T ) for any p ≥ and ∂ ζ ¯ Q ∈ C ( g M T ) (SeeFriedman (1975)).2. Let us denote e Q ( t, z ) = Q ( t, ηz ) for any η > . Then e Q satisfies following VariationalInequality: − ∂ t e Q − L e Q = 11 − γ (cid:16) ( ηz ) γ − − (cid:17) , if e Q ( t, z ) > and ( t, z ) ∈ M T , − ∂ t e Q − L e Q ≥ − γ (cid:16) ( ηz ) γ − − (cid:17) , if e Q ( t, z ) = 0 and ( t, z ) ∈ M T , e Q ( T, z ) = 0 ∀ z ∈ (0 , + ∞ ) . (39) or any η > , we can easily check that the inhomogeneous term of e Q is greater than Q : − γ (cid:16) ( ηz ) γ − − (cid:17) > − γ (cid:16) z γ − − (cid:17) , for ∀ γ > γ = 1) . In addition, since the terminal value of Q and e Q are the same, the comparison theory forVI implies that e Q ( t, z ) = Q ( t, ηz ) ≥ Q ( t, z ) for any η > and ( t, z ) ∈ M T . So we obtain ∂ z Q ≥ in M T .Define ˆ Q ( t, z ) = Q ( t − δ, z ) with δ > being sufficiently small. Then ˆ Q follows: − ∂ t ˆ Q − L ˆ Q = 11 − γ (cid:16) z γ − − (cid:17) , if ˆ Q ( t, z ) > and ( t, z ) ∈ M T , − ∂ t ˆ Q − L ˆ Q ≥ − γ (cid:16) z γ − − (cid:17) , if ˆ Q ( t, z ) = 0 and ( t, z ) ∈ M T , ˆ Q ( T, z ) = 0 ∀ z ∈ (0 , + ∞ ) . (40)Since ˆ Q ( T, z )(= Q ( T − δ, z )) ≥ Q ( T, z ) for any δ > , z > , the following can be deducedby the comparison theory for the same reason: ˆ Q ( t, z ) = Q ( t − δ, z ) ≤ Q ( t, z ) . Thereforewe can prove that ∂ t Q ≤ , a.e. (cid:3) We can define two regions derived from the Variational Inequality (37): Ω = { ( t, z ) | Q ( t, z ) = 0 } , Ω = { ( t, z ) | Q ( t, z ) > } . (41)Since ∂ z Q ≥ , we can define the free boundary z ⋆ ( t ) as follows: z ⋆ ( t ) = inf { z ≥ | Q ( t, z ) > } , ∀ t ∈ [0 , T ) . (42)The two regions can be defined according to the boundary as follows. Ω = { ( t, z ) | < z ≤ z ⋆ ( t ) , t ∈ [0 , T ] } , Ω = { ( t, z ) | z > z ⋆ ( t ) , t ∈ [0 , T ] } . (43) Lemma 4.2.
The free boundary z ⋆ is smooth, i.e., z ⋆ ( t ) ∈ C [0 , T ] ∩ C ∞ ([0 , T )) . More-over, the solution Q ≡ in Ω , and Q ∈ C ∞ ( { ( t, z ) | z ≥ z ⋆ ( t ) , t ∈ [0 , T ] } ) , and ∂ t Q ∈ C ( f M T ) . Proof.
By Lemma 4.1, we obtain ∂ t Q ≤ a.e. in f M T . Moreover, the coefficientfunctions in the operator L , the lower obstacle function, the terminal function, and thenon-homogeneous term − γ (cid:16) z γ − − (cid:17) are all smooth. Therefore, the regularity resultsin the lemma follow from Theorem 3.1 in Friedman (1975). (cid:3) Consider the following function Q ∞ as follow: Q ∞ ( t, z ) = (cid:18) − γ K α − ( z ∞ ) γ − − α − (cid:19) z α − + 11 − γ (cid:18) K z γ − − ρ (cid:19) , if ( t, z ) ∈ Ω ∞ , , if ( t, z ) ∈ Ω ∞ , (44)where Ω ∞ = { ( t, z ) | < z ≤ z ∞ , t ∈ [0 , T ] } , Ω ∞ = { ( t, z ) | z > z ∞ , t ∈ [0 , T ] } (45) nd z ∞ = (cid:18) ˆ ρ ( α − γ + γ − Kα − γ (cid:19) γγ − . (46) α + and α − are the positive and negative root of the following quadratic equation f ( α ) = 0 ,respectively: f ( α ) = γ σ α + (ˆ r − ˆ ρ + γ σ α − ˆ ρ = 0 . (47)It is easy to confirm that f ( −
1) = − ˆ r = µ − r < , and α − < − . Lemma 4.3.
The solution Q satisfy the following statements:1. Q ( t, z ) ≡ in Ω ∞ .2. Q ( t, z ) > in [0 , T ) × ( z T , ∞ ) , where z T = 1 . Proof.
1. It is easy to check that Q ∞ ∈ W , p,loc ( M T ) ∩ C ( g M T ) for p ≥ . Since α − < − , we deduce ∂Q ∞ ∂z = 1 γK z γ − − (cid:18) zz ∞ (cid:19) α − +1 − γ ! > , in Ω ∞ = { ( t, z ) | z > z ∞ , t ∈ [0 , T ] } . We can see that Q ∞ satisfies the following equality. − ∂ t Q ∞ − L Q ∞ = − γ (cid:16) z γ − − (cid:17) , in Ω ∞ = { ( t, z ) | z > z ∞ , t ∈ [0 , T ] } , , in Ω ∞ = { ( t, z ) | < z ≤ z ∞ , t ∈ [0 , T ] } . (48)Prior to proving . of Lemma 4.3 , we can first show the following inequality: z ∞ < z T ⇐⇒ (cid:18) ˆ ρ ( α − γ + γ − Kα − γ (cid:19) γγ − < . (49)If < γ < , the above inequality is equivalent to ˆ ρ ( α − γ + γ − < Kα − γ ⇐⇒ (ˆ ρ − K ) γα − < ˆ ρ (1 − γ ) . By Assumption 4, ˆ ρ − K = ( ρ − r ) (cid:18) − γ (cid:19) − (1 − γ ) µ + γ (1 − γ ) σ < . It is enough to show α − > ˆ ρ (1 − γ ) γ (ˆ ρ − K ) . ince α − is negative root of quadratic equation f ( α ) = 0 in (47), we need to show f (cid:16) ˆ ρ (1 − γ ) γ (ˆ ρ − K ) (cid:17) > . By simple calculations, f (cid:18) ˆ ρ (1 − γ ) γ (ˆ ρ − K ) (cid:19) = K ˆ ρσ (1 − γ ) γ (ˆ ρ − K ) > . The following γ > case is also the same as the previously proven case < γ < andthus is omitted here.By using (48) and the inequality z ∞ < z T in (49), − ∂ t Q ∞ − L Q ∞ ≥ − γ (cid:16) z γ − − (cid:17) ∀ ( t, z ) ∈ M T Note that ∂Q ∞ ∂z ( t, z ∞ ) = Q ∞ ( t, z ∞ ) = 0 and ∂Q ∞ ∂z ( t, z ) > in Ω . Therefore Q ∞ ( t, z ) > in Ω and Q ∞ satisfies the following VI: − ∂ t Q ∞ − L Q ∞ = 11 − γ (cid:16) z γ − − (cid:17) , in Q ∞ > and ( t, z ) ∈ M T , − ∂ t Q ∞ − L Q ∞ ≥ − γ (cid:16) z γ − − (cid:17) , in Q ∞ = 0 and ( t, z ) ∈ M T ,Q ∞ ( t, z ) ≥ , ∀ ( t, z ) ∈ M T . (50)By the comparison principle for VI, we have Q ( t, z ) ≤ Q ∞ ( t, z ) , ∀ ( t, z ) in M T . So we can get the first result of this lemma Q = 0 in Ω ∞ . ≤ Q ( t, z ) ≤ Q ∞ ( t, z ) ≤ , in Ω ∞ = { ( t, z ) | < z ≤ z ∞ , t ∈ [0 , T ] } . Furthermore, this means that Ω = { ( t, z ) | < z ≤ z ⋆ ( t ) , t ∈ [0 , T ] } ⊇ Ω ∞ = { ( t, z ) | < z ≤ z ∞ , t ∈ [0 , T ] } . Thus, we obtain that z ⋆ ( t ) ≥ z ∞ in [0 , T ] .2. Finally, let us prove the second property of this Lemma. VI (37) implies that Q = 0 , − ∂ t Q − L Q − − γ (cid:16) z γ − − (cid:17) ≥ , in Ω . This leads to Ω ⊆ [0 , T ] × (0 , z T ] . Hence, it is obvious that Ω = { ( t, z ) | Q ( t, z ) > } ⊇ [0 , T ) × ( z T , ∞ ) . Thus, Q ( t, z ) > in [0 , T ) × ( z T , ∞ ) . Moreover, z ⋆ ( t ) ≤ z T (= 1) in [0 , T ] . This is the end of the lemma. (cid:3)
Lemma 4.4.
The free boundary z ⋆ ( t ) , t ∈ [0 , T ] is strictly increasing with the terminalpoint z ⋆ ( T ) = lim t → T − z ⋆ ( t ) = z T . And z ∞ < z ⋆ ( t ) < z T , ∀ t ∈ [0 , T ) . roof. By Lemma 4.1, ∂ z Q ≥ , ∂ t Q ≤ a.e. in M T , Q ∈ C ( f M T ) . For any fixed t ∈ [0 , T ) , ≤ Q ( t, z ) ≤ Q ( t, z ⋆ ( t )) ≤ Q ( t , z ⋆ ( t )) = 0 , ∀ t ∈ ( t , T ] and ∀ z ∈ (0 , z ⋆ ( t )] . By the definition of the free boundary z ⋆ ( t ) , z ⋆ ( t ) ≥ z ⋆ ( t ) , ≤ t ≤ t < T. The above inequality shows that z ⋆ ( t ) is just increasing function with respect to t . (Westill need to show the strict increasing property of z ⋆ ( t ) ).Since z ⋆ ( t ) is increasing, the z ⋆ ( T ) = lim t → T − z ⋆ ( t ) exists. We know that z ⋆ ( t ) ≤ z T in ∀ t ∈ [0 , T ] . It is therefore sufficient to show that z ⋆ ( T ) ≥ z T . Otherwise, there existsinterval ( z ⋆ ( T ) , z T ) such that [0 , T ) × ( z ⋆ ( T ) , z T ) ⊂ Ω . So, we have shown that − ∂ t Q − L Q = 11 − γ (cid:16) z γ − − (cid:17) , in Ω Q ( T, z ) = 0 , ∀ z ≥ z ⋆ ( T ) , Q ( t, z ⋆ ( t )) = 0 ∀ t ∈ [0 , T ] . By using the above fact, we can deduce that at time T∂ t Q ( T, z ) = −L Q ( T, z ) − − γ (cid:16) z γ − − (cid:17) = − − γ (cid:16) z γ − − (cid:17) > ∀ z ∈ ( z ⋆ ( T ) , z T ) . However, it is easy to see that the above inequality is inconsistent with the results ofLemma 4.1.Finally, we show that z ⋆ ( t ) is strictly increasing. Otherwise, there exists t , t and z such that z ⋆ ( t ) = z for all t ∈ [ t , t ] where ≤ t < t ≤ T and z ∈ [ z ∞ , z T ] . Then,it is clear that Q ( t, z ) = 0 for all ( t, z ) ∈ [ t , t ] × (0 , z ) . Because of the continuity atthe free boundary of ∂ z Q , ∂ z Q ( t, z ) = 0 for all t ∈ [ t , t ] . Therefore, we obtain that ∂ t Q ( t, z ) = ∂ z ∂ z Q ( t, z ) = 0 for all t ∈ [ t , t ] . In domain [ t , t ) × ( z , ∞ ) , ∂ t Q satisfies − ∂ t ∂ t Q − L ∂ t Q = 0 , ∂ t Q ≤ , in [ t , t ) × ( z , ∞ ) ,∂ t Q ( t, z ) = 0 , ∀ t ∈ ( t , t ) . According to Hopf’s boundary point lemma (See Liu (2015)), we obtain that ∂ z ( ∂ t Q ) < ,which contradicts the ∂ z ∂ t Q ( t, z ) = 0 in t ∈ [ t , t ] . So, the free boundary z ⋆ ( t ) is strictlyincreasing. Thus, We conclude that z ∞ < z ⋆ ( t ) < z T for all t ∈ [0 , T ) (cid:3) Lemma 4.5.
For ( t, z ) ∈ M T , ≤ ∂ z Q ( t, z ) ≤ γ · − e − K ( T − t ) K z γ − . (51) roof. From VI (37), Q ( t, z ) satisfies − ∂ t Q − L Q = 11 − γ (cid:16) z γ − − (cid:17) , in Ω ,Q ( T, z ) = 0 , ∀ z ≥ z ⋆ ( T ); Q ( t, z ⋆ ( t )) = 0 , ∀ t ∈ [0 , T ] . Since L ( z∂ z Q ) = z∂ z ( L Q ) and ∂ z Q ( t, z ⋆ ( t )) = 0 , we have − ∂ t ( z∂ z Q ) − L ( z∂ z Q ) = 1 γ z γ − , in Ω , ( z∂ z Q )( T, z ) = 0 , ∀ z ≥ z ⋆ ( T ); ( z∂ z Q )( t, z ⋆ ( t )) = 0 , ∀ t ∈ [0 , T ] . Let us temporarily denote Q ( t, z ) = 1 γ · − e − K ( T − t ) K z γ − . Then, Q ( t, z ) satisfies − ∂ t Q − L Q = 1 γ z γ − , in Ω ,Q ( T, z ) = 0 , ∀ z ≥ z ⋆ ( T ); Q ( t, z ⋆ ( t )) = 1 γ · − e − K ( T − t ) K ( z ⋆ ( t )) γ − , ∀ t ∈ [0 , T ] . By the comparison principle for PDEs(see Lieberman (1996)), z∂ z Q ( t, z ) ≤ Q ( t, z ) . From Lemma 4.1, we can conclude ≤ ∂ z Q ( t, z ) ≤ γ · − e − K ( T − t ) K z γ − . (cid:3) We provide the integral equation representation of Q ( t, z ) in the following lemma. Lemma 4.6.
In the region Ω , the value function Q ( t, z ) has the following integralequation representation: Q ( t, z ) = z γ − − γ Z Tt e − K ( s − t ) N (cid:18) d γ (cid:18) s − t, zz ⋆ ( s ) (cid:19)(cid:19) ds − − γ Z Tt e − ˆ ρ ( s − t ) N (cid:18) d (cid:18) s − t, zz ⋆ ( s ) (cid:19)(cid:19) ds, where d ( t, z ) = log z + (ˆ r − ˆ ρ + ( γσ ) ) tγσ √ t , d γ ( t, z ) = log z + (ˆ r − ˆ ρ − ( γσ ) + γ ( γσ ) ) tγσ √ t , and N ( · ) is a standard normal distribution function.Moreover, the free boundary z ⋆ ( t ) satisfies the following integral equation: z ⋆ ( t )) γ − − γ Z Tt e − K ( s − t ) N (cid:18) d γ (cid:18) s − t, z ⋆ ( t ) z ⋆ ( s ) (cid:19)(cid:19) ds − − γ Z Tt e − ˆ ρ ( s − t ) N (cid:18) d (cid:18) s − t, z ⋆ ( t ) z ⋆ ( s ) (cid:19)(cid:19) ds. roof. From Lemma 4.1, Q ∈ W , p,loc ( M T ) ∩ C ( g M T ) , p ≥ . By applying Itó lemma to e − ˆ ρs Q ( s. H s ) (see Krylov (1980)), Z Tt d (cid:0) e − ˆ ρs Q ( s, H s ) (cid:1) = Z Tt e − ˆ ρs (cid:18) ∂Q∂s + L Q (cid:19) ds − γσ Z Tt e − ˆ ρs H s ∂Q∂z dB Q s . (52)By Lemma 4.5, E Q "Z T (cid:18) γσe − ˆ ρt H t ∂Q∂z (cid:19) dt ≤ σ K E Q "Z T (cid:16) e − ˆ ρt H t γ − γ (cid:17) dt < ∞ . This implies that γσ Z Tt e − ˆ ρs H s ∂Q∂z dB Q s is a martingale under Q measure and E Q t " γσ Z Tt e − ˆ ρs H s ∂Q∂z dB Q s = 0 . (see Chapter 3 in Oksendal (2005))By taking expectation to both-side of the equation (52), Q ( t, z ) = E Q t h e − ˆ ρ ( T − t ) Q ( T, H T ) i − E Q t "Z Tt e − ˆ ρ ( s − t ) (cid:18) ∂Q∂s + L Q (cid:19) ds = − E Q t "Z Tt e − ˆ ρ ( s − t ) (cid:18) ∂Q∂s + L Q (cid:19) { ( s, H s ) ∈ Ω } ds = − E Q t "Z Tt e − ˆ ρ ( s − t ) (cid:18) ∂Q∂s + L Q (cid:19) {H s ≥ z ⋆ ( s ) } ds = 11 − γ E Q t "Z Tt e − ˆ ρ ( s − t ) (cid:16) H s − γγ − (cid:17) {H s ≥ z ⋆ ( s ) } ds = 11 − γ E Q t "Z Tt e − ˆ ρ ( s − t ) H s − γγ {H s ≥ z ⋆ ( s ) } ds − − γ E Q t "Z Tt e − ˆ ρ ( s − t ) {H s ≥ z ⋆ ( s ) } ds . Since d P d Q = exp (cid:26) −
12 (1 − γ ) σ ( s − t ) − (1 − γ ) σ ( B Q s − B Q t ) (cid:27) and B s = B Q t + (1 − γ ) σs, for s ∈ [ t, T ] , E Q t "Z Tt e − ˆ ρ ( s − t ) H s − γγ {H s ≥ z ⋆ ( s ) } ds = z γ − E t "Z Tt e − K ( s − t ) {H s ≥ z ⋆ ( s ) } ds = z γ − Z Tt e − K ( s − t ) P ( H s ≥ z ⋆ ( s )) ds = z γ − Z Tt e − K ( s − t ) N log zz ⋆ ( s ) + (ˆ r − ˆ ρ − ( γσ ) + γ ( γσ ) )( s − t ) γσ √ s − t ! ds. imilarly, E Q t "Z Tt e − ˆ ρ ( s − t ) {H s ≥ z ⋆ ( s ) } ds = Z Tt e − ˆ ρ ( s − t ) Q ( H s ≥ z ⋆ ( s )) ds = Z Tt e − ˆ ρ ( s − t ) N log zz ⋆ ( s ) + (ˆ r − ˆ ρ + ( γσ ) )( s − t ) γσ √ s − t ! ds. Thus, Q ( t, z ) = z γ − − γ Z Tt e − K ( s − t ) N log zz ⋆ ( s ) + (ˆ r − ˆ ρ − ( γσ ) + γ ( γσ ) )( s − t ) γσ √ s − t ! ds − − γ Z Tt e − ˆ ρ ( s − t ) N log zz ⋆ ( s ) + (ˆ r − ˆ ρ + ( γσ ) )( s − t ) γσ √ s − t ! ds. By the smooth-pasting condition (Lemma 4.2), z ⋆ ( t )) γ − Z Tt e − K ( s − t ) N log z ⋆ ( t ) z ⋆ ( s ) + (ˆ r − ˆ ρ − ( γσ ) + γ ( γσ ) )( s − t ) γσ √ s − t ds − Z Tt e − ˆ ρ ( s − t ) N log z ⋆ ( t ) z ⋆ ( s ) + (ˆ r − ˆ ρ + ( γσ ) )( s − t ) γσ √ s − t ds. (cid:3) We will show that the value function Q ( t, z ) converges when the time-to-maturity goesto infinity. We will need the following lemma. Lemma 4.7.
For arbitrary c > and d ∈ R , Z ∞ e − cξ N ( d p ξ ) dξ = 12 c (cid:18) d √ d + 2 c (cid:19) . Proof.
By integration by parts, Z ∞ e − cξ N ( d p ξ ) dξ = (cid:20) − c e − cξ N ( d p ξ ) (cid:21) ξ = ∞ ξ =0 + d c √ π Z ∞ e − cξ − d ξ √ ξ dξ. (53)By Abramowitz and Stegun (1972) (p.304, equation (7.4.33)), for any a, b ∈ R , Z ∞ exp (cid:26) − a x − b x (cid:27) dx = √ π | a | e − | a || b | . Therefore, Z ∞ e − ( c + d ) ξ √ ξ dξ = √ π q c + d . (cid:3) We now provide the convergence of Q ( t, z ) in the following lemma. emma 4.8. lim T − t →∞ Q ( t, z ) = Q ∞ ( z ) and lim T − t →∞ z ⋆ ( t ) = z ∞ , where Q ∞ ( z ) and z ∞ are defined in (44) and (45) , respectively. Proof.
Let e z ⋆ define such that e z ⋆ ( T − t ) ≡ z ⋆ ( t ) , then z ∞ = lim T − t →∞ z ⋆ ( t ) = lim T − t →∞ e z ⋆ ( T − t ) . From the integral equation of z ⋆ in Lemma 4.6 and T − t → ∞ , z ∞ γ − γ Z ∞ e − Kξ N ˆ r − ˆ ρ − ( γσ ) + γ ( γσ ) γσ p ξ ! dξ − Z ∞ e − ˆ ρξ N (cid:18) ˆ r − ˆ ρ + ( γσ ) γσ p ξ (cid:19) dξ. (54)By applying Lemma 4.7 and direct computation, we can get z ∞ as follows: z ∞ = (cid:18) ˆ ρ ( α − γ + γ − Kα − γ (cid:19) γγ − . Consider the integral equation representation of Q ( t, y ) in Lemma 4.6, then Q ∞ ( z ) ≡ lim T − t →∞ Q ( t, z )= lim T − t →∞ z γ − − γ Z T − t e − Kξ N (cid:18) d γ (cid:18) ξ, z e z ⋆ ( T − t − ξ ) (cid:19)(cid:19) dξ − lim T − t →∞ − γ Z T − t e − ˆ ρξ N (cid:18) d (cid:18) ξ, z e z ⋆ ( T − t − ξ ) (cid:19)(cid:19) dξ = z γ − − γ Z ∞ e − Kξ N (cid:18) d γ (cid:18) ξ, zz ∞ (cid:19)(cid:19) dξ − − γ Z ∞ e − ˆ ρξ N (cid:18) d (cid:18) ξ, zz ∞ (cid:19)(cid:19) dξ = z γ − − γ Z ∞ e − Kξ N (cid:18) log zz ∞ γσ (cid:19) ξ − + ˆ r − ˆ ρ − ( γσ ) + γ ( γσ ) γσ ! ξ ! dξ − − γ Z ∞ e − ˆ ρξ N (cid:18)(cid:18) log zz ∞ γσ (cid:19) ξ − + (cid:18) ˆ r − ˆ ρ + ( γσ ) γσ (cid:19) ξ (cid:19) dξ. The exact solution of the above integral equation can be easily calculated by integrationby parts and some characteristic solutions.First, consider the following integral equation: Z ∞ e − Kξ N (cid:18) log zz ∞ γσ (cid:19) ξ − + ˆ r − ˆ ρ − ( γσ ) + γ ( γσ ) γσ ! ξ ! dξ. Let c = (cid:18) log zz ∞ γσ (cid:19) , and c = (cid:18) ˆ r − ˆ ρ + ( γσ ) γσ (cid:19) . (55) ince we consider the value of Q in the Ω , z ≥ z ⋆ ( ξ + t ) ≥ z ∞ implies c > .By integration by parts, Z ∞ e − Kξ N (cid:16) c ξ − + c ξ (cid:17) = − K e − Kξ N (cid:16) c ξ − + c ξ (cid:17)(cid:21) ∞ ξ =0 + 1 K √ π Z ∞ e − Kξ · e ( c ξ − + c ξ ) · (cid:16) − c ξ − + c ξ − (cid:17) dξ = 1 K + e − c c K √ π Z ∞ e − (cid:16) c √ (cid:17) x − − √ c K √ ! x ( − c x − + c ) dx. ( ξ = x ) Since Abramowitz and Stegun (1972) (p.304, equation (7.4.33)), Z ∞ exp (cid:26) − a x − b x (cid:27) dx = √ π | a | e − | a || b | , Z ∞ x exp (cid:26) − a x − b x (cid:27) dx = √ π | b | e − | a || b | . By using above equations , we can get Z ∞ e − Kξ N (cid:16) c ξ − + c ξ (cid:17) = 1 K − K − c p c + 2 K ! e − c ( √ c +2 K + c ) = 1 K − α + + (cid:16) γ − γ (cid:17) α + − α − · (cid:18) zz ∞ (cid:19) α − + (cid:18) γ − γ (cid:19) , (56)where q c + 2 K = γσ ( α + − α − )2 , q c + 2 K + c = − γσ ( α − + (cid:16) γ − γ (cid:17) ) , q c + 2 K − c = γσ ( α + + (cid:16) γ − γ (cid:17) ) . Similarly, we can derive Z ∞ e − ˆ ρξ N (cid:18)(cid:18) log zz ∞ γσ (cid:19) ξ − + (cid:18) ˆ r − ˆ ρ + ( γσ ) γσ (cid:19) ξ (cid:19) dξ = 1ˆ ρ (cid:18) − (cid:18) α + α + − α − (cid:19) (cid:18) zz ∞ (cid:19) α − (cid:19) . (57)By the equation (56) and (57), we can get Q ∞ as follows: Q ∞ ( z ) = (cid:18) − γ K α − ( z ∞ ) γ − − α − (cid:19) z α − + 11 − γ (cid:18) K z γ − − ρ (cid:19) , in Ω ∞ . (cid:3) By analyzing the variational inequality (36) in Section 4, we derive the following integralequation representation of value function of g ( t, z ) of Problem 4: g ( t, z ) = 11 − γ Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d (cid:18) s − t, zz ⋆ ( s ) (cid:19)(cid:19) ds − z γ − − γ Z Tt e − K ( s − t ) N (cid:18) − d γ (cid:18) s − t, zz ⋆ ( s ) (cid:19)(cid:19) ds, here z ⋆ ( t ) is the free boundary of Problem 4, N ( · ) is a standard normal distributionfunction, and d ( t, z ) = log z + (ˆ r − ˆ ρ + ( γσ ) ) tγσ √ t , d γ ( t, z ) = log z + (ˆ r − ˆ ρ − ( γσ ) + γ ( γσ ) ) tγσ √ t . Also, in terms of free boundary z ⋆ ( t ) , we can define the jump region JR and the no-jumpregion NR as follows: JR = { ( t, λ, y ) | λ ≤ z ⋆ ( t ) y γ } and NR = { ( t, λ, y ) | λ > z ⋆ ( t ) y γ } . As seen in Figure 1, the free boundary z ⋆ ( t ) partitions the ( t, z ) -region into the jumpregion and no-jump region. Time(year) λ / y γ no-jump region NR jump region JR Figure 1: The jump-region and the no-jump region in ( t, z ) -domain. Remark 5.1.
In terms of te regions Ω and Ω defined in Section 4, JR = { ( t, λ, y ) | ( t, λy γ ) ∈ Ω } and NR = { ( t, λ, y ) | ( t, λy γ ) ∈ Ω } . If initially ( t, λ, y ) ∈ JR , then X should increase immediately, such that λy γ reachesthe free boundary z ⋆ ( t ) . That is, the principal should increase the agent’s consumptionprocess. On the other hand, if ( t, λ, y ) ∈ NR , X must stay constant and this implies thatthe principal does not adjust the agent’s consumption process. Thus, we call JR and NR the jump region and the no-jump region, respectively.Moreover, the free boundary z ⋆ ( t ) satisfies the following integral equation: z ⋆ ( t )) γ − − γ Z Tt e − K ( s − t ) N (cid:18) d γ (cid:18) s − t, z ⋆ ( t ) z ⋆ ( s ) (cid:19)(cid:19) ds − − γ Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d (cid:18) s − t, z ⋆ ( t ) z ⋆ ( s ) (cid:19)(cid:19) ds. (58)Then, we can directly obtain the following lemma. emma 5.1. The optimal stopping time τ ∗ for Problem 4 is given by τ ∗ ≡ inf (cid:26) s ≥ t (cid:12)(cid:12)(cid:12)(cid:12) H s ≤ z ⋆ ( t ) (cid:27) ∧ T, where z ⋆ ( t ) satisfies the integral equation (58) . By using Lemma 3.1 and Lemma 5.1, we provide a solution to Problem 3 in thefollowing proposition.
Proposition 5.1. (a) The infinite series of optimal stopping times { τ ∗ ( x ) } x ≥ λ in Lemma 3.1 is given by τ ∗ ( x ) = inf { s ≥ t | x H s ≤ z ⋆ ( t ) } ∧ T. (b) The dual value function is given by J ( t, λ, y ) = − y − γ Z ∞ λ g ( t, xy γ ) dx + J ( t, λ, y )= − y − γ − γ Z ∞ λ "Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d γ ( s − t, xz ⋆ ( s ) y γ ) (cid:19) ds − (cid:18) xy γ (cid:19) γ − Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d ( s − t, xz ⋆ ( s ) y γ ) (cid:19) ds dx + γ − γ − e − K ( T − t ) K λ γ + 1 − e − ˆ r ( T − t ) ˆ r y. Remark 5.2.
It is easy to see that the infinite series of optimal stopping times { τ ∗ ( x ) } x ≥ λ defined in Proposition 5.1 is non-decreasing, left continuous with right limits as functionof x . Thus, the optimal stopping problem in (26) can be expressed as follows: sup τ ∗ ( x ) ∈ [ t,T ] E Q t h e − ˆ ρ ( τ ∗ ( x ) − t ) h ( τ ∗ ( x ) , x H τ ∗ ( x ) ) i = g ( t, xH t ) . (59)Proposition 5.1 (a) characterizes the optimal time to adjust the process X as wediscussed earlier. Proposition 5.1 (b) provides the dual value function by using the time-varying function z ⋆ ( t ) determining the free boundary for the optimal stopping problem.By Proposition 5.1, ∂ λ J ( t, λ, y ) = y − γ g (cid:18) t, λy γ (cid:19) + ∂ λ J ( t, λ, y ) . (60)From (41) and (43) in Section 4, ∂ λ J ( t, λ, y ) > y − γ h (cid:18) t, λy γ (cid:19) + ∂ λ J ( t, λ, y ) = U d ( t, y ) , for λ > y γ z ⋆ ( t ) ,∂ λ J ( t, λ, y ) = y − γ h (cid:18) t, λy γ (cid:19) + ∂ λ J ( t, λ, y ) = U d ( t, y ) , for λ ≤ y γ z ⋆ ( t ) . Hence, we can rewrite JR = { ( t, λ, y ) | ∂ λ J ( t, λ, y ) = U d ( t, y ) } , NR = { ( t, λ, y ) | ∂ λ J ( t, λ, y ) > U d ( t, y ) } . Now we can state the following corollary. orollary 5.1. The dual value function J ( t, λ, y ) can be rewritten by(a) In the no-jump region NR , J ( t, λ, y ) = − y − γ − γ Z ∞ λ "Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d (cid:18) s − t, xz ⋆ ( s ) y γ (cid:19)(cid:19) ds − (cid:18) xy γ (cid:19) γ − Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d γ (cid:18) s − t, xz ⋆ ( s ) y γ (cid:19)(cid:19) ds dx + γ − γ · − e − K ( T − t ) K λ γ + 1 − e − ˆ r ( T − t ) ˆ r y. (b) In the jump-region JR , J ( t, λ, y ) = J ( t, z ⋆ ( t ) y γ , y ) + ( λ − z ⋆ ( t ) y γ ) U d ( t, y ) . By applying a standard method of singular control problem developed by Davis and Norman(1990) or Fleming and Soner (2006) to Problem 3, the dual value function J ( t, λ, y ) satis-fies the certain Hamilton-Jacobi-Bellman(HJB) equation. In fact, Miao and Zhang (2015)study the infinite-horizon problem by solving the linear Hamilton-Jacobi-Bellman equa-tion. The following proposition provides that our derived dual value function J ( t, λ, y ) satisfies the associated HJB equation. Proposition 5.2.
The dual value function J ( t, λ, y ) satisfies the following HJB equation: min { ∂ t J + A J + y + ˜ u ( λ ) , ∂ λ J − U d ( t, y ) } = 0 , ( t, λ, y ) ∈ [0 , T ] × R + × R + ,J ( T, λ, y ) = 0 , where A = σ y ∂ yy + µy∂ y + ( r − ρ ) λ∂ λ − r. Proof.
It is sufficient to ∂ t J + A J + y + ˜ u ( λ ) = 0 in NR , and ∂ t J + A J + y + ˜ u ( λ ) ≥ in JR . From the representation of J ( t, λ, y ) in Proposition 5.1, we have ∂ t J = y − γ Z ∞ λ − ∂ t g (cid:18) t, xy γ (cid:19) dx − γ − γ e − K ( T − t ) λ γ − e ˆ r ( T − t ) yy∂ y J = y − γ Z ∞ λ h − (1 − γ ) g (cid:18) t, xy γ (cid:19) + γ xy γ ∂ z g (cid:18) t, xy γ (cid:19) i dxy ∂ yy J = y − γ Z ∞ λ h γ (1 − γ ) g (cid:18) t, xy γ (cid:19) − (cid:0) ( γ − γ ) + 2 γ (cid:1) xy γ ∂ z g (cid:18) t, xy γ (cid:19) − γ (cid:18) xy γ (cid:19) ∂ zz g (cid:18) t, xy γ (cid:19) i dxλ∂ λ J = y − γ λg (cid:18) t, λy γ (cid:19) + 11 − γ − e − K ( T − t ) K λ γ . (61) ince the value function g ( t, z ) satisfies the variational inequality (36), − ∂ t g ( t, xy γ ) − L g ( t, xy γ ) = 0 , for ( x, y ) ∈ NR . (62)By using (61) and (62), A J ( t, λ, y ) can be derived as follows: ∂ t J ( t, λ, y ) + A J ( t, λ, y )= y − γ Z ∞ λ h − ∂ t g − γ σ (cid:18) xy γ (cid:19) ∂ zz g − (ˆ r − ˆ ρ + γ σ + ( ρ − r )) xy γ ∂ z g + (ˆ ρ − ( ρ − r )) g i dx − y − γ ( ρ − r ) λg (cid:18) t, λy γ (cid:19) − γ − γ λ γ − y = y − γ Z ∞ λ h − ∂ t g − L g ( t, xy γ ) − ( ρ − r ) (cid:16) xy γ ∂ z g ( t, xy γ ) + g ( t, xy γ ) (cid:17)i dx − y − γ ( ρ − r ) λg (cid:18) t, λy γ (cid:19) − γ − γ λ γ − y = − y − γ ( ρ − r ) (cid:18)Z ∞ λ (cid:20) xy γ ∂ z g (cid:18) t, xy γ (cid:19) + g (cid:18) t, xy γ (cid:19)(cid:21) dx + λg (cid:18) t, λy γ (cid:19)(cid:19) − γ − γ λ γ − y = − y − γ ( ρ − r ) (cid:18)(cid:20) xg ( t, xy γ ) (cid:21) ∞ x = λ + λg (cid:18) t, λy γ (cid:19)(cid:19) − γ − γ λ γ − y. (63)By Lemma 4.6 and g ( t, z ) = Q ( t, z ) + h ( t, z ) , we have g ( t, z ) = z γ − − γ Z Tt e − K ( s − t ) N (cid:18) − d γ (cid:18) s − t, zz ⋆ ( s ) (cid:19)(cid:19) ds − − γ Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d (cid:18) s − t, zz ⋆ ( s ) (cid:19)(cid:19) ds, Since N (cid:16) − d γ (cid:16) s − t, zz ⋆ ( s ) (cid:17)(cid:17) exponentially converge to as z goes to infinity, it is easyto show that lim z →∞ zg ( t, z ) = 0 . This implies that ∂ t J ( t, λ, y ) + A J ( t, λ, y ) = − y − γ ( ρ − r ) (cid:18)(cid:20) xg ( t, xy γ ) (cid:21) ∞ x = λ + λg (cid:18) t, λy γ (cid:19)(cid:19) − γ − γ λ γ − y = − γ − γ λ γ − y. and ∂ t J ( t, λ, y ) + A J ( t, λ, y ) + y + ˜ u ( λ ) = 0 . Since − ∂ t g ( t, xy γ ) − L g ( t, xy γ ) ≥ for ( x, y ) ∈ JR , (64)we can similarly show that ∂ t J ( t, λ, y ) + A J ( t, λ, y ) + y + ˜ u ( λ ) ≥ in JR . (cid:3) We will now state and prove the main theorem of this paper. heorem 5.1. (a) For given w satisfying Assumption 3, the value function V ( t, w, y ) of Problem 2and the dual value function J ( t, λ, y ) derived in Proposition 5.1 satisfy the followingduality relationship : V ( t, w, y ) = min λ> ( J ( t, λ, y ) − λw ) . (65) There exists a unique solution λ ∗ with ( t, λ ∗ , y ) ∈ NR for the minimization problem (65) .(b) For s ∈ [ t, T ] , the optimal costate process X ∗ s is given by X ∗ s = max λ ∗ , sup t ≤ ξ ≤ s e ( ρ − r )( ξ − t ) Y γξ z ⋆ ( ξ ) ! , (66) where λ ∗ is the unique solution to the minimization problem (65) in (a). Moreover,the optimal costate process { X ∗ s } Ts = t satisfies the integrability condition (20) .(c) For s ∈ [ t, T ] , the optimal consumption plan c ∗ s and continuation value w ∗ t are,respectively, given by C ∗ s = ( λ ∗ s ) γ ,w ∗ s = ( λ ∗ s ) − γ − γ Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d (cid:18) s − t, λ ∗ s y γs z ⋆ ( t ) (cid:19)(cid:19) ds + ( λ ∗ s ) γ − − γ Z Tt e − K ( s − t ) N (cid:18) d γ (cid:18) s − t, λ ∗ s y γs z ⋆ ( s ) (cid:19)(cid:19) ds with λ ∗ s ≡ e − ( ρ − r )( s − t ) X ∗ s . Proof.
We will prove the duality relationship in the theorem in the following steps. (Step 1)
First, we will show that the dual value function J ( t, λ, y ) is strictly convex in λ in NR . Proof of (Step 1)
Consider λ , λ > λ = λ ) with ( t, λ , y ) , ( t, λ , y ) ∈ NR . Let λ = αλ + (1 − α ) λ with α ∈ (0 , . Then, clearly ( t, λ , y ) ∈ NR . Also, let X ∗ ,j be the optimal process for the minimization problem (25) with { X ∗ ,js } Ts = t ∈ D ( λ j ) , j = 1 , , . In other words, for j = 1 , , , J ( t, λ j , y ) = inf X ∈N D ( λ j ) ( E "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) X s ) + Y s − e − ( ρ − r )( s − t ) X s u ( Y s ) (cid:17) ds + λ j E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds = E "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) X ∗ ,js ) + Y s − e − ( ρ − r )( s − t ) X ∗ ,js u ( Y s ) (cid:17) ds + λ j E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds . Then, αJ ( t, λ , y ) + (1 − α ) J ( t, λ , y )= E "Z Tt e − r ( s − t ) (cid:16) α ˜ u ( e − ( ρ − r )( s − t ) X ∗ , s ) + (1 − α )˜ u ( e − ( ρ − r )( s − t ) X ∗ , s )+ Y s − e − ( ρ − r )( s − t ) ( αX ∗ , s + (1 − α ) X ∗ , s ) u ( Y s ) (cid:17) ds i + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds By (b) in Proposition 5.1, the optimal processes X ∗ ,j ( j = 1 , , ) are given by X ∗ ,js = max λ j , sup t ≤ ξ ≤ s e ( ρ − r )( s − t ) Y γξ z ⋆ ( ξ ) ! . Since λ = λ and X ∗ ,jt = λ j , we can deduce that X ∗ , s = X ∗ , s a.s.Thus, by strict convexity of ˜ u ( · ) , α ˜ u ( e − ( ρ − r )( s − t ) X ∗ , s )+(1 − α )˜ u ( e − ( ρ − r )( s − t ) X ∗ , s ) > ˜ u ( e − ( ρ − r )( s − t ) ( αX ∗ , s +(1 − α ) X ∗ , s )) . Let us temporarily denote ¯ X ∗ = ( αX ∗ , s + (1 − α ) X ∗ , s ) . Then, αJ ( t, λ , y ) + (1 − α ) J ( t, λ , y ) > E "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) ¯ X ∗ s ) + Y s − e − ( ρ − r )( s − t ) ¯ X ∗ s u ( Y s ) (cid:17) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds ≥ inf X ∈N D ( λ ) ( E "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) X s ) + Y s − e − ( ρ − r )( s − t ) X s u ( Y s ) (cid:17) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds = E "Z Tt e − r ( s − t ) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) X ∗ , s ) + Y s − e − ( ρ − r )( s − t ) X ∗ , s u ( Y s ) (cid:17) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds = J ( t, λ , y ) . his implies that J ( t, λ, y ) is strictly convex in λ in NR . (cid:3) Now, we rewrite the Lagrangian L in (19) as L ( t, λ, y, X ) = E t "Z Tt e − r ( s − t )) (cid:16) ˜ u ( e − ( ρ − r )( s − t ) X s ) + Y s − e − ( ρ − r )( s − t ) X s u ( Y s ) (cid:17) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds − λw. (67)with X t = λ, Y t = y . For simplicity, let L ( t, λ, y, X ) = L ( λ, X ) . (Step 2) For every enforceable plan C ∈ Γ( t, y, w ) , every x > , and every X ∈ N D ( λ ) ,the following inequality is established: L ( λ, X ) ≥ U Pt ( y, C ) = E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds The equality holds if and only if for all s ∈ [ t, T ] , X s e − ( ρ − r )( s − t ) u ′ ( C s ) − , Z Ts e − ρ ( ξ − s ) ( U aξ ( C ) − U d ( ξ, Y ξ )) dX ξ = 0 . This leads to the following weakly duality relationship: V ( t, w, y ) ≤ inf λ> ( J ( t, λ, y ) − λw ) . (68) Proof of (Step 2)
By the definition of ˜ u ( · ) , ˜ u ( e − ( ρ − r )( s − t ) X s ) ≥ X s e − ( ρ − r )( s − t ) u ( C s ) − C s . This leads to L ( λ, X ) ≥ E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds + E t "Z Tt e − r ( s − t ) (cid:16) X s e − ( ρ − r )( s − t ) ( u ( C s ) − u ( Y s ) (cid:17) ds + λ E t "Z Tt e − ρ ( s − t ) u ( Y s ) ds − w ! = E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds + E t "Z Tt e − ρ ( s − t ) Z Ts e − ρ ( ξ − s ) ( u ( C ξ ) − u ( Y ξ )) dξ · dX s + λ E t "Z Tt e − ρ ( s − t ) u ( C s ) ds − w ! ≥ E t "Z Tt e − r ( s − t ) ( Y s − C s ) ds , where the middle equation follows from integration by parts, and the last inequality followsfrom the fact C is enforceable and X is in N D ( λ ) . learly, the equality holds if and only if X s e − ( ρ − r )( s − t ) u ′ ( C s ) − , Z Ts e − ρ ( ξ − s ) ( U aξ ( C ) − U d ( ξ, Y ξ )) dX ξ = 0 . Moreover, we can immediately obtain V ( t, w, y ) ≤ inf λ> ( J ( t, λ, y ) − λw ) . (cid:3) (Step 3) The consumption plan C ∗ defined in (18) is enforceable and optimal. Proof of (Step 3)
Let us assume that ( X ∗ s ) Ts = t ∈ N D ( λ ∗ ) minimize the Lagrangian L in(24). Under this assumption we first prove that ( c ∗ s ) Ts = t is enforceable and optimal. Thenwe will show the existence and uniqueness of λ ∗ and derive ( X ∗ s ) Ts = t . Finally, we derivethe optimal continuation process.Since ( X ∗ s ) Ts = t ∈ N D ( λ ∗ ) , we know that E t "Z Tt e − r ( s − t ) | ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) | < ∞ . From this, it is easy to check that E t "Z Tt e − r ( s − t ) C ∗ s ds < + ∞ . For sufficiently small δ > and h ∈ (0 , δ ) , we consider λ h ≡ λ ∗ + h, X hs ≡ X ∗ s + h, and X δs ≡ X ∗ s + δ Then E t "Z Tt e − r ( s − t ) | ˜ u (( X ∗ s + δ ) e − ( ρ − r )( s − t ) ) | ds < (cid:12)(cid:12)(cid:12)(cid:12) γ − γ (cid:12)(cid:12)(cid:12)(cid:12) E t "Z Tt e − r ( s − t ) (cid:16) e − ( ρ − r )( s − t ) ( X ∗ s (1 + δ/X ∗ t )) (cid:17) γ ds < (1 + δ/λ ) γ E t "Z Tt e − r ( s − t ) | ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) | < ∞ . (69)Similarly, E t "Z Tt e − r ( s − t ) | ˜ u (( X ∗ s (1 ± δ )) e − ( ρ − r )( s − t ) ) | ds < ∞ . (70)Since E t "Z Tt e − ρ ( s − t ) | U d ( Y s ) | X ∗ s ds < ∞ , learly, we can have E t "Z Tt e − ρ ( s − t ) | U d ( Y s ) | ( X s + δ ) ds < ∞ . The convexity of ˜ u ( · ) implies e − ( ρ − r )( s − t ) u ( C ∗ s ) ≤ ˜ u ( X hs e − ( ρ − r )( s − t ) ) − ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) h ≤ ˜ u ( X δs e − ( ρ − r )( s − t ) ) − ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) δ (71)Thus, E "Z Tt e − ρ ( s − t ) | u ( C ∗ s ) | ds ≤ E "Z Tt e − r ( s − t ) | u ( C ∗ s ) | e − ( ρ − r )( s − t ) X ∗ s λ ∗ ds ≤ λ ∗ E "Z Tt e − r ( s − t ) (cid:16) | ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) | + C ∗ s (cid:17) ds < ∞ . (72)(69),(71), and (72) imply X h ∈ N D ( λ ∗ + h ) .Since ( X ∗ s ) Ts = t ∈ N D ( λ ∗ ) minimizes the Lagrangian L , L ( λ h , X h ) ≥ L ( λ ∗ , X ∗ ) . Hence, lim h ↓ L ( λ h , X h ) − L ( λ ∗ , X ∗ ) h ≥ . or, equivalently, lim h ↓ E t "Z Tt e − r ( s − t ) ˜ u ( X hs e − ( ρ − r )( s − t ) ) − ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) h ds − w ≥ . The Dominated Convergence Theorem implies lim h ↓ E t "Z Tt e − r ( s − t ) ˜ u ( X hs e − ( ρ − r )( s − t ) ) − ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) h ds − w = E t " lim h ↓ Z Tt e − r ( s − t ) ˜ u ( X hs e − ( ρ − r )( s − t ) ) − ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) h ds − w = E t "Z Tt e − ρ ( s − t ) u ( C ∗ s ) ds − w ≥ . Thus, C ∗ satisfies the promise-keeping constraints.Similar to Miao and Zhang (2015), define X h ( w, ξ ) ≡ X ∗ ( t, ξ ) + h A × ( s,T ] ( w, ξ ) for h ∈ (0 , δ ) , s ∈ [ t, T ] and A ∈ F s . Note X ht = X ∗ t = λ ∗ .By a similar argument, we can obtain X h ∈ N D ( λ ∗ ) . ince lim h ↓ L ( λ ∗ , X h ) − L ( λ ∗ , X ∗ ) h ≥ . This leads to E t " A Z Ts e − ρ ( ξ − t ) U ( C ∗ ξ ) dξ ≥ E t h A e − ρ ( s − t ) U d ( s, Y s ) i . Since A was an arbitrary set in F s , we deduce that U s ( C ) = E s "Z Ts e − ρ ( ξ − s ) U ( C ∗ ξ ) dξ ≥ U d ( s, Y s ) , for any s ∈ [ t, T ] .Therefore, the consumption plan C ∗ is enforceable. Now we will show that C ∗ isoptimal.For h ∈ (0 , δ ) , let us consider X ± h ≡ X ∗ (1 ± h ) . By the following convexity of ˜ u ( · ) , ˜ u ( X − δs e − ( ρ − r )( s − t ) ) − ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) − δ ≤ ˜ u ( X ± hs e − ( ρ − r )( s − t ) ) − ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) ± h ≤ ˜ u ( X δs e − ( ρ − r )( s − t ) ) − ˜ u ( X ∗ s e − ( ρ − r )( s − t ) ) δ and the condition (70), we can deduce X ± h ∈ N D ( λ ∗ (1 + h )) .Since L ( λ ∗ (1 ± h ) , X ± h ) ≥ L ( λ ∗ , X ∗ ) , lim h ↓ L ( λ ∗ (1 + h ) , X h ) − L ( λ ∗ , X ∗ ) h ≥ , lim h ↑ L ( λ ∗ (1 − h ) , X − h ) − L ( λ ∗ , X ∗ ) − h ≥ By the Dominated Convergence theorem and integration by parts, λ ∗ ( U t ( C ∗ ) − w ) + E t "Z Tt e − ρ ( s − t ) Z Ts e − ρ ( ξ − s ) (cid:0) u ( C ∗ ξ ) − u ( Y ξ ) (cid:1) dξ · dX s = 0 . Thus, the promise keeping constraint and the participation constraint must hold withequality for the consumption plan C ∗ .Since U Pt ( y, C ∗ ) ≤ sup C ∈ Γ( t,y,w ) U Pt ( t, y, C ) ≤ inf X ∈N D ( λ ) L ( t, λ, y, X ) − λw ≤ L ( t, λ ∗ , X ∗ ) − λ ∗ w, we conclude that the consumption plan C ∗ is optimal and the following duality relationshipholds: V ( t, w, y ) = inf λ> { J ( t, λ, y ) − λw } . (cid:3) Step 4)
Determination of λ ∗ in the duality relationship (65) and the optimal process ( X ∗ s ) Ts = t .By Lemma 4.6 and (60), ∂ λ J ( t, λ, y ) = y − γ g ( t, λy γ ) + ∂ λ J ( t, λ, y )= y − γ − γ Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d ( s − t, λy γ z ⋆ ( s ) ) (cid:19) ds + λ γ − − γ Z Tt e − K ( s − t ) N (cid:18) d γ ( s − t, λy γ z ⋆ ( s ) ) (cid:19) ds. and we deduce that lim λ →∞ ∂ λ J ( t, λ, y ) = ∞ if < γ < , if < γ. Moreover, by Lemma 4.2 or the value matching condition for g ( t, z ) , ∂ λ J ( t, z ⋆ ( t ) y γ , y ) = U d ( t, y ) . By (Step 1) , J ( t, λ, y ) is strictly convex in λ in NR , i.e., ∂ λλ J ( t, λ, y ) > in NR . This implies that ∂ λ J ( t, λ, y ) is strictly increasing in λ in NR .Thus, for given w satisfying Assumption 3, there exists a unique λ ∗ such that w = ∂ λ J ( t, λ ∗ , y ) and ( t, λ ∗ , y ) ∈ NR .Now, we will determine the optimal shadow process ( X ∗ s ) Ts = t ∈ N D ( λ ∗ ) .Since the optimal stopping time τ in Problem 4 is given by τ = inf { s | ≤ e ( ρ − r )( s − t ) z ⋆ ( s ) Y γs } , the optimal stopping time τ ( x ) in Lemma 3.1 can be written by τ ( x ) = inf { s | x ≤ e ( ρ − r )( s − t ) z ⋆ ( s ) Y γs } . Since { X ∗ s < x } = { s < τ ( x ) } = (cid:26) max t ≤ θ ≤ s e ( ρ − r )( θ − t ) z ⋆ ( θ ) Y γθ < x (cid:27) , the optimal process ( X ∗ s ) Ts = t can be expressed explicitly as X ∗ s = max (cid:18) λ ∗ , max t ≤ θ ≤ s e ( ρ − r )( θ − t ) z ⋆ ( θ ) Y γθ (cid:19) . y defining λ ∗ s = e − ( ρ − r )( s − t ) X ∗ s , since Problem 5.1 is time consistent , we can deducethat dual value function J ( s, λ ∗ s , Y s ) at time s ∈ [ t, T ] is given by J ( s, λ ∗ s , Y s ) = − Y − γs − γ Z ∞ λ ∗ s "Z Ts e − ˆ ρ ( ξ − s ) N (cid:18) − d ( ξ − s, xz ⋆ ( ξ ) Y γs ) (cid:19) dξ − (cid:18) xY γs (cid:19) γ − Z Ts e − ˆ ρ ( ξ − s ) N (cid:18) − d ( ξ − s, xz ⋆ ( ξ ) Y γs ) (cid:19) dξ dx + γ − γ (cid:18) − e − K ( T − s ) K ( λ ∗ s ) γ + 1 − e − ˆ r ( T − s ) ˆ r Y s (cid:19) . and satisfies the duality-relationship: V ( s, Y s , w s ) = inf λ> { J ( s, λ, Y s ) − w s λ } = J ( s, λ ∗ s , Y s ) − w s λ ∗ s . By the first-order condition, we obtain w s = ∂ λ J ( s, λ ∗ s , Y s )= Y − γs − γ Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d ( s − t, λ ∗ s Y γs z ⋆ ( s ) ) (cid:19) ds + ( λ ∗ s ) γ − − γ Z Tt e − K ( s − t ) N (cid:18) d γ ( s − t, λ ∗ s Y γs z ⋆ ( s ) ) (cid:19) ds. (Step 5) The optimal process ( X ∗ s ) Ts = t satisfies the integrability condition (20). Proof of (Step 5)
It is enough to show that E t "Z Tt e − r ( s − t ) ( e − ( ρ − r )( s − t ) X ∗ s ) γ ds < ∞ and E t "Z Tt e − ρ ( s − t ) Y − γs X ∗ s ds < ∞ First, we will utilize the idea in Lemma 3.1. E t "Z Tt e − r ( s − t ) ( e − ( ρ − r )( s − t ) X ∗ s ) γ ds = E t "Z Tt e − K ( s − t ) ( X ∗ s ) γ ds = E t "Z Tt e − K ( s − t ) Z X ∗ s X ∗ t γ x γ − dx + X ∗ t ! ds = 1 − e − K ( T − t ) K λ ∗ + 1 γ E t "Z Tt e − K ( s − t ) (cid:18)Z ∞ λ ∗ x γ − { X ∗ s ≥ x } dx (cid:19) ds = 1 − e − K ( T − t ) K λ ∗ + 1 γ Z ∞ λ ∗ Z Tt e − K ( s − t ) x γ − E t (cid:2) { s ≥ τ ( x ) ∗ } (cid:3) dsdx, where the last equality is obtained from Fubini’s theorem.By Proposition 5.1, we can obtain P ( s ≥ τ ( x ) ∗ ) = P ( x H s ≤ z ⋆ ( s )) . ince d H s = (ˆ r − ˆ ρ + γσ ) H s ds − γσ H s dB s under the measure P with H t = 1 /y γ , we can easily obtain that P ( s ≥ τ ( x ) ∗ ) = N (cid:18) − d γ ( s − t, xz ⋆ ( s ) y γ ) (cid:19) . For e z ⋆ ( s ) = z ⋆ ( T − s ) , ξ = s − t and τ = T − t , E t "Z Tt e − r ( s − t ) ( e − ( ρ − r )( s − t ) X ∗ s ) γ ds = 1 − e − K ( T − t ) K λ ∗ + 1 γ Z ∞ λ ∗ Z τ e − Kξ x γ − N (cid:18) − d γ ( ξ, x e z ⋆ ( τ − ξ ) y γ ) (cid:19) dξdx< λ ∗ K + 1 γ Z ∞ λ ∗ lim τ →∞ (cid:20)Z τ e − Kξ x γ − N (cid:18) − d γ ( ξ, x e z ⋆ ( τ − ξ ) y γ ) (cid:19) dξ (cid:21) dx = λ ∗ K + 1 γ Z ∞ λ ∗ x γ − Z ∞ e − Kξ N (cid:18) − d γ ( ξ, xz ∞ y γ ) (cid:19) dξdx. (73)By using the result in Proof of Lemma 4.8, we can derive Z ∞ e − Kξ N (cid:18) − d γ ( ξ, xz ∞ y γ ) (cid:19) dξ = 1 K α + + (cid:16) γ − γ (cid:17) α + − α − · (cid:18) xz ∞ y γ (cid:19) α − + ( γ − γ ) (74)By (73) and (74), we deduce E t "Z Tt e − r ( s − t ) ( e − ( ρ − r )( s − t ) X ∗ s ) γ ds < λ ∗ K + 1 γ K α + + (cid:16) γ − γ (cid:17) α + − α − ( z ∞ y γ ) γ − − α − Z ∞ λ ∗ x α − dx < ∞ . Similarly, E t "Z Tt e − ρ ( s − t ) Y − γs X ∗ s ds = E t "Z Tt e − ρ ( s − t ) Y − γs Z X ∗ s X ∗ t dx + X ∗ t ! = 1 − e − ˆ ρ ( T − t ) ˆ ρ y − γ λ ∗ + E t "Z Tt e − ρ ( s − t ) Y − γs Z X ∗ s X ∗ t dx ! ds = 1 − e − ˆ ρ ( T − t ) ˆ ρ y − γ λ ∗ + y − γ E Q "Z Tt e − ˆ ρ ( s − t ) (cid:18)Z ∞ λ ∗ { X ∗ s ≥ x } (cid:19) ds = 1 − e − ˆ ρ ( T − t ) ˆ ρ y − γ λ ∗ + y − γ Z ∞ λ ∗ Z Tt e − ˆ ρ ( s − t ) E Q (cid:2) { s ≥ τ ( x ) ∗ } (cid:3) dsdx, where the measure Q is defined in Lemma 3.1.Since d H s = (ˆ r − ˆ ρ + γ σ ) H s ds − γσ H s dB Q s , e have Q ( s ≥ τ ( x ) ∗ ) = Q ( x H s ≤ z ⋆ ( t )) = N (cid:18) − d ( s − t, xz ⋆ ( s ) y γ ) (cid:19) . By using the result in Proof of Lemma 4.8, we deduce that E t "Z Tt e − ρ ( s − t ) Y − γs X ∗ s ds = 1 − e − ˆ ρ ( T − t ) ˆ ρ y − γ λ ∗ + y − γ Z ∞ λ ∗ Z Tt e − ˆ ρ ( s − t ) N (cid:18) − d ( s − t, xz ⋆ ( s ) y γ ) (cid:19) dsdx< y − γ λ ∗ ˆ ρ + y − γ Z ∞ λ ∗ Z ∞ e − ˆ ρξ N (cid:18) − d ( ξ, xz ∞ y γ ) (cid:19) dξdx = y − γ λ ∗ ˆ ρ + 1ˆ ρ (cid:18) α + α + − α − (cid:19) ( z ∞ ) − α − y − γ − α − γ Z ∞ λ ∗ x α − dx < ∞ . Thus, we can conclude that the optimal process ( X ∗ s ) Ts = t satisfies the integrability condi-tion (20). (cid:3) From (Step 1) – (Step 5) , we have just proved Theorem 5.1. (cid:3) We will now show the convergence of the optimal strategies for a finite horizon problemto those for the infinite horizon problem.
Theorem 5.2.
For t ≥ , as time to maturity goes to infinity, i.e., T − t → ∞ , theagent’s optimal consumption C ∗ t, ∞ and the optimal promised value w ∗ t, ∞ at time t are givenby C ∗ t, ∞ =( λ ∗ t, ∞ ) γ ,w ∗ t, ∞ = − γ K α − ( z ∞ Y γt ) γ − − α − ( λ ∗ t, ∞ ) α − + 11 − γ K ( λ ∗ t, ∞ ) γ − , where α − and z ∞ are defined in (47) and (45) in Section 4, respectively, λ ∗ t, ∞ = e − ( ρ − r ) t X ∗ t, ∞ ,and X ∗ t, ∞ = max λ ∗∞ , z ∞ sup ≤ ξ ≤ t e ( ρ − r ) ξ Y γξ ! . Moreover, λ ∗∞ is a unique solution to the following algebraic equation: w = − γ K α − ( z ∞ y γ ) γ − − α − ( λ ∗∞ ) α − + 11 − γ K ( λ ∗∞ ) γ − . Proof.
By Lemma 4.8, lim T − t →∞ Q ( t, z ) = Q ∞ ( z ) and lim T − t →∞ z ⋆ ( t ) = z ∞ , where Q ∞ ( z ) and z ∞ are defined in (44) and (45), respectively.Thus, we have g ∞ ( z ) ≡ lim T − t →∞ g ( t, z ) = Q ∞ ( z ) + 11 − γ (cid:18) ρ − K z γ − (cid:19) , nd J ∞ ( y, w ) = lim T − t →∞ J ( t, y, w ) = − y − γ Z ∞ λ g ∞ (cid:18) xy γ (cid:19) dx + γ − γ K λ γ + 1ˆ r y. From Theorem 5.1, we can derive C ∗ t, ∞ =( λ ∗ t, ∞ ) γ w ∗ t, ∞ = ∂ λ J ∞ ( λ ∗ t, ∞ Y γt )= − γ K α − ( z ∞ Y γt ) γ − − α − ( λ ∗ t, ∞ ) α − + 11 − γ K ( λ ∗ t, ∞ ) γ − . where λ ∗ t, ∞ = e − ( ρ − r ) t X ∗ t, ∞ and X ∗ t, ∞ = max λ ∗∞ , z ∞ sup ≤ ξ ≤ t e ( ρ − r ) ξ Y γξ ! . Here, λ ∗∞ is a unique solution to the following algebraic equation: w = − γ K α − ( z ∞ y γ ) γ − − α − ( λ ∗∞ ) α − + 11 − γ K ( λ ∗∞ ) γ − . For this infinite horizon case, the uniqueness of λ ∗∞ was proven by Miao and Zhang (2015). (cid:3) Remark 5.3.
For α = 1 − γ and β = 1 − γ − γα − , the results in Theorem 5.2 areconsistent with those of Section 2 in Miao and Zhang (2015). In this section, we illustrate numerical simulation results for optimal contracting policies.The optimal contracting policies stated in Theorem 5.1 is not fully explicit, since it requiressolving the integral equation (58) for the free boundary z ⋆ ( t ) . Thus, we should solve forthe free boundary by numerical methods. Thus, we solve the integral equation (58) bythe recursive integration method proposed by Huang et al. (1996).Figure 2 presents simulation paths of the optimal consumption C ∗ , the optimal costateprocess X ∗ , and the regulated process λ ∗ /Y γ . The optimal costate process X ∗ s ensuresthat ( s, λ ∗ s /Y γs ) will never leave the no-jump region for s ∈ [ t, T ] as shown in Figure 2(a). Whenever λ ∗ /Y γ low enough to hit the free boundary z ⋆ , the non-decreasing process X ∗ increases in Figure 2 (b). The reason is that the optimal costate process X ∗ havethe property that it increases only when λ ∗ /Y γ hits the free boundary, at which time theparticipation constraints (5) bind. This leads to increase s in the optimal consumption C ∗ . In other words, if income process Y increases enough and thus λ ∗ /Y γ decreases andhits the free boundary, the principal should increase the agent’s consumption C ∗ so thatthe agent does not walk away from the contract, as shown in Figure 2 (c).Figure 3 which is representative of the case where ρ > r , show ing a simulation path ofoptimal consumption allocation. In the case of ρ > r , however, the optimal consumption Time(year) z Regulated process λ ∗ Y γ Free boundary z ⋆ (a) λ ∗ /Y γ Time(year) X ∗ Optimal process X ∗ (b) X ∗ Time(year) C o n s u m p t i o n C ∗ Optimal consumption C ∗ Income process Y (c) C ∗ Figure 2: Simulation of the optimal consumption C ∗ , the optimal process X ∗ and the regulatedprocess λ ∗ /Y γ . The parameter values are as follows: ρ = 0 . , r = 0 . , µ = 0 . σ =0 . , γ = 3 , w = − , and T = 30 . C ∗ gradually decreases even when the regulated process λ ∗ /Y γ does not hit the freeboundary.Figure 4 shows the comparison results of the optimal consumption C ∗ and the first-bestconsumption C F B in two different scenarios. Two scenarios correspond to two differentsample paths of the income process generated with the same parameter values. In the firstscenario (scenario 1), the income process steadily increases whereas the income processtends to decrease in the second scenario (scenario 2). Since the first-best consumptiondefined in (10) is a deterministic function, C F B is the same in both scenarios. In contrastto the first-best allocation, the optimal consumption C ∗ in limited commitments dependson the entire history of income process Y t . In other words, the optimal consumption C ∗ also increases as income steadily increases in the scenario 1. That is, although C ∗ References Abramowitz M, Stegun I (1972) Handbook of Mathematical Function. Dover.Ai H, Hartman-Glaser B (2014) A Theory of Optimal Capital Structure and EndogenousBankruptcy. Working Paper .Ai H, Li R (2015) Investment and CEO Compensation under Limited Commitment. Jour-nal of Financial Economics Working Paper .Alvarez F, Jermann U (2000) Efficiency, Equilibrium, and Asset Pricing with Risk ofDefault. Econometrica Working Paper. Carr P, Jarrow R, Myneni R (1992) Alternative Characterizations of American Put Op-tions. Math. Finance J. Econ. Theory leming W, Soner H (2006) Controlled Markov Processes and Viscosity Solutions.(Springer-Verlag) New York.Friedman A (1975) Parabolic variational inequalities in one space dimension and smooth-ness of the free boundary. J. Funct. Anal. J. Econ. Theory Economic Theory Rev. Financ. Stud. Math. Finance J. Econ.Dyn. Control SIAM J. Control Optim. Rev. Financ. Stud. Rev. Econ. Stud. New York: Springer-Verlag. Lieberman G (1996) Second Order Parabolic Differential Equations. World Scientific Sin-gapore.Liu H (2015) Properties of American Volatility Options in the Mean-Reverting / Volatil-ity Model. SIAM J.Financial Math. J. Econ. Theory Amer. Econ. Rev. Rev. Econ. Stud. ang Z, Koo H (2018) Optimal Consumption and Portfolio Selection with Early Retire-ment Option. Math. Oper. Res. forthcoming .Zhang Y (2013) Characterization of a risk sharing contract with one-sided commitment. J. Econ. Dyn. Control37:794–809.