Solutions to a system of first order H-J equations related to a debt management problem
aa r X i v : . [ m a t h . O C ] F e b Solutions to a system of first order H-J equationsrelated to a debt management problem
Antonio Marigonda (1) and Khai T. Nguyen (2) (1)
Department of Computer Science, University of Verona, ITALY (2)
Department of Mathematics, North Carolina State University, USAE-mails: [email protected], [email protected]
February 9, 2021
Abstract
The paper studies a system of first order Hamilton-Jacobi equations with discontinuouscoefficients, arising from a model of deterministic optimal debt management in infinitetime horizon, with exponential discount and currency devaluation. The existence of anequilibrium solution is obtained by a suitable concatenation of backward solutions to thesystem of Hamilton-Jacobi equations. A detailed analysis of the behavior of the solutionas the debt-ratio-income x ∗ → + ∞ is also provided. Key words.
Hamilton-Jacobi equations, optimal debt management, equilibrium solutions
Consider a system of Hamilton-Jacobi equation rV = H ( x, V ′ , p ) + σ x · V ′′ , ( r + λ + v ( x )) · p − ( r + λ ) = H ξ ( x, V ′ , p ) · p ′ + σ x · p ′′ ,v ( x ) = argmin w ≥ { c ( v ) − wxV ′ ( x ) } , (1.1)with the boundary conditions V (0) = 0 , V ( x ∗ ) = B and p (0) = 1 , p ( x ∗ ) = θ ( x ∗ ) , motivated by an optimal debt management problem in infinite time horizon with exponentialdiscount. As in [4, 6, 5, 8, 12, 13], this modeled as a noncooperative interaction between aborrower and a pool of risk-neutral lenders. Here, the independent variable x is the debt-to-income ratio, x ∗ is a threshold of the debt-to-income ratio where the borrower must declarebankruptcy, the salvage function θ ∈ [0 ,
1] determines the fraction of capital that can berecovered by lenders when bankruptcy occurs, and1 V is the value function for the borrower who is a sovereign state that can decide thedevaluation rate of its currency v and the fraction of its income u which is used to repaythe debt, • p is the discounted rate at which the lenders buy bonds to offset the possible loss of partof their investment.Since p is determined by the expected evolution of the debt-to-income ratio at all future times,it depends globally on the entire feedback controls u and v . This leads to a highly nonstandardoptimal control problem, and a “solution” must be understood as a Nash equilibrium, wherethe strategy implemented by the borrower represents the best reply to the strategy adoptedby the lenders, and conversely.For the stochastic model ( σ > V σ , p σ ) as a steady state of an auxiliary parabolic system. The proof requires acareful analysis to construct an invariant domain and apply a fixed-point result to derive theexistence of a steady state for the auxiliary parabolic system. Moreover, they also establishedthe upper (lower) bound of discounted bond price p σ and the expected total optimal cost forservicing the debt V σ . Here, a natural question is trying to understand whether a solutionexists and its structure remains unchanged in the deterministic case ( σ = 0). A classicalapproach for a solution in this case is the vanishing viscosity method. More precisely, onestudies the limit ( V σ , p σ ) → ( V, p ) as the diffusion coefficient σ →
0+ and show that this limitthe limits ( V σ , p σ ) yields a solution to (1.1) with σ = 0. However, this is a highly nontrivialproblem and still remains open.The present paper aims to provide a direct study to the deterministic case of the system (1.1)by looking at the corresponding system of differential inclusions ( V ′ ( x ) ∈ { F − ( x, V ( x ) , p ( x )) , F + ( x, V ( x ) , p ( x )) } p ′ ( x ) ∈ { G − ( x, V ( x ) , p ( x )) , G + ( x, V ( x ) , p ( x )) } (1.2)where F ± ( x, V ( x ) , p ( x )) solves the equation rV ( x ) = H ( x, ξ, p ( x )) with variable ξ , and G ± ( x, η, p ) = ( r + λ + v ∗ ( x, F ± ( x, η, p ))) p − ( r + λ ) H ξ ( x, F ± ( x, η, p ) , p ) . In Theorem 3.5, we first construct a solution (
V, p ) of (1.2) with boundary conditions by asuitable concatenation of backward solutions, and then determine an equilibrium solution tothe corresponding differential game with deterministic dynamics. Moreover, we show thatthere exists a semi-equilibrium point x ∈ [0 , x ∗ [ such that if the debt-ratio-income less than x , then the optimal strategy will reach a steady state, otherwise bankruptcy in finite time isunavoidable. In our construction, the main technical difficulties in the analysis stem from thefact that, the system (1.2) is not monotone and F ± are just H¨older continuous at points where H ξ vanishes. Moreover, p ( · ) may well have many discontinuities x k . At these points, backwardsolutions is not necessarily unique and does not a detail analysis. Thereafter, in Proposition4.1, using the the analysis of sub- and super-solutions, we study in an asymptotic behaviourof ( V, p ) as the maximum debt-to-income threshold x ∗ is pushed to + ∞ . Consequently, • if the salvage rate decay sufficiently slowly, i.e., the lenders can still recover a sufficientlyhigh fraction of their investment after the bankruptcy, then the best choice for theborrower is to implement the Ponzi’s scheme;2 otherwise, if the salvage rate θ ( x ∗ ) decays sufficiently fast, then Ponzi’s scheme is nolonger an optimal solution for the borrower; • for sufficiently large initial debt-to-income and bankruptcy threshold and recovery frac-tion after bankruptcy, the optimal strategy for the borrower will use currency devaluation v to deflate the debt-to-income.The remainder of the paper is organized as follows. In Section 2, we provide a more detaileddescription of the model and the system of Hamilton-Jacobi equation satisfied by ( V, p ), andstudy basic properties of H . In Section 3, we construct a solution to (1.1) with σ = 0, andthen derive an equilibrium solutions to the model of optimal debt management. In Section 4,we perform a detailed analysis of the behavior of the optimal feedback controls as x ∗ → ∞ .We close by an appendix which contains some concepts of convex analysis and collect somefurther technical results related to the Hamiltonian function. In this subsection, we shall recall our deterministic optimal debt management problem withcurrency devaluation with exponential discount in [12]. Here, the borrower is a sovereign state,that can decide to devaluate its currency, and its total income Y ( t ) and total debt X ( t ) aregoverned by the control dynamics ˙ Y ( t ) = ( µ + v ( t )) Y ( t ) , ˙ X ( t ) = − λX ( t ) + ( λ + r ) X ( t ) − U ( t ) p ( t ) , (2.1)where µ is the average growth rate of the economy, λ is the rate at which the borrower paysback the principal, r is the interest rate paid on bonds, and • U ( t ) is the rate of payments that the borrower chooses to make to the lenders at time t ; • v ( t ) ≥ t , regarded as an additional control;We define the debt-to-income ratio x . = XY , and set u . = UY . The system (2.1) yields˙ x ( t ) = (cid:18) λ + rp ( t ) − λ − µ − v ( t ) (cid:19) x ( t ) − u ( t ) p ( t ) . (2.2)In this model, the borrower is forced to declare bankruptcy when the debt-ratio-income x reaches threshold x ∗ . The bankruptcy time is denoted by T b . = inf { t > x ( t ) = x ∗ } ∈ R ∪ { + ∞} . (2.3)Without the presence of the devaluation of currency ( v = 0), when a foreign investor buys abond of unit nominal value, he will receive a continuous stream of payments with intensity3 r + λ ) e − λt . If bankruptcy never occurs, the payoff for a foreign investor (exponentiallydiscounted in time) is Ψ = Z ∞ e − rt · ( r + λ ) e − λt dt = 1 . Otherwise, the lenders recover only a fraction θ ( x ∗ ) ∈ [0 ,
1] of their outstanding capital. Inthis case, taking account of the presence of the devaluation of currency, the pay of for a foreigninvestor will beΨ = Z T b ( r + λ ) · exp (cid:26) − Z t ( r + λ + v ( s )) ds (cid:27) dt + exp (cid:26) − Z T b ( r + λ + v ( s )) ds (cid:27) · θ ( x ∗ ) . If the outstanding capital is recovered in full (i.e., θ ( x ∗ ) = 1) and v = 0, then again Ψ = 1. Ingeneral, however, θ ( x ∗ ) < , v = 0, and thus Ψ <
1. To offset this possible loss, the investorsbuy a bond with unit nominal value at a discounted price p ( t ) = Z T b ( r + λ ) · exp (cid:26) − Z t ( r + λ + v ( s )) ds (cid:27) dt + exp (cid:26) − Z T b ( r + λ + v ( s )) ds (cid:27) · θ ( x ∗ ) . (2.4)Given an initial size x of the debt-to-income ratio, the borrower wants to find a pair of optimalcontrols ( u, v ) which minimizes his total expected cost, exponentially discounted in time:minimize J . = Z T b e − rt [ L ( u ( t )) + c ( v ( t ))] dt + e − rT b B (2.5)where c ( v ) is the social cost resulting from devaluation, L ( u ) is the cost to the borrower forputting income towards paying the debt, and B is the cost of bankruptcy. Throughout thepaper we shall assume the following structural conditions on the cost functions L , c : (A1) The implementing cost function L is twice continuously differentiable for u ∈ [0 , , andsatisfies L (0) = 0 , L ′ ( u ) > , L ′′ ( u ) > u → − L ( u ) = + ∞ . (A2) The social cost c is twice continuously differentiable for v ∈ [0 , + ∞ [ ,and satisfies c (0) = 0 , c ′ ( v ) > , c ′′ ( v ) > v →∞ c ( v ) = + ∞ . The control system (2.2)–(2.4) is not standard. Indeed, the discount price p in (2.4) dependson the debt-to-income ratio not only at the present time t but also at all future times. Here,we are mainly interested in construct optimal controls ( u ∗ , v ∗ ) in feedback form:( u, v ) = ( u ∗ ( x ) , v ∗ ( x )) for x ∈ [0 , x ∗ ] . Definition 2.1 ( Equilibrium solution in feedback form ) . A couple of piecewise Lipschitzcontinuous functions ( u ∗ ( · ) , v ∗ ( · )) and l.s.c. p ∗ ( · ) provide an equilibrium solution to the debtmanagement problem (2.2)-(2.5), with continuous value function V ∗ ( · ), if4i) Given the price p ∗ = p ∗ ( x ), one has that V ∗ is the value function and ( u ∗ ( x ) , v ∗ ( x )) isthe optimal feedback control, in connection with the deterministic control problemminimize: Z T b e − rt [ L ( u ( t )) + c ( v ( t ))] dt + e − rT b B, (2.6)subject to ˙ x ( t ) = (cid:18) λ + rp ∗ ( x ) − λ − µ − v ( t ) (cid:19) x ( t ) − u ( t ) p ∗ ( x ) , x (0) = x , (2.7)where the time T b is determined by (2.3).(ii) Given the feedback control ( u ∗ ( x ) , v ∗ ( x )) in (2.7), for every x ∈ [0 , x ∗ ] one has p ∗ ( x ) = Z T b ( r + λ ) exp (cid:26) − Z t (cid:0) λ + r + v ∗ ( x ( s ) (cid:1) ds (cid:27) dt ++ exp (cid:26) − Z T b ( r + λ + v ∗ ( x ( t ))) dt (cid:27) · θ ( x ∗ ) . (2.8)Under the assumptions (A1)-(A2) , the Hamiltonian function H : [0 , x ∗ ] × R × [0 , → R associated to the dynamics (2.2) and to the cost functions L, c is defined by H ( x, ξ, p ) := min u ∈ [0 , (cid:26) L ( u ) − u ξp (cid:27) + min v ≥ n c ( v ) − vxξ o + (cid:18) λ + rp − λ − µ (cid:19) x ξ. (2.9)The Debt Management Problem leads to the following implicit system of first order ODEssatisfied by the value function V and the discounted rate p rV ( x ) = H ( x, V ′ ( x ) , p ( x ))( r + λ + v ( x )) p ( x ) − ( r + λ ) = H ξ ( x, V ′ ( x ) , p ( x )) · p ′ ( x ) v ( x ) = argmin ω ≥ { c ( ω ) − ωxV ′ ( x ) } (2.10)with the boundary conditions V (0) = 0 , V ( x ∗ ) = B and p (0) = 1 , p ( x ∗ ) = θ ( x ∗ ) . (2.11) H and normal form of the system In this subsection, we present some basic properties of the Hamiltonian function which will beused to provide a semi-explicit formula for the optimal feed back strategy ( u ∗ , v ∗ ). Let L ◦ , c ◦ are the convex conjugate of L and c (see in the Appendix for the notation). We have that − H ( x, ξ, p ) . = L ◦ (cid:18) ξp (cid:19) + c ◦ ( xξ ) − (cid:18) λ + rp − λ − µ (cid:19) x ξ, (2.12)5nd the map ξ
7→ − H ( x, ξ, p ) is convex and lower semicontinuous. Moreover, given x > p ∈ ]0 , ξ ≥
0, we denote by u ∗ ( ξ, p ) ∈ [0 ,
1] and v ∗ ( x, ξ ) ∈ [0 , + ∞ [ the unique elements of ∂L ◦ (cid:18) ξp (cid:19) and ∂c ◦ ( xξ ), respectively, provided by Lemma 5.3. u ∗ ( ξ, p ) . = argmin u ∈ [0 , (cid:26) L ( u ) − u ξp (cid:27) = ( ≤ ξ < pL ′ (0)( L ′ ) − ( ξ/p ) if ξ ≥ pL ′ (0) > v ∗ ( x, ξ ) . = argmin v ≥ n c ( v ) − vxξ o = ( ≤ xξ < c ′ (0)( c ′ ) − ( xξ ) if xξ ≥ c ′ (0) > . It is clear that • for every p ∈ ]0 ,
1] the map ξ u ∗ ( ξ, p ) is strictly increasing in [ pL ′ (0) , + ∞ [, and u ∗ ( · , p ) ≡ , pL ′ (0)]; • for every ξ ≥ p u ∗ ( ξ, p ) is strictly decreasing in [ ξ/L ′ (0) , u ∗ ( ξ, · ) ≡ , ξ/L ′ (0)]; • for every ξ > x v ∗ ( x, ξ ) is strictly increasing in [ c ′ (0) /ξ, + ∞ [, and v ∗ ( · , ξ ) ≡ , c ′ (0) /ξ ]; • for every x > ξ v ∗ ( x, ξ ) is strictly increasing in [ c ′ (0) /x, + ∞ [, and v ∗ ( x, · ) ≡ , c ′ (0) /x ].From Lemma 5.4, the gradient of the Hamiltonian function H ( · ) can be expressed in terms of u ∗ ( ξ, p ) and v ∗ ( x, ξ ) at any point ( x, ξ, p ) ∈ [0 , + ∞ [ × [0 , + ∞ [ × ]0 ,
1] by H x ( x, ξ, p ) = h ( λ + r ) − p ( λ + µ + v ∗ ( x, ξ )) i · ξpH ξ ( x, ξ, p ) = 1 p · h x (cid:0) ( λ + r ) − p ( λ + µ + v ∗ ( x, ξ )) (cid:1) − u ∗ ( ξ, p ) i H p ( x, ξ, p ) = ( u ∗ ( ξ, p ) − x ( λ + r )) · ξp . (2.13)The following Lemma will catch some relevant properties of H ( · ) needed to study the system(2.10). Lemma 2.2.
Let x ≥ and < p ≤ be fixed, and set H max ( x, p ) . = max ξ ≥ H ( x, ξ, p ) . Then1. there exists ξ ♯ ( x, p ) > such that, given η > , the equation rη = H ( x, ξ, p ) admits • no solutions ξ ∈ [0 , + ∞ ) if rη > H max ( x, p ) , • ξ ♯ ( x, p ) as unique solution if rη = H max ( x, p ) , exactly two distinct solutions { F − ( x, η, p ) , F + ( x, η, p ) } with < F − ( x, η, p ) < ξ ♯ ( x, p ) < F + ( x, η, p ) if < rη < H max ( x, p ) ,2. we extend the definition of η F ± ( x, η, p ) by setting F ± (cid:18) x, r H max ( x, p ) , p (cid:19) = ξ ♯ ( x, p ) , thus for fixed x > , p ∈ ]0 , , the maps η F − ( x, η, p ) and η F + ( x, η, p ) arerespectively strictly increasing and strictly decreasing in (cid:20) , H max ( x, p ) r (cid:21) .3. for all < η < H max ( x, p ) /r with x > and p ∈ ]0 , , we have ∂∂η F ± ( x, η, p ) = rH ξ ( x, F ± ( x, η, p ) , p ) ,
4. The map p H max ( x, p ) is strictly decreasing on ]0 , for every fixed x ∈ ]0 , x ∗ [ .Proof. Since for all fixed x >
0, 0 < p ≤ ξ H ( x, ξ, p ) is the minimum of afamily of affine functions of ξ , we have that the map ξ H ( x, ξ, p ) is concave down. Recalling(2.13), and the monotonicity properties of u ∗ ( · , p ) and v ∗ ( x, · ), since • H ξ ( x, ξ, p ) = H ξ ( x, , p ) > , for all ξ ∈ [0 , min { pL ′ (0) , c ′ (0) /x } ], • ξ H ξ ( x, ξ, p ) , is strictly decreasing for all ξ > min { pL ′ (0) , c ′ (0) /x } , • lim ξ → + ∞ H ξ ( x, ξ, p ) = −∞ ,we have that ξ H ξ ( x, ξ, p ) vanishes in at most one point in [0 , + ∞ ), so ξ H ( x, ξ, p )reaches its maximum value H max ( x, p ) on [0 , + ∞ ) at a unique point ξ ♯ ( x, p ), moreover it isstrictly increasing for 0 < ξ < ξ ♯ ( x, p ) and strictly decreasing for ξ > ξ ♯ ( x, p ), with ξ ♯ ( x, p ) ≥ min { pL ′ (0) , c ′ (0) /x } . We define • the strictly increasing map η F − ( x, η, p ), for 0 < η < r · H max ( x, p ), to be the inverseof ξ r · H ( x, ξ, p ) for 0 < ξ < ξ ♯ ( x, p ); • the strictly decreasing map η F + ( x, η, p ), for 0 < η < r · H max ( x, p ), to be the inverseof ξ r · H ( x, ξ, p ) for ξ > ξ ♯ ( x, p ).This proves (1) and (2). Now, set u ♯ ( x, p ) . = u ∗ ( ξ ♯ ( x, p ) , p ) , v ♯ ( x, p ) . = v ∗ ( x, ξ ♯ ( x, p )) . u ♯ ( x, p ) = (cid:2) ( λ + r ) − ( λ + µ + v ♯ ( x, p )) p (cid:3) · x, (2.14) H max ( x, p ) = L ( u ♯ ( x, p )) + c ( v ♯ ( x, p )) . (2.15)Moreover, • if ξ ♯ ( x, p ) ≥ pL ′ (0) then ξ ♯ ( x, p ) = pL ′ ( u ♯ ( x, p )) = pL ′ (cid:16)(cid:2) ( λ + r ) − ( λ + µ + v ♯ ( x, p )) p (cid:3) · x (cid:17) , (2.16) • if ξ ♯ ( x, p ) ≥ c ′ (0) x then ξ ♯ ( x, p ) = c ′ ( v ♯ ( x, p )) x . (2.17)Conversely, for any fixed x ≥ < p ≤
1, if u ∗ ( ξ, p ) = x (cid:0) ( λ + r ) − p ( λ + µ + v ∗ ( x, ξ )) (cid:1) , then ξ = ξ ♯ ( x, p ), v ∗ ( x, ξ ) = v ♯ ( x, p ) and u ∗ ( ξ, p ) = u ♯ ( x, p ). Indeed, this follows from the factthat H ξ ( x, ξ, p ) = 0 iff ξ = ξ ♯ ( x, p ).Consider the equation η = H ( x, ξ, p ) /r for a given η >
0, and, noticing that, given 0 < ξ <ξ ♯ ( x, p ) we have F − ( x, η, p ) = F − (cid:18) x, r H ( x, ξ, p ) , p (cid:19) = ξ for all 0 < ξ < ξ ♯ ( x, p ) ,F + ( x, η, p ) = F + (cid:18) x, r H ( x, ξ, p ) , p (cid:19) = ξ for all ξ ♯ ( x, p ) > ξ, and so (3) follows from the Inverse Function Theorem. To prove item (4), we notice that ddp H max ( x, p ) = ddp H ( x, ξ ♯ ( x, p ) , p ) = H p ( x, ξ ♯ ( x, p ) , p ) . Recalling (2.13), we have H p ( x, ξ ♯ ( x, p ) , p ) = h u ♯ ( x, p ) − ( r + λ ) x i · ξ ♯ ( x, p ) p = − ( λ + µ + v ♯ ( x, p )) xξ ♯ ( x, p ) < , since for x, p = 0 we have ξ ♯ ( x, p ) > Definition 2.3 (Normal form of the system) . Given ( x, p ) ∈ ]0 , x ∗ ] × ]0 , such that < rη ≤ H max ( x, p ) we define the maps G ± ( x, η, p ) = ( r + λ + v ∗ ( x, F ± ( x, η, p ))) p − ( r + λ ) H ξ ( x, F ± ( x, η, p ) , p ) . (2.18)8 − ( x, η, p ) ξ ♯ ( x, p ) F + ( x, η, p ) OrηH max ( x, p ) ξ Figure 1: For x ≥ p ∈ ]0 , ξ H ( x, ξ, p ) has a unique global maximum H max ( x, p ) attained at ξ = ξ ♯ ( x, p ). For 0 < rη ≤ H max , the values F − ( x, η, p ) ≤ ξ ♯ ( x, p ) ≤ F + ( x, η, p ) are well defined. Moreover, F ± ( x, r H max ( x, p ) , p ) = ξ ♯ ( x, p ). Notice that if rV ( x ) > H max ( x, p ) , then the first equation of (2.10) has no solution. Otherwise,if < rV ( x ) < H max ( x, p ) this equation splits into V ′ ( x ) = F − ( x, V ( x ) , p ( x )) ,p ′ ( x ) = G − ( x, V ( x ) , p ( x )) , or V ′ ( x ) = F + ( x, V ( x ) , p ( x )) ,p ′ ( x ) = G + ( x, V ( x ) , p ( x )) . Remark 2.4.
Recalling (2.2) and (2.14) , we observe that • The value V ′ ( x ) = F + ( x, V ( x ) , p ) ≥ ξ ♯ ( x, p ) corresponds to the choice of an optimalcontrol such that ˙ x ( t ) < . The total debt-to-ratio is decreasing. • The value V ′ ( x ) = F − ( x, V ( x ) , p ) ≤ ξ ♯ ( x, p ) corresponds to the choice of an optimalcontrol such that ˙ x ( x ) > . The total debt-to-ratio is increasing. • When rV ( x ) = H max ( x, p ) , then the value V ′ ( x ) = F + ( x, V ( x ) , p ) = F − ( x, V ′ ( x ) , p ) = ξ ♯ ( x, p ) corresponds to the unique control strategy such that ˙ x ( t ) = 0 . Remark 2.5.
We notice that if ≤ xξ < min { xpL ′ (0) , c ′ (0) } , since u ∗ = v ∗ = 0 , we have ξ = F − ( x, η, p ) = prη ( λ + r − p ( λ + µ )) x , in particular, if ≤ xξ < min { xpL ′ (0) , c ′ (0) } we have that η F − ( x, η, p ) is Lipschitzcontinuous, uniformly for ( x, p ) ∈ [ x , x ∗ ] × [ p , , for all x ∈ ]0 , x ∗ ] , p ∈ ]0 , . If xξ > min { xpL ′ (0) , c ′ (0) } , we have instead H ξξ ( x, ξ, p ) ≤ − p · min (cid:26) pL ′′ ( u ∗ ( ξ, p )) , x pc ′′ ( v ∗ ( x, ξ )) (cid:27) . emma 2.6. Given x ∈ ]0 , x ∗ ] , p ∈ ]0 , , there exists a constant C = C ( x , p ) such that | F − ( x, η , p ) − F − ( x, η , p ) | ≤ C · | η − η | / , for all x ∈ [ x , x ∗ ] , p ∈ [ p , , < η , η ≤ r H max ( x, p ) .Proof. We distinguish two cases:1. if 0 ≤ xξ < min { xpL ′ (0) , c ′ (0) } , since u ∗ = v ∗ = 0, we have ξ = F − ( x, η, p ) = prη ( λ + r − p ( λ + µ )) x , and so | F − ( x, η , p ) − F − ( x, η , p ) | ≤ pr ( λ + r − p ( λ + µ )) x | η − η |≤ √ Br ( r − µ ) x | η − η | / . for all x ∈ [ x , x ∗ ], p ∈ [ p , < η , η ≤ r H max ( x, p ).2. If xξ > min { xpL ′ (0) , c ′ (0) } , we have instead H ξξ ( x, ξ, p ) ≤ − p min (cid:26) pL ′′ ( u ∗ ( x, ξ, p )) , x pc ′′ ( v ∗ ( x, ξ )) (cid:27) , thus, recalling that by assumption we have L ′′ ( u ) ≥ δ and c ′′ ( v ) ≥ δ for 0 < u < v ≥
0, we obtain − H ξξ ( x, ξ, p ) ≥ min { , x p } δ . By applying Lemma 5.6 to f ( · ) = − r H ( x, · , p ), we have | F − ( x, η , p ) − F − ( x, η , p ) | ≤ s rδ min { , x p } | η − η | / . The proof is complete by choosing C ( x , p ) . = s rδ min { , x p } + √ Br ( r − µ ) x . In this section, we will provide a detail analysis on the existence of a solution to the system ofHamilton-Jacobi equation (2.10) with boundary conditions (2.11) which yields an equilibriumsolution to the Debt Management Problem (2.2)-(2.5). A solution to will be constructed inthe next following subsections. 10 .1 Constant strategies
We begin our analysis from the control strategies keeping the DTI constant in time, i.e., suchthat the corresponding solution x ( · ) of (2.2) is constant. In this case, there is no bankruptcyrisk, i.e., T b = + ∞ . Definition 3.1 (Constant strategies) . Let ¯ x > be given. We say that a pair (¯ u, ¯ v ) ∈ [0 , × [0 , + ∞ [ is a constant strategy for ¯ x if (cid:20)(cid:18) λ + r ¯ p − λ − µ − ¯ v (cid:19) ¯ x − ¯ u ¯ p (cid:21) = 0 , ¯ p = r + λr + λ + ¯ v , where the second relation comes from taking T b = + ∞ in (2.3) . From these equations, if a couple (¯ u, ¯ v ) ∈ [0 , × [0 , + ∞ [ is a constant strategy then it holds( r + λ )( r − µ )¯ x = ( r + λ + ¯ v )¯ u . In this case, the borrower will never go bankrupt and thus thecost of this strategy in (2.5) is computed by1 r · h L (¯ u ) + c (¯ v ) i = 1 r · (cid:20) L (cid:18) ( r + λ )( r − µ )¯ xr + λ + ¯ v (cid:19) + c (¯ v ) (cid:21) = 1 r · (cid:20) L (( r − µ )¯ x · ¯ p ) + c (cid:18)(cid:18) − p (cid:19) ( r + λ ) (cid:19)(cid:21) . Notice that if ¯ x ( r − µ ) >
1, since 0 ≤ ¯ u < v > p <
1, in particular ifDTI is sufficiently large, every constant strategy needs to implement currency devaluation. Amore precise estimate will be provided in Proposition 3.4.We are now interested in the minimum cost of a strategy keeping the debt constant. To thisaim, we first characterize the cost of a constant strategy in terms of the variables x, p . Lemma 3.2.
Given any ( x, p ) ∈ ]0 , + ∞ [ × ]0 , , we have H max ( x, p ) = min n L ( u ) + c ( v ) : u ∈ [0 , , v ≥ , u = (cid:2) ( λ + r ) − ( λ + µ + v ) p (cid:3) · x o . (3.19) Moreover, (ˆ u, ˆ v ) realizes the minimum in the right hand side of (3.19) if and only if c (ˆ v ) + px ˆ vξ ♯ ( x, p ) = min ζ ≥ n pxξ ♯ ( x, p ) ζ + c ( ζ ) o ,L (ˆ u ) + ˆ uξ ♯ ( x, p ) = min u ∈ [0 , n ξ ♯ ( x, p ) u + L ( u ) o . Proof.
Set F ( v ) := f ( v ) + g (Λ v ) where f ( ζ ) = c ( ζ ) for ζ ≥ f ( ζ ) = + ∞ if ζ < C ( x, p ) = (cid:2) ( λ + r ) − ( λ + µ ) p (cid:3) · x , g ( ζ ) = L ( C ( x, p ) + ζ ) if C ( x, p ) + ζ ∈ [0 ,
1] and g ( ζ ) = + ∞ if C ( x, p ) + ζ / ∈ [0 , − xp . By standard argument in convex analysis (see e.g. Theorem11.2 and Remark 4.2 p. 60 of [9]), denoted by f ◦ , g ◦ the convex conjugates of f, g respectively,we have inf v ∈ R F ( v ) = sup ν ∈ R [ − f ◦ (Λ ν ) − g ◦ ( − ν )]= sup ν ∈ R (cid:20) min ζ ≥ n c ( ζ ) + xpνζ o + min C ( x,p )+ ζ ∈ [0 , n L ( C + ζ ) + νζ o(cid:21) = sup ν ∈ R (cid:20) min ζ ≥ n c ( ζ ) + xpνζ o + min u ∈ [0 , n L ( u ) + νu o − Cν (cid:21) = sup ξ ∈ R (cid:20) min ζ ≥ n c ( ζ ) − xξζ o + min u ∈ [0 , n L ( u ) − u · ξp o + C ( x, p ) p · ξ (cid:21) = sup ξ ∈ R H ( x, ξ, p ) = H max ( x, p ) . Moreover, since sup ξ ∈ R H ( x, ξ, p ) is attained only at ξ = ξ ♯ ( x, p ) according to the strict concavityof ξ H ( x, ξ, p ), (ˆ u, ˆ v ) realizes the minimum in the right hand side of (3.19) if and only if f (ˆ v ) + f ◦ (Λ ξ ♯ ( x, p )) − Λˆ vξ ♯ ( x, p ) = 0 ,g (Λˆ v ) + g ◦ ( − ξ ♯ ( x, p )) + Λˆ vξ ♯ ( x, p ) = 0 , which implies ˆ v ≥ C ( x, p ) − px ˆ v ∈ [0 , c (ˆ v ) + px ˆ vξ ♯ ( x, p ) = min ζ ≥ { pxξ ♯ ( x, p ) ζ + c ( ζ ) } ,L ( C ( x, p ) − px ˆ v ) − px ˆ vξ ♯ ( x, p ) = min ν ∈ R n ξ ♯ ( x, p ) ν + L ( C ( x, p ) + ν ) o . The second relation can be rewritten as L (ˆ u ) + ˆ uξ ♯ ( x, p ) = min u ∈ [0 , n ξ ♯ ( x, p ) u + L ( u ) o , and this complete the proof.Formula (3.19) allows us to give a simpler characterization of the minimum cost of a strat-egy keeping the debt-to-income ratio constant in time. Indeed, given x ∈ [0 , x ∗ ], we select( u ( x ) , v ( x )) keeping the debt-to-income ratio constant in time. This defines uniquely a value p = p ( x ) by Definition 3.1 and impose a relation between u ( x ) and v ( x ). Then we take theminimum over all the costs of such strategies, i.e., the right hand side of formula (3.19). Thisnaturally leads to the following definition. Definition 3.3 (Optimal cost for constant strategies) . Given x ∈ [0 , x ∗ ], we define W ( x ) = 1 r · H max ( x, p c ( x ))where p c ( x ) = r + λr + λ + v c ( x ) ,v c ( x ) = argmin v ≥ (cid:20) L (cid:18) ( r + λ )( r − µ ) xr + λ + v (cid:19) + c ( v ) (cid:21) . (3.20)12or every x ∈ [0 , x ∗ ], W ( x ) denotes the minimum cost of a strategy keeping the DTI ratioconstant in time.The next results proves that if the debt-to-income ratio is sufficiently small, the optimalstrategy keeping it constant does not use the devaluation of currency. Proposition 3.4 (Non-devaluating regime for optimal constant strategies) . Let x c ≥ be theunique solution of the following equation in x ( r + λ ) c ′ (0) = ( r − µ ) xL ′ (( r − µ ) x ) . Then • for all x ∈ [0 , min { x c , x ∗ } ] we have W ( x ) = 1 r · L (( r − µ ) x ) and p c ( x ) = 1 , • for all x ∈ ] min { x c , x ∗ } , x ∗ ] we have W ( x ) = 1 r (cid:20) L (cid:18) ( r + λ )( r − µ ) xr + λ + v c ( x ) (cid:19) + c ( v c ( x )) (cid:21) ,p c ( x ) = r + λr + λ + v c ( x ) < , where v c ( x ) > solves the following equation in vc ′ ( v ) = ( r + λ )( r − µ ) x ( r + λ + v ) · L ′ (cid:18) ( r + λ )( r − µ ) xr + λ + v (cid:19) . • for every x ∈ ]0 , x ∗ [ we have W ′ ( x ) = r − µr p c ( x ) L ′ ( p c ( x )( r − µ ) x ) < ξ ♯ ( x, p c ( x )) . (3.21) Proof.
Given x ∈ ]0 , x ∗ [, we define the convex function F x ( v ) . = r · (cid:20) L (cid:18) ( r + λ )( r − µ ) xr + λ + v (cid:19) + c ( v ) (cid:21) , if v ≥ , ( r + λ )( r − µ ) xr + λ + v ∈ [0 , , + ∞ , otherwise . We compute ddv F x ( v ) = 1 r · (cid:20) c ′ ( v ) − L ′ (cid:18) ( r + λ )( r − µ ) xr + λ + v (cid:19) ( r + λ )( r − µ ) x ( r + λ + v ) (cid:21) , which is monotone increasing and satisfies lim v → + ∞ ddv F x ( v ) = + ∞ , ddv F x ( v ) ≥ ddv F x (0) = 1 r · (cid:20) c ′ (0) − L ′ (( r − µ ) x ) ( r − µ ) xr + λ (cid:21) . Two cases may occur: 13 If ddv F x (0) ≥
0, we have that v = 0 realizes the minimum of F on [0 , + ∞ [. This occourswhen x ∈ [0 , min { x c , x ∗ } ] where x c is the unique solution of( r + λ ) c ′ (0) = ( r − µ ) xL ′ (cid:18) ( r + λ )( r − µ ) xr + λ (cid:19) , and it implies W ( x ) = 1 r · L (( r − µ ) x ) and p c ( x ) = 1. • If we have min { x c , x ∗ } < x ≤ x ∗ , then there exists a unique point v c ( x ) > F ′ ( v c ( x )) = 0, and this point is characterized by c ′ ( v c ( x )) = ( r + λ )( r − µ ) x ( r + λ + v c ( x )) · L ′ (cid:18) ( r + λ )( r − µ ) xr + λ + v c ( x ) (cid:19) . The remaining statements follows noticing that for min { x c , x ∗ } < x ≤ x ∗ we have W ′ ( x ) = ∂F x ∂x ( v c ( x )) + ∂∂v F x ( v c ( x )) · v ′ c ( x ) = ∂F x ∂x ( v c ( x ))= r − µr p c ( x ) L ′ ( p c ( x )( r − µ ) x ) , and deriving the explicit expression of W ( x ) for [0 , min { x c , x ∗ } ] yields the same formula.Notice that, by (2.16), we have ξ ♯ ( x, p c ( x )) = p c ( x ) L ′ (cid:16)(cid:2) ( λ + r ) − ( λ + µ + v ♯ ( x, p c ( x ))) p c ( x ) (cid:3) · x (cid:17) = p c ( x ) L ′ (cid:18)(cid:20) ( λ + r ) − ( λ + µ + v ♯ ( x, p c ( x ))) · λ + rλ + r + v ♯ ( x, p c ( x )) (cid:21) · x (cid:19) = p c ( x ) L ′ ( p c ( x )( r − µ ) · x ) > W ′ ( x ) , where we used the fact that L ′ is strictly increasing and, since the argument of L ′ mustbe nonnegative, we have λ + rλ + µ + v ♯ ( x, p c ( x )) ≥ p c ( x ) , and the proof is complete. We are now ready to establish an existence result of a equilibrium solution to the debt manage-ment problem (2.2) - (2.5). Before going to state our main theorem, we recall from Proposition3.4 that v c is the unique solution to c ′ ( v ) = ( r + λ )( r − µ ) x ( r + λ + v ) · L ′ (cid:18) ( r + λ )( r − µ ) xr + λ + v (cid:19) , and p c ( x ∗ ) = r + λr + λ + v c ( x ∗ ) < ,W ( x ∗ ) = 1 r (cid:20) L (cid:18) ( r + λ )( r − µ ) x ∗ r + λ + v c ( x ∗ ) (cid:19) + c ( v c ( x ∗ )) (cid:21) . (3.22)14 heorem 3.5. Assume that the cost functions L and c satisfies the assumptions (A1)-(A2) ,and moreover W ( x ∗ ) > B and θ ( x ∗ ) ≤ p c ( x ∗ ) . (3.23) Then the debt management problem (2.2) - (2.5) admits an equilibrium solution ( u ∗ , v ∗ , p ∗ ) as-sociated to Lipschitz continuous value functions V ∗ in feedback form such that p ∗ is decreasing, V ∗ is strictly increasing and V ∗ ( x ) ≤ W ∗ ( x ) for all x ∈ [0 , x ∗ ] . Toward the proof of this theorem, we first study basic properties of the backward solutions ofthe system of implicit ODEs (2.10). In fact, an equilibrium solution will be constructed by asuitable concatenation of backward solutions.
We first define the backward solution to the system (2.10) starting from x ∗ . Definition 3.6 (Backward solution for x ∗ ) . Let x ( Z ( x, x ∗ ) , q ( x, x ∗ )) be the backwardsolution of the system of ODEs Z ′ ( x ) = F − ( x, Z ( x ) , q ( x )) ,q ′ ( x ) = G − ( x, Z ( x ) , q ( x )) , with Z ( x ∗ ) = B ,q ( x ∗ ) = θ ( x ∗ ) . (3.24) with H ξ ( x, F − ( x, Z ( x ) , q ( x )) , q ( x )) = 0 . The following Lemma states some basic properties of the backward solution. In particular,the backward solution Z ( · , x ∗ ), starting from B at x ∗ with W ( x ∗ ) < B , survives backward atleast until the first intersection with the graph of W ( · ). Moreover, in this interval is monotoneincreasing and positive. In the same way, q ( · , x ∗ ) is always in ]0 , Proposition 3.7. [Basic properties of the backward solution] Set x ∗ W := , if Z ( x, x ∗ ) < W ( x ) for all x ∈ ]0 , x ∗ [ , sup { x ∈ ]0 , x ∗ [: Z ( x, x ∗ ) ≥ W ( x ) } , otherwise . Assume that W ( x ∗ ) > B and θ ( x ∗ ) < r + λr + λ + v ∗ ( x ∗ , F − ( x ∗ , B, θ ( x ∗ ))) . (3.25) Denote by I x ∗ ⊆ [0 , x ∗ ] the maximal domain of the backward equation (3.24) , define y ( x ) to bethe maximal solution of dydx ( x ) = 1 H ξ ( x, Z ′ ( x, x ∗ ) , q ( x, x ∗ )) ,y ( x ∗ ) = 0 , and let J x ∗ the intersection of its domain with [0 , x ∗ ] . Then . I x ∗ ⊇ J x ∗ ⊇ ] x ∗ W , x ∗ [ ;2. Z ( · , x ∗ ) is strictly monotone increasing in ] x ∗ W , x ∗ [ , and Z ( x, x ∗ ) > for all x ∈ ] x ∗ W , x ∗ ] ;3. q ( x, x ∗ ) ∈ ]0 , for all x ∈ ] x ∗ W , x ∗ ] .Proof. We first claim that q ( · , x ∗ ) is non-increasing on J x ∗ T ] x ∗ W , x ∗ [ and thus q ′ ( x, x ∗ ) = [ r + λ + v ∗ ( x, Z ′ ( x, x ∗ ))] · q ( x, x ∗ ) − ( r + λ ) H ξ ( x, Z ′ ( x, x ∗ ) , q ( x, x ∗ )) ≤ x ∈ J x ∗ ∩ ] x ∗ W , x ∗ [ . (3.26)By contradiction, assume that there exists x ∈ J B ∩ ] x BW , x ∗ [ such that q ′ ( x , x ∗ ) = [ r + λ + v ∗ ( x , Z ′ ( x , x ∗ ))] · q ( x , x ∗ ) − ( r + λ ) H ξ ( x , Z ′ ( x, x ∗ ) , q ( x, x ∗ )) = 0 , q ′′ ( x , x ∗ ) < . (3.27)This yields r + λ = [ r + λ + v ∗ ( x , Z ′ ( x , x ∗ ))] · q ( x , x ∗ ) , q ( x , x ∗ ) > . Two cases are considered: • if x Z ′ ( x , x ∗ ) ≤ c ′ (0) then, recalling the monotonicity of Z ′ ( · , x ∗ ), we have that xV ′ ( x, x ∗ ) ≤ c ′ (0) for all x ∈ J x ∗ ∩ ] x ∗ W , x ∗ [ satisfying x ≤ x , and so v ∗ ( x, Z ′ ( x, x ∗ )) = 0 for all x ∈ J x ∗ ∩ ] x ∗ W , x ∗ [ with x ≤ x . Thus, q ( x , x ∗ ) = 1 and q ′ ( x, x ∗ ) = [ r + λ ] · [ q ( x, x ∗ ) − H ξ ( x, Z ′ ( x, x ∗ ) , q ( x, x ∗ )) for all x ∈ J x ∗ ∩ ] x ∗ W , x ∗ [ with x ≤ x . This implies that q ( x, x ∗ ) = 1 for all x ∈ J x ∗ ∩ ] x ∗ W , x ∗ [ with x ≤ x . In particular, wehave q ′′ ( x , x ∗ ) = 0, which yields a contradiction. • If x Z ′ ( x , x ∗ ) > c ′ (0) then ddx ( v ∗ ( x , Z ′ ( x , x ∗ ))) = Z ′′ ( x , x ∗ ) x + Z ′ ( x , x ∗ ) c ′′ ( x Z ′ ( x , x ∗ )) > . From the first equation of (2.10) and (2.13), it holds rZ ′ ( x , x ∗ ) = H x ( x , Z ′ , q ) + H ξ ( x , Z ′ , q ) · Z ′′ ( x , x ∗ ) + H p ( x , Z, q ) · q ′ ( x , x ∗ )= h ( λ + r ) − q ( x , x ∗ )( λ + µ + v ∗ ( x, Z ′ ) i · Z ′ q + H ξ ( x , Z ′ , q ) · Z ′′ ( x , x ∗ )= ( r − µ ) · Z ′ ( x , x ∗ ) + H ξ ( x , Z ′ , q ) · Z ′′ ( x , x ∗ ) . Observe that Z ′ ( x , x ∗ ) > H ξ ( x , Z ′ ( x , x ∗ ) , q ( x , x ∗ )) >
0, one obtains that Z ′′ ( x , x ∗ ) = µZ ′ ( x , x ∗ ) H ξ ( x , Z ′ ( x , x ∗ ) , q ( x , x ∗ )) > . x in both sides of the second equation of (2.10), we have (cid:2) r + λ + ( v ∗ ( x, Z ′ ( x, x ∗ )) (cid:3) · q ′ ( x, x ∗ ) + q ( x, x ∗ ) · ddx v ∗ ( x, Z ′ ( x, x ∗ ))= q ′′ ( x, x ∗ ) H ξ ( x, Z ′ ( x, x ∗ ) , q ( x, x ∗ )) + q ′ ( x, x ∗ ) ddx H ξ ( x, Z ′ ( x, x ∗ ) , q ( x, x ∗ )) . Recalling (3.27), we obtain that q ′′ ( x , x ∗ ) = q ( x , x ∗ ) H ξ ( x , Z ′ , q ) · ddx v ∗ ( x , Z ′ ( x , x ∗ )) > . (3.28)and it yields a contradiction.Assume that there exists x ∈ J x ∗ ∩ ] x ∗ W , x ∗ [ such that H ξ ( x , Z ′ ( x , x ∗ ) , q ( x , x ∗ )) = 0. Then ξ ♯ ( x , q ( x , x ∗ )) = Z ′ ( x , x ∗ ) , Z ( x , x ∗ ) = 1 r · H max ( x , q ( x , x ∗ )) , and u ♯ ( x , q ( x , x ∗ )) = (cid:2) ( λ + r ) − ( λ + µ + v ♯ ( x , q ( x , x ∗ ))) q ( x , x ∗ ) (cid:3) · x . Since q ( x , x ∗ ) ≤ r + λr + λ + v ♯ ( x, Z ′ ( x , x ∗ )) , we estimate H max ( x , q ( x , x ∗ )) = L ( u ♯ ( x , q ( x , x ∗ ))) + c ( v ♯ ( x , q ( x , x ∗ )))= L (cid:16)(cid:2) ( λ + r ) − ( λ + µ + v ♯ ( x , q ( x , x ∗ ))) q ( x , x ∗ ) (cid:3) · x (cid:17) + c ( v ♯ ( x , q ( x , x ∗ ))) ≥ L (cid:18) r + λ ( r − µ ) x λ + µ + v ♯ ( x , q ( x , x ∗ )) (cid:19) + c ( v ♯ ( x , q ( x , x ∗ ))) ≥ H max ( x , p c ( x )) . Thus, Z ( x , x ∗ ) = 1 r · H max ( x , q ( x , x ∗ )) ≥ r · H max ( x , p c ( x )) = W ( x ) , and this yields a contradiction. By construction, y ( · ) is strictly monotone and invertible in ] x ∗ W , x ∗ ], let x = x ( y ) be itsinverse, from the inverse function theorem we get ddy Z ( x ( y ) , x ∗ ) = Z ′ ( x ( y ) , x ∗ ) · H ξ ( x ( y ) , Z ′ ( x ( y ) , x ∗ ) , q ( x ( y ) , x ∗ )) ,ddy q ( x ( y ) , x ∗ ) = q ′ ( x ( y ) , x ∗ ) · H ξ ( x ( y ) , Z ′ ( x ( y ) , x ∗ ) , q ( x ( y ) , x ∗ )) . Since the map ξ H ( x, ξ, q ) is concave, it holds H ξ ( x, , q ( x, x ∗ )) ≥ H ξ ( x, ξ, q ( x, x ∗ )) ≥ H ξ (cid:0) x, Z ′ ( x, x ∗ ) , q ( x, x ∗ ) (cid:1) , ξ ∈ [0 , Z ′ ( x, x ∗ )]. Thus, rZ ( x ( y ) , x ∗ ) = H (cid:0) x ( y ) , Z ′ ( x ( y ) , x ∗ ) , q ( x ( y ) , x ∗ ) (cid:1) = Z Z ′ ( x ( y ) ,x ∗ )0 H ξ ( x, ξ, q ( x ( y ) , x ∗ )) dξ ≥ Z ′ ( x ( y ) , x ∗ ) · H ξ ( x, Z ( x ( y ) , x ∗ ) , q ( x ( y ) , x ∗ )) = ddy Z ( x ( y ) , x ∗ ) , and this implies that Z ( x, x ∗ ) ≥ Be ry ( x ) > x ∈ ] x ∗ W , x ∗ ] . With a similar argument for q ( · , x ∗ ), we obtain (cid:2) r + λ + v ∗ ( x ( y ) , Z ′ ( x ( y ) , x ∗ ) (cid:3) · q ( x ( y ) , x ∗ ) − ( r + λ ) = ddy q ( x ( y ) , x ∗ )) . Hence,( r + λ )( q ( x ( y ) , x ∗ ) − ≤ ddy q ( x ( y ) , x ∗ ) ≤ (cid:2) r + λ + v ∗ ( x ( y ) , Z ′ ( x ( y ) , x ∗ ) (cid:3) · q ( x ( y ) , x ∗ ) , and this yields q ( x, x ∗ ) ≤ q ( x, x ∗ ) ≥ θ ( x ∗ ) · e ( r + λ + v ∗ ( x,Z ′ ( x,x ∗ )) y ( x ) > x ∈ I x ∗ ∩ [0 , x ∗ ]. In particular, q ( x, x ∗ ) ∈ ]0 ,
1] for all x ∈ ] x ∗ W , x ∗ ].As far as the graph of Z ( · , x ∗ ) intersects the graph of W ( · ), Z ( · , x ∗ ) is no longer optimal. Thefollowing lemma is to investigate the local behavior of Z ( · , x ∗ ) and W ( · ) near to an intersectionof their graphs. Lemma 3.8 (Comparison between optimal constant strategy and backward solution) . Let I ⊆ ]0 , x ∗ [ be an open interval, ( Z, q ) : I → [0 , + ∞ [ × ]0 , be a backward solution, and ¯ x ∈ ¯ I .If lim x → ¯ xx ∈ I Z ( x ) = W (¯ x ) then p c (¯ x ) ≥ lim sup I ∋ x → ¯ x q ( x ) and W ′ ( x ) < F − ( x, W ( x ) , p c ( x )) .Proof. Let { x j } j ∈ N ⊆ I be a sequence converging to ¯ x and q ¯ x ∈ [0 ,
1] be such that q ¯ x =lim sup x → ¯ x + q ( x ) = lim j →∞ q ( x j ). We have H max ( x, p c ( x )) = lim j → + ∞ H (cid:0) x j , Z ′ ( x j ) , q ( x j ) (cid:1) ≤ lim j → + ∞ H max ( x j , q ( x j )) = H max (¯ x, q ¯ x ) . From 2.2 (4), it holds that p c (¯ x ) ≥ q ¯ x . By Proposition 3.4, we have W ′ (¯ x ) < ξ ♯ (¯ x, p c (¯ x )), andso H (¯ x, W ′ (¯ x ) , p c (¯ x )) < H max (¯ x, p c (¯ x )) = rW (¯ x ) . Thus, by applying the strictly increasing map F − (¯ x, · , p c ( x )) on both sides, we obtain W ′ ( x )
0, we denote by Z ε ( · , x ) , q ε ( · , x ) the backward solution to(3.29) with the terminal data Z ε ( x , x ) = W ( y ) − ε and q ε ( x , x ) = p c ( x ) . With the same argument in the proof of Proposition 3.7, the solution is uniquely defined ona maximal interval [ a ε ( x ) , x ] such that Z ε ( · , x ) is increasing, q ε ( · , x ) is decreasing and Z ε ( a ε ( x ) , x ) = W ( a ε ( x )) , q ε ( a ε ( x ) , x ) ≤ p c ( a ε ( x )) . Let x ♭ be the unique solution to the equation c ′ (0) = x · L ′ (( r − µ ) x ) . (3.30)It is clear that 0 < x ♭ < x c where x c is defined in Proposition 3.4 as the unique solution tothe equation ( r + λ ) c ′ (0) = ( r − µ ) xL ′ (( r − µ ) x ) . Two cases are considered:
CASE 1:
For any x ∈ ]0 , x ♭ ], we claim that a ε ( x ) = 0 , q ε ( x, x ) = 1 for all x ∈ [0 , x ] , and Z ε ( · , x ) solves backward the following ODE Z ′ ( x ) = F − ( x, Z ( x ) , , Z ( x ) = W ( x ) − ε (3.31)for ε > Z be the unique backward solution of (3.31). From(2.16), it holds F − ( x, W ( x ) ,
1) = ξ ♯ ( x,
1) = L ′ (( r − µ ) x ) > r − µr · L ′ (( r − µ ) x ) = W ′ ( x )for all x ∈ ]0 , x ♭ ]. As in [5], a contradiction argument yields0 < Z ( x ) < W ( x ) for all x ∈ ]0 , x ] . Thus, Z is well-defined on [0 , x ] and Z (0) = 0. On the other hand, it holds Z ′ ( x ) = F − ( x, Z ( x ) , ≤ ξ ♯ ( x,
1) = L ′ (( r − µ ) x ) ≤ L ′ (( r − µ ) x ♭ )19or all x ≤ x ♭ and (3.30) implies that v ∗ ( x, Z ′ ( x )) = 0 for all x ∈ [0 , x ♭ ] . Therefore, ( Z ( x ) ,
1) solves (3.29) and the uniqueness yields Z ε ( x, x ) = Z ( x ) and q ε ( x, x ) = 1 for all x ∈ [0 , x ] . Thanks to the monotone increasing property of the map ξ → F − ( x, ξ, Z ( · , x ) , q ( · , x )) denoted by q ( x, x ) = 1 and Z ( x, x ) = sup ε> Z ε ( x, x ) for all x ∈ [0 , x ]is the unique solution of (3.29). If the initial size of the debt is ¯ x ∈ [0 , x ] we think of Z (¯ x, x )is as the expected cost of (2.6)-(2.7) with p ( · , x ) = 1, x (0) = x achieved by the feedbackstrategies u ( x, x ) = argmin w ∈ [0 , (cid:8) L ( w ) − Z ′ ( x, x ) · w (cid:9) , v ( x, x ) = 0 (3.32)for all x ∈ [0 , x ]. With this strategy, the debt has the asymptotic behavior x ( t ) → x as t → ∞ . CASE 2:
For x ∈ ( x ♭ , x ∗ W ], system of ODEs (3.29) does not admit a unique solution ingeneral since it is not monotone. The following lemma will provide the existence result of(3.29) for all x ∈ ( x ♭ , x ∗ W ]. Lemma 3.9.
There exists a constant δ ♭ > depending only on x ♭ such that for any x ∈ (cid:0) x ♭ , x ∗ W (cid:1) , it holds x − a ε ( x ) ≥ δ x ♭ for all ε ∈ (0 , ε ) for some ε > sufficiently small.Proof. From (3.21) and (2.16), it holdsinf x ∈ [ x ♭ ,x ∗ W ] n ξ ♯ ( x, p c ( x )) − W ′ ( x ) o = δ ,♭ > . In particular, we have F − ( x , W ( x ) , p c ( x )) − W ′ ( x ) = δ ,♭ . By continuity of the map η F − ( x , η, p c ( x )) on [0 , W ( x )], we can find a constant ε > F − ( x , η, p c ( x )) ≥ W ′ ( x ) + δ ,♭ ξ ∈ [ W ( x ) − ε , W ( x )] . On the other hand, the continuity of W ′ yields δ ,♭ = sup (cid:26) s ≥ (cid:12)(cid:12)(cid:12) W ′ ( x − τ ) < W ′ ( x ) + δ ,♭ τ ∈ [0 , s ] (cid:27) > . ε ∈ (0 , ε ), denote by x := inf n s ∈ (0 , x ] (cid:12)(cid:12)(cid:12) F − (cid:0) x, Z ε ( x, x ) , q ε ( x, x ) (cid:1) > W ′ ( x ) for all x ∈ ( s, x ] o . If x > x − δ , ¯ x then it holds F − (cid:0) x , Z ε ( x , x ) , q ε ( x , x ) (cid:1) = W ′ ( x ) ≤ W ′ ( x ) + δ ,♭ x ∈ ( x , x ] such that F − (cid:0) x , Z ε ( x , x ) , q ε ( x , x ) (cid:1) = W ′ ( x ) + δ ,♭ F − ( x, Z ε ( x, x ) , q ε ( x, x )) ≤ W ′ ( x ) + δ ,♭ x ∈ [ x , x ] . (3.35)Recalling that ( x, η, p ) F − ( x, η, p ) is defined by H ( x, F − ( x, η, p ) , p ) = rη , by the implicitfunction theorem, set ξ = F − ( x, η, p ), we have ∂∂p F − ( x, η, p ) = − H p ( x, ξ, p ) H ξ ( x, ξ, p )= ξp · u ∗ ( x, ξ, p ) − x ( λ + r ) u ∗ ( x, ξ, p ) − x ( λ + r ) + xp ( λ + µ + v ∗ ( x, ξ ))= (cid:18) x ( λ + µ + v ∗ ( x, ξ ) H ξ ( x, ξ, p ) (cid:19) ξp > F − ( x, η, p ) p > . Since q ε ( · , x ) is decreasing, it holds F − (cid:0) x , Z ε ( x , x ) , q ε ( x , x ) (cid:1) ≥ F − (cid:0) x , Z ε ( x , x ) , q ε ( x , x ) (cid:1) , and (3.33)-(3.34) yield F − (cid:0) x , Z ε ( x , x ) , q ε ( x , x ) (cid:1) − F − (cid:0) x , Z ε ( x , x ) , q ε ( x , x ) (cid:1) ≥ δ ,♭ . On the other hand, from (2.13) it follows that the map x → F − ( x, η, p ) is monotone decreasingand thus F − (cid:0) x , Z ε ( x , x ) , q ε ( x , x ) (cid:1) − F − (cid:0) x , Z ε ( x , x ) , q ε ( x , x ) (cid:1) ≥ δ ,♭ . (3.36)Observe that the map η → F − ( x, η, p ) is H¨older continuous due to Lemma 2.6. More precisely,there exist a constant C x ♭ > (cid:12)(cid:12) F − ( x, η , p ) − F − ( x, η , p ) (cid:12)(cid:12) ≤ C x ♭ · (cid:12)(cid:12) η − η (cid:12)(cid:12) for all η , η ∈ (0 , W ( x )], x ∈ [¯ x, x ∗ ], p ∈ [ θ ( x ∗ ) , | Z ε ( x , x ) − Z ε ( x , x ) | ≥ δ ,♭ C x ♭ . Z ′ ε ( x, x ) = F − ( x, Z ε ( x, x ) , q ε ( x, x )) ≤ W ′ ( x ) + δ ,x ♭ x ∈ [ x , x ] , and this yields | x − x | ≥ δ ,x ♭ C x ♭ [2 W ′ ( x ) + δ ,x ♭ ] . Therefore, x − a ε ( x ) ≥ δ x ♭ := min ( δ ,x ♭ , δ ,x ♭ C x ♭ [2 W ′ ( x ) + δ ,x ♭ ] ) > , and the proof is complete. Remark 3.10.
In general, the backward Cauchy problem (3.29) may admit more than onesolution.
As a consequence of Lemma 3.9, there exists a sequence { ε n } n ≥ →
0+ such that the sequenceof backwards solutions { ( Z ε n ( · , x ) , q ε n ( · , x )) } n ≥ converges to ( Z ( · , x ) , q ( · , x )) which is asolution of (3.29). With the same argument in the proof of Proposition 3.7, we can extendbackward the solution ( Z ( · , x ) , q ( · , x )) until a ( x ) such thatlim x → a ( x )+ Z ( a ( x ) , x ) = W ( a ( x )) , and Lemma 3.8 yields lim x → a ( x )+ q ( a ( x ) , x ) ≤ p c ( a ( x )). If the initial size of the debt is¯ x ∈ [ a ( x ) , x ] we think of Z (¯ x, x ) is as the expected cost of (2.6)-(2.7) with p ( · , x ), x (0) = x achieved by the feedback strategies u ( x, x ) = argmin w ∈ [0 , (cid:26) L ( w ) − Z ′ ( x, x ) p ( x, x ) · w (cid:27) ,v ( x, x ) = argmin v ≥ n c ( v ) − vxZ ′ ( x, x ) o . (3.37)With this strategy, the debt has the asymptotic behavior x ( t ) → x as t → ∞ . We are now ready to construct an solution to the system of Hamilton-Jacobi equation (2.10)with boundary conditions (2.11). By induction, we define a family of back solutions as follows: x := x ∗ W , ( Z ( x ) , q ( x )) = ( Z ( x, x ∗ ) , q ( x, x ∗ )) for all x ∈ [ x , x ∗ ]and x n +1 := a ( x n ) , ( Z ( x, x n ) , q ( x, x n )) for all x ∈ [ x n +1 , x n ] . x x x x ∗ B x ♭ W ( x ) Z ( x ) Z ( x ) Z ( x ) Z ( x )Figure 2: Construction of a solution: starting from ( x ∗ , B ) we solve backward the system untilthe first touch with the graph of W at ( x , W ( x )). Then we restart by solving backward thesystem with the new terminal conditions ( W ( x ) , p c ( x )), until the next touch with the graphof W at ( x , W ( x )) and so on. In a finite number of steps we reach the origin. If a touchoccurs at x n < x ♭ then the backward solution from x n reaches the origin with q ≡
1. Givenan initial value ¯ x of the DTI, if 0 ≤ x n +1 < ¯ x < x n < x the the optimal strategy let the DTIincrease asymptotically to x n (no banktuptcy), while if x < ¯ x < x ∗ then the optimal strategylet the DTI increase to x ∗ , thus providing bankruptcy in finite time.From Case 1 and Lemma 3.9, there exists a natural number N < x ∗ − x ♭ δ x ♭ such that ourconstruction will be stop in N step, i.e., x N > , a ( x N ) = 0 and lim x → a ( x N ) Z ( x, x N ) = 0 . We will show that a feedback equilibrium solution to the debt management problem is obtainedas follows( V ∗ ( x ) , p ∗ ( x )) = ( Z ( x, x ∗ ) , q ( x, x ∗ )) for all x ∈ ( x W , x ∗ ] , ( Z ( x, x k ) , q ( x, x k )) for all x ∈ ( a ( x k ) , x k ] , k ∈ { , , . . . , N } , , (3.38)and u ∗ ( x ) = argmin w ∈ [0 , (cid:26) L ( w ) − ( V ∗ ) ′ ( x ) p ∗ ( x ) · w (cid:27) ,v ∗ ( x ) = argmin v ≥ { c ( v ) − vx ( V ∗ ) ′ ( x ) } . (3.39)23 roof of Theorem 3.5. From the monotone increasing property of the maps ξ v ∗ ( x ∗ , ξ ), η F − ( x ∗ , η, θ ( x ∗ )) and p F − ( x ∗ , W ( x ∗ ) , p ), we have θ ( x ∗ ) · ( r + λ + v ∗ ( x ∗ , F − ( x ∗ , B, θ ( x ∗ ))) < p c ( x ∗ ) · ( r + λ + v ∗ ( x ∗ , F − ( x ∗ , W ( x ∗ ) , p c ( x ∗ ))) = r + λ and it yields (3.25). By Proposition 3.7 and Lemma 3.9, a pair V ∗ ( · ) , p ∗ ( · ) in (3.38) is well-defined on [0 , x ∗ ]. In the remaining steps, we show that V ∗ , p ∗ , u ∗ , v ∗ provide an equilibriumsolution. Namely, they satisfy the properties (i)-(ii) in Definition 2.1. To prove (i) in Definition 2.1, let V ( · ) be the value function for the optimal control problem(2.6)-(2.7). For any initial value, x (0) = x ∈ [0 , x ∗ ], the feedback controls u ∗ and v ∗ in (3.39)yield the cost V ∗ ( x ). This implies V ( x ) ≤ V ∗ ( x ) . To prove the converse inequality we need to show that, for any measurable control u :[0 , + ∞ [ [0 ,
1] and v : [0 , + ∞ [ → [0 , + ∞ [, calling t x ( t ) the solution to˙ x ( t ) = (cid:18) λ + rp ∗ ( x ( t )) − λ − µ − v ( t ) (cid:19) x ( t ) − u ( t ) p ∗ ( x ( t )) , x (0) = x , (3.40)it holds Z T b e − rt [ L ( u ( x ( t ))) + c ( v ( x ( t )))] dt + e − rT b B ≥ V ∗ ( x ) (3.41)where T b = inf (cid:8) t ≥ x ( t ) = x ∗ (cid:9) is the bankruptcy time (possibly with T b = + ∞ ).For t ∈ [0 , T b ], consider the absolutely continuous function φ u,v ( t ) := Z t e − rs · [ L ( u ( s )) + c ( v ( s ))] ds + e − rt V ∗ ( x ( t )) . At any Lebesgue point t of u ( · ) and v ( · ), recalling that ( V ∗ , p ∗ ) solves the system (2.10), wecompute ddt φ u,v ( t ) = e − rt · h L ( u ( t )) + c ( v ( t )) − rV ∗ ( x ( t )) + ( V ∗ ) ′ ( x ( t )) · ˙ x ( t ) i = e − rt · h L ( u ( t )) + c ( v ( t )) − rV ∗ ( x ( t ))+ ( V ∗ ) ′ ( x ( t )) (cid:18)(cid:18) λ + rp ∗ ( x ( t )) − λ − µ − v ( t ) (cid:19) x ( t ) − u ( t ) p ∗ ( x ( t )) (cid:19) i ≥ e − rt · h min ω ∈ [0 , (cid:26) L ( ω ) − ( V ∗ ) ′ ( x ( t )) p ∗ ( x ( t )) ω (cid:27) + min ζ ∈ [0+ ∞ [ (cid:8) c ( ζ ) − ( V ∗ ) ′ ( x ( t )) x ( t ) ζ (cid:9) + (cid:18) λ + rp ∗ ( x ( t )) − λ − µ (cid:19) x ( t )( V ∗ ) ′ ( x ( t )) − rV ∗ ( x ( t )) i = e − rt · h H (cid:0) x ( t ) , ( V ∗ ) ′ ( x ( t )) , p ∗ ( x ( t )) (cid:1) − rV ∗ ( x ( t )) i = 0 . V ∗ ( x ) = φ u,v (0) ≤ lim t → T b − φ u,v ( t ) = Z T b e − rt · [ L ( u ( t )) + c ( v ( t ))] dt + e − rT b B, and this yields (3.41). It remains to check (ii) in Definition 2.1. The case x = 0 is trivial. Two remain cases willbe considered. CASE 1: If x ∈ ] x , x ∗ ] then x ( t ) > x for all t ∈ [0 , T b ]. This implies˙ x ( t ) = H ξ ( x ( t ) , Z ( x ( t ) , x ∗ ) , q ( x ( t ) , x ∗ )) . From the second equation in (2.10) it follows ddt p ( x ( t )) = p ′ ( x ( t )) ˙ x ( t ) = ( r + λ + v ∗ ( x ( t ))) p ( x ( t )) − ( r + λ ) , Thus, for every t ∈ [0 , T b ] it holds p ( x (0)) = p ( x ( t )) · Z t e − ( r + λ + v ∗ ( x ( τ ))) dτ + Z t ( r + λ ) Z τ e − ( r + λ + v ∗ ( x ( s ))) ds dτ By letting t → T b , we obtain p ( x ) = Z T b ( r + λ ) Z τ e − ( r + λ + v ∗ ( x ( s ))) ds dτ + θ ( x ∗ ) · Z T b e − ( r + λ + v ∗ ( x ( τ ))) dτ CASE 2:
Assume that x ∈ [ a ( x k ) , x k [ for some k ∈ { , , ..., N } . In this case, T b = + ∞ and x ( t ) ∈ [ a x k , x k [ such that lim t → + ∞ x ( t ) = x k . With a similar computation, we obtain p ( x ) = θ ( x ∗ ) · Z ∞ e − ( r + λ + v ∗ ( x ( τ ))) dτ proving (ii). x ∗ In this section, we study the behavior of the total cost for servicing when the maximum size x ∗ of the debt-ratio-income, at which bankruptcy is declared, becomes very large. It turnsout that a crucial role in the asymptotic behavior of V as x ∗ → + ∞ is played by the speed ofdecay of the salvage rate θ ( x ∗ ) as x ∗ → + ∞ , which represents the fraction of the investmentthat can be recovered by the investors after the bankruptcy (and the unitary bond discountedprice at the bankruptcy threshold). More precisely, the following proposition show that25 if the salvage rate decay sufficiently slowly, i.e., the lenders can still recover a sufficientlyhigh fraction of their investment after the bankruptcy, then the best choice for theborrower is to implement the Ponzi’s scheme; • otherwise, if the salvage rate θ ( x ∗ ) decays sufficiently fast, then Ponzi’s scheme is nolonger an optimal solution for the borrower. Proposition 4.1.
Let ( V ( x, x ∗ ) , p ( x, x ∗ )) be constructed in Theorem 3.5. The following holds:(i) if lim sup s → + ∞ θ ( s ) s = R < + ∞ then lim inf x ∗ → + ∞ V ( x, x ∗ ) ≥ B · (cid:18) − Rx (cid:19) rr + λ (4.42) for all x ≥ r − µ · max (cid:26) , BL ′ (0) , C Bc ′ (0) , C c − ( rB ) (cid:27) . (ii) if lim s → + ∞ θ ( s ) s = + ∞ then lim sup x ∗ →∞ V ( x, x ∗ ) = 0 for all x ∈ [0 , x ∗ [ . (4.43) Proof. We first provide an upper bound on v ( · , x ∗ ). From (2.10) and (2.9), we estimate H ( x, ξ, p ) ≥ min v ≥ { c ( v ) − xξv } + [( r − µ ) x − · ξp ≥ min v ≥ { c ( v ) − xξv } + ( r − µ ) x · ξp := K ( x, ξ, p )for all ξ, p > x ≥ r − µ . We compute K ξ ( x, ξ, p ) = ( r − µ ) x p − xv K where v K = ≤ xξ < c ′ (0) , ( c ′ ) − ( xξ ) if xξ ≥ c ′ (0) > . This implies that the maximum of K is achieved for v K = r − µ p and its value ismax ξ ≥ K ( x, ξ, p ) = K ( x, ξ K , p ) = c (cid:18) r − µ p (cid:19) , with ξ K = c ′ ( v K ) x . Thus, the monotone increasing property of the map ξ → H ( x, ξ, p ( x, x ∗ )) on the interval (cid:2) , ξ ♯ ( x, p ( x, x ∗ )) (cid:3) implies that F − ( x, V ( x, x ∗ ) , p ( x, x ∗ )) < ξ K = ⇒ v ( x, x ∗ ) ≤ r − µ p ( x, x ∗ ) . (4.44)26rovided that c (cid:18) r − µ p ( x, x ∗ ) (cid:19) ≥ rB . From (2.10)) and (2.9), it follows rB ≥ − xV ′ ( x, x ∗ ) v ( x, x ∗ ) + [( r − µ ) x − u ( x, x ∗ )] · V ′ ( x, x ∗ ) p ( x, x ∗ ) ≥ (cid:20) ( r − µ ) x − (cid:21) · V ′ ( x, x ∗ ) p ( x, x ∗ ) ≥ ( r − µ ) x · V ′ ( x, x ∗ ) p ( x, x ∗ ) . Thus, if p ( x, x ∗ ) ≤ min (cid:26) r − µ c − ( rB ) , ( r − µ ) c ′ (0)4 B (cid:27) and x ≥ max (cid:26) r − µ , B ( r − µ ) L ′ (0) (cid:27) (4.45)then V ′ ( x, x ∗ ) p ( x, x ∗ ) ≤ B ( r − µ ) x ≤ L ′ (0) = ⇒ u ( x, x ∗ ) = 0 , (4.46)and V ′ ( x, x ∗ ) x ≤ Br − µ · p ( x, x ∗ ) ≤ c ′ (0) = ⇒ v ( x, x ∗ ) = 0 . (4.47)In this case, from (2.10), (2.9) and (2.13), it holds( r + λ )( p ( x, x ∗ ) −
1) = (cid:18) λ + rp ( x, x ∗ ) − λ − µ (cid:19) xp ′ ( x, x ∗ ) . Thus, p ( x, x ∗ ) = θ ( x ∗ ) x ∗ x · (cid:18) − p ( x, x ∗ )1 − θ ( x ∗ ) (cid:19) r − µr + λ provided that (4.45) holds. Assume that lim sup s ∈ [0 , + ∞ ) θ ( s ) s = R < + ∞ , there exists a constant C < + ∞ such that sup s ∈ [0 , + ∞ ) θ ( s ) s = C . Since p ( · , x ∗ ) is increasing,it holds p ( x, x ∗ ) ≤ θ ( x ∗ ) x ∗ x ≤ C x if (4.45) holds . (4.48)Denote by M := 1 r − µ · max (cid:26) , BL ′ (0) , C Bc ′ (0) , C c − ( rB ) (cid:27) , we then have u ( x, x ∗ ) = v ( x, x ∗ ) = 0 for all x ∈ [ M, x ∗ ] , x ∗ ≥ M .
From (2.10), (2.9) and (2.13), (
V, p ) solves the system of ODEs V ′ ( x, x ∗ ) = rp [( λ + r ) − ( λ + µ ) p ( x, x ∗ )] x · Vp ′ ( x, x ∗ ) = ( λ + r ) · p ( x, x ∗ )( p ( x, x ∗ ) − λ + r ) − ( λ + µ ) p ( x, x ∗ )] x (4.49)27or all x ∈ [ M, x ∗ ] with x ∗ ≥ M . Solving the above system of ODEs (see in Section 5 of [5]),we obtain that V ( x, x ∗ ) = B · (cid:18) − p ( x, x ∗ )1 − θ ( x ∗ ) (cid:19) rr + λ , p ( x, x ∗ ) = θ ( x ∗ ) x ∗ x · (cid:18) − p ( x, x ∗ )1 − θ ( x ∗ ) (cid:19) r − µr + λ for all x ≥ [ M, x ∗ ]. Thus,lim inf x ∗ → + ∞ V ( x, x ∗ ) ≥ B · (cid:18) − Rx (cid:19) rr + λ for all x ≥ M and this yields (4.42). We are now going to prove (ii). Assume thatlim sup s → + ∞ θ ( s ) s = + ∞ . (4.50)Set γ := min (cid:26) r − µ c − ( rB ) , ( r − µ ) c ′ (0)4 B (cid:27) and M := max (cid:26) r − µ , B ( r − µ ) L ′ (0) (cid:27) . For any x ∗ > M , denote by τ ( x ∗ ) := x ∗ if θ ( x ∗ ) ≥ γ , inf n x ≥ M (cid:12)(cid:12)(cid:12) p ( x, x ∗ ) ≤ γ o if θ ( x ∗ ) < γ . From (4.45)–(4.47), the decreasing property of p yields p ( x, x ∗ ) ≥ γ for all x ∈ [ M , τ ( x ∗ )[ (4.51)and p ( x, x ∗ ) < γ = ⇒ u ( x, x ∗ ) = v ( x, x ∗ ) for all x ∈ [ τ ( x ∗ ) , x ∗ ] . As in the step 2, for any x ∈ [ τ ( x ∗ ) , x ∗ ], we have V ( x, x ∗ ) = B · (cid:18) − p ( x, x ∗ )1 − θ ( x ∗ ) (cid:19) rr + λ , p ( x, x ∗ ) = θ ( x ∗ ) x ∗ x · (cid:18) − p ( x, x ∗ )1 − θ ( x ∗ ) (cid:19) r − µr + λ This implies that V ( x, x ∗ ) = B · (cid:18) p ( x, x ∗ ) xθ ( x ∗ ) x ∗ (cid:19) rr − µ ≤ B · (cid:18) xθ ( x ∗ ) x ∗ (cid:19) rr − µ (4.52)for all x ∈ [ τ ( x ∗ ) , x ∗ ]. On the other hand, for any x ∈ [ M , τ ( x ∗ )], from (2.10), (2.9) and(4.51), it holds rV ( x, x ∗ ) ≤ r + λp ( x, x ∗ ) xV ′ ( x, x ∗ ) ≤ ( r + λ ) xγ · V ′ ( x, x ∗ ) . V ( x, x ∗ ) ≤ V ( τ ( x ∗ ) , x ∗ ) · (cid:18) xτ ( x ∗ ) (cid:19) rγr + λ ≤ B · (cid:18) xτ ( x ∗ ) (cid:19) rγr + λ for all x ∈ [ M , τ ( x ∗ )] . (4.53)For any fix x ≥ M , we will prove thatlim sup x ∗ → + ∞ V ( x , x ∗ ) = 0 . (4.54)Two cases are considered: • If lim sup x ∗ → + ∞ τ ( x ∗ ) = + ∞ then (4.53) yieldslim x ∗ → + ∞ V ( x , x ∗ ) ≤ lim inf x ∗ → + ∞ B · (cid:18) x τ ( x ∗ ) (cid:19) rγr + λ = 0 . • If lim sup x ∗ → + ∞ τ ( x ∗ ) < + ∞ then τ ( x ∗ ) < M for all x ∗ > M >
0. Recalling (4.52) and (4.50), we obtain thatlim x ∗ →∞ V ( x , x ∗ ) ≤ lim x ∗ →∞ V ( x + M , x ∗ ) ≤ lim x ∗ →∞ B · (cid:18) x + M θ ( x ∗ ) x ∗ (cid:19) rr − µ = 0 . Thus, (4.54) holds and the increasing property of V ( · , x ∗ ) yields (4.43).We complete this section by showing that for sufficiently large initial debt-ratio-income andbankruptcy threshold and recovery fraction after bankruptcy, the optimal strategy for theborrower will use currency devaluation to deflate the debt-ratio-income. For simplicity, let usconsider x ∗ and B ∗ sufficiently large such that x ∗ > L ′ (0) + BrL ′ (0) · ( r − µ ) and B ≥ r − µ ) c ′ (0) r . (4.55)In this case, the following holds: Proposition 4.2 (Devaluating strategies) . Let x ( V ( x, x ∗ ) , p ( x, x ∗ )) be an equilibriumsolution of (2.10) with boundary conditions (2.11) . If θ ( x ∗ ) x ∗ > r + λ ) c ′ (0) r − µ · (cid:18) rB + 1 L ′ (0) (cid:19) (4.56) then the function v ∗ ( x, x ∗ ) = argmin ω ≥ (cid:8) c ( ω ) − ωxV ′ ( x, x ∗ ) (cid:9) is not identically zero. roof. Set M := L ′ (0) + BrL ′ (0) · ( r − µ ) . Assume by a contradiction that v ∗ ( x, x ∗ ) = 0 for all x ∈ [ M, x ∗ ]. In particular, we have0 ≤ xV ′ ( x, x ∗ ) ≤ c ′ (0) x ∈ [ M, x ∗ ] . (4.57)The system (2.10) in [ M, x ∗ ] reduces to rV ( x ) = ˜ H ( x, V ′ ( x ) , p ( x ))( r + λ )( p ( x ) −
1) = ˜ H ξ ( x, V ′ ( x ) , p ( x )) · p ′ ( x ) (4.58)with ˜ H ( x, ξ, p ) = min u ∈ [0 , (cid:26) L ( u ) − up ξ (cid:27) + (cid:18) λ + rp − λ − µ (cid:19) x ξ. Since r > µ and p ∈ [0 , H ( x, ξ, p ) ≥ − ξp + ( λ + r − p ( λ + µ )) x ξp ≥ (( r − µ ) x − · ξp and (4.58) yields rB ≥ rV ( x, x ∗ ) ≥ (( r − µ ) x − · V ′ ( x, x ∗ ) p ( x, x ∗ ) . Thus, for x ∈ [ M, x ∗ ] we obtain V ′ ( x, x ∗ ) p ( x, x ∗ ) ≤ rB ( r − µ ) x − ≤ L ′ (0) , which immediately implies u ∗ ( x, x ∗ ) := argmin u ∈ [0 , (cid:26) L ( u ) − u · V ′ ( x, x ∗ ) p ( x, x ∗ ) (cid:27) = 0 . Hence, ( V ( · , x ∗ ) , p ( · , x ∗ )) solves (2.10) on [ M, x ∗ ] and V ( x, x ∗ ) = B · (cid:18) − p ( x, x ∗ )1 − θ ( x ∗ ) (cid:19) rr + λ ≥ B · (cid:18) − rr + λ · p ( x, x ∗ ) (cid:19) , (4.59) p ( x, x ∗ ) = θ ( x ∗ ) x ∗ x · (cid:18) − p ( x, x ∗ )1 − θ ( x ∗ ) (cid:19) r − µr + λ ≥ θ ( x ∗ ) x ∗ x · (cid:18) − r − µr + λ · p ( x, x ∗ ) (cid:19) , for all x ∈ [ M, x ∗ ]. From the above inequality, we derive p ( x, x ∗ ) ≥ ( r + λ ) θ ( x ∗ ) x ∗ ( r + λ ) x + ( r − µ ) θ ( x ∗ ) x ∗ . Thus, (4.57) and the first equation in (4.59) imply c ′ (0) ≥ xV ′ ( x, x ∗ ) = rp ( x, x ∗ ) · V ( x, x ∗ )( λ + r ) − ( λ + µ ) p ( x, x ∗ ) ≥ rp ( x, x ∗ ) Br + λ · r + λ − rp ( x, x ∗ )( λ + r ) − ( λ + µ ) p ( x, x ∗ ) ≥ rp ( x, x ∗ ) Br + λ ≥ rBθ ( x ∗ ) x ∗ ( r + λ ) x + ( r − µ ) θ ( x ∗ ) x ∗ for all x ∈ [ M, x ∗ ]. In particular, choose x = M and recall (4.55), we get M ≥ rB − ( r − µ ) c ′ (0)( r + λ ) c ′ (0) · θ ( x ∗ ) x ∗ ≥ rB r + λ ) c ′ (0) · θ ( x ∗ ) x ∗ and it contradicts (4.56). 30 Appendix
We first introduce now some concepts of convex analysis, referring the reader to [9] and [14]for a comprehensive introduction to the subject.
Definition 5.1 (Convex conjugate and subdifferential) . We recall that the convex conjugate F ◦ : R d → R ∪ {±∞} of a map F : R d → R ∪ { + ∞} is the lower semicontinuous convexfunction defined by F ◦ ( z ∗ ) = sup z ∈ R d n h z ∗ , z i − F ( z ) o . Let F : R d → R ∪ { + ∞} be proper (i.e., not identically + ∞ ), convex, lower semicontinuousfunctions, x ∈ dom F := { x ∈ R d : F ( x ) ∈ R } . We define the subdifferential in the sense ofconvex analysis of F at x by setting ∂F ( x ) := { v x ∈ R d : F ( y ) − F ( x ) ≥ h v x , y − x i for all y ∈ R d } . The following result provide a list of some properties of the sub-differential in the sense ofconvex analysis.
Lemma 5.2 (Properties of the subdifferential) . Let
F, G : R d → R ∪ { + ∞} be proper (i.e.,not identically + ∞ ), convex, lower semicontinuous functions,1. If F is classically (Fr´echet) differentiable at x , then ∂F ( x ) = { F ′ ( x ) } .2. z ∗ ∈ ∂F ( z ) if and only if z ∈ ∂F ◦ ( z ∗ ) .3. F ( x ) = min x ∈ R d F ( x ) if and only if ∈ ∂F ( x ) z ∗ ∈ ∂F ◦ ( z ) if and only if F ( z ) + F ◦ ( z ∗ ) = h z ∗ , z i . In this case z ∗ ∈ dom F ◦ ;5. λ ≥ we have ∂ ( λF )( z ) = λ∂F ( z ) ;6. if there exists z ∈ dom( F ) ∩ dom( G ) such that F is continuous at z then ∂ ( F + G )( x ) = ∂F ( x ) + ∂G ( x ) for all x ∈ dom( F ) ∩ dom( G ) ;7. let ¯ y ∈ R m , Λ : R m → R d be a linear map, G be continuous and finite at Λ(¯ y ) ; Then ∂ ( G ◦ Λ)( y ) = Λ T ∂G (Λ y ) for all y ∈ R m , where Λ T : R d → R m is the adjoint of Λ . We now collect some technical results related to the Hamiltonian function:
Lemma 5.3. If (A1) - (A2) hold then L ◦ , c ◦ : R → R are continuously differentiable such that L ◦ ( ρ ) ≤ max { , ρ } , c ◦ ( ρ ) ≤ max { , v max ρ } , and ( L ◦ ) ′ ( ρ ) = , if ρ < L ′ (0) , ( L ′ ) − ( ρ ) , if ρ ≥ L ′ (0) , ( c ◦ ) ′ ( ρ ) = , if ρ < c ′ (0) , ( c ′ ) − ( ρ ) , if ρ ≥ c ′ (0) . roof. Recalling the assumptions (A1) − (A2) on L, c , the equations L ◦ ( ρ ) + L ( u ) = uρ , c ◦ ( ρ ) + c ( v ) = vρ , admits as unique solutions u ( ρ ) = , if ρ < L ′ (0) , ( L ′ ) − ( ρ ) , if ρ ≥ L ′ (0) > , v ( ρ ) = , if ρ < c ′ (0) , ( c ′ ) − ( ρ ) , if ρ ≥ c ′ (0) ≥ . The result now follows from Theorem 23.5, Theorem 25.1, and Theorem 26.3 in [14]. Forthe second part, set I C ( s ) = 0 if s ∈ C and 0 otherwise, since L ( u ) ≥ I [0 , ( u ) and c ( v ) ≥ I [0 ,v max ] ( v ), we have L ◦ ( ρ ) ≤ I ◦ [0 , ( ρ ) = max u ∈ [0 , h u, ρ i = max { , ρ } ,c ◦ ( ρ ) ≤ I ◦ [0 ,v max ] ( ρ ) = max v ∈ [0 ,v max ] h v, ρ i = max { , v max · ρ } and this complete the proof.As a consequence of Lemma 5.3, the following holds: Lemma 5.4.
Assume (A1) - (A2) , and let H be defined as in (2.9) . Then H is continuousdifferentiable and its gradient at points ( x, ξ, p ) ∈ [0 , + ∞ [ × [0 , + ∞ [ × ]0 , can be expressed interms of u ∗ ( ξ, p ) := ( L ◦ ) ′ ( ξ/p ) and v ∗ ( x, ξ ) := ( c ◦ ) ′ ( xξ ) by H x ( x, ξ, p ) = h ( λ + r ) − p ( λ + µ + v ∗ ( x, ξ )) i · ξp ,H ξ ( x, ξ, p ) = 1 p · h x (cid:0) ( λ + r ) − p ( λ + µ + v ∗ ( x, ξ )) (cid:1) − u ∗ ( ξ, p ) i ,H p ( x, ξ, p ) = ( u ∗ ( ξ, p ) − x ( λ + r )) · ξp , (5.60) and u ∗ ( ξ, p ) = argmin u ∈ [0 , (cid:26) L ( u ) − u ξp (cid:27) ,v ∗ ( x, ξ ) = argmin v ≥ { c ( v ) − vxξ } . Moreover, for all x > , < p ≤ , it holds ∇ u ∗ ( ξ, p ) = (1 , − L ′ ( u ∗ ( x, ξ, p ))) pL ′′ ( u ∗ ( x, ξ, p )) if ξ > pL ′ (0) , (5.61) ∇ v ∗ ( x, ξ ) = ( ξ, x ) c ′′ ( v ∗ ( x, ξ )) if xξ > c ′ (0) , lim ξ → + ∞ v ∗ ( x, ξ ) = v max . Lemma 5.5.
Let the assumptions (A1) - (A2) hold. Then . for all ξ ≥ and p ∈ ]0 , , the function H satisfies H ( x, ξ, p ) ≤ (cid:18) λ + rp − ( λ + µ ) (cid:19) xξ ; H ( x, ξ, p ) ≥ (cid:18) ( λ + r ) x − p − ( λ + µ + v ∗ ( x, ξ )) x (cid:19) · ξ ≥ (cid:18) ( λ + r ) x − p − ( λ + µ + v max ) x (cid:19) · ξ ; H ξ ( x, ξ, p ) ≤ (cid:18) λ + rp − ( λ + µ ) (cid:19) x ; H ξ ( x, ξ, p ) ≥ ( λ + r ) x − p − ( λ + µ + v ∗ ( x, ξ )) x ≥ ( λ + r ) x − p − ( λ + µ + v max ) x ;
2. for every x, p > the map ξ H ( x, ξ, p ) is concave down and satisfies H ( x, , p ) = 0 , H ξ ( x, , p ) = (cid:18) λ + rp − ( λ + µ ) (cid:19) x. Proof.
The concavity of ξ H ( x, ξ, p ) for every x, p > H in (2.9). The equalities in item (2) are immediate from Lemma 5.3. The upper boundon H ( x, ξ, p ) follows from the positivity of L ◦ and c ◦ . By concavity, the map ξ H ξ ( x, ξ, p )is monotone decreasing, thus H ξ ( x, ξ, p ) ≤ H ξ ( x, , p ), which proves the upper bound on H ξ ( x, ξ, p ) together with item (2). The lower estimate for H ( x, ξ, p ) comes from the secondpart of Lemma 5.3, in particular from the upper estimate on L ◦ ( · ). The lower estimate for H ξ ( x, ξ, p ) comes from Lemma 5.4, noticing thatlim ξ → + ∞ u ∗ ( ξ, p ) = lim ρ → + ∞ ( L ′ ) − ( ρ ) = 1 , lim ξ → + ∞ v ∗ ( x, ξ ) = lim ρ → + ∞ ( c ′ ) − ( ρ ) = v max , and using the decreasing property of ξ H ξ ( x, ξ, p ), i.e., the fact thatlim ζ → + ∞ H ξ ( x, ζ, p ) ≤ H ξ ( x, ξ, p )for all x ≥ p ∈ ]0 , ξ ∈ R . Lemma 5.6.
Assume that f : I → R is a C convex strictly increasing function definedon a real interval I , and satisfying f ′′ ≥ δ > . Then, denoted by g its inverse function, g : f ( I ) → I , we have that g is / -H¨older continuous.Proof. Indeed, let x , x ∈ f − ( I ) with x ≤ x , and set y = g ( x ) and y = g ( x ). f ( y ) − f ( y ) = Z y y f ′ ( t ) dt = Z y y [ f ′ ( t ) − f ′ ( y )] dt = f ′ ( y ) · ( y − y ) + Z y y Z ty f ′′ ( s ) ds dt ≥ f ′ ( y ) · ( y − y ) + δ y − y ) ≥ δ y − y ) , f is strictly increasing, f ′′ ( s ) ≥ δ , and y ≤ y . Thus if x ≥ x we have | g ( x ) − g ( x ) | ≤ r δ | x − x | / . By switching the roles of x and x , the same holds true if x ≥ x . Acknowledgments.
The research by K. T. Nguyen was partially supported by a grant fromthe Simons Foundation/SFARI (521811, NTK).
References [1] M. Bardi and I. Capuzzo Dolcetta,
Optimal Control and Viscosity Solutions ofHamilton-Jacobi-Bellman Equations , Birkh¨auser, 1997.[2] H. Amann, Invariant sets and existence theorems for semilinear parabolic equation andelliptic system,
J. Math. Anal. Appl. (1978), 432–467.[3] T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory , 2 nd Edition, Aca-demic Press, London 1995.[4] A. Bressan and Khai T. Nguyen, A game theoretical model of debt and bankruptcy,
ESAIM: COCV , Volume 22, Number 4, October-December 2016, 953 – 982[5] A. Bressan, A. Marigonda, Khai T. Nguyen, and M. Palladino, Optimal strategies ina debt management problem,
SIAM J. Financial Math. , vol. 8, n. 1 (2017), pp. 841 –873.[6] A. Bressan and J. Yiang, The vanishing viscosity limit for a system of H-J equationsrelated to a debt management problem,
Discr. Cont. Dyn. Syst. - Series S
11 (2018),793–824.[7] A. Bressan and B. Piccoli,
Introduction to the Mathematical Theory of Control , AIMSSeries in Applied Mathematics, Springfield Mo. 2007.[8] R. Capuani, S. Gilmore, and Khai T. Nguyen, A model of debt with bankruptcy riskand currency devaluation,
Minimax Theory and its Applications , 5 (2020), no. 2, 251– 274.[9] I. Ekeland and R. Temam,
Convex Analysis and Variational Problems , SIAM Classicsin Applied Mathematics , 1999.[10] L. C. Evans, Partial Differential Equations , Second Edition, American MathematicalSociety, Providence, 2010.[11] W. Fleming and R. Rishel,
Deterministic and stochastic optimal control , Springer-Verlag, Berlin-New York, 1975.[12] A. Marigonda, and Khai T. Nguyen,
A Debt Management Problem with CurrencyDevaluation , submitted (https://arxiv.org/abs/1805.05043).3413] G. Nu˜no and C. Thomas, Monetary policy and sovereign debt vulnerability,
Workingdocument n. 1517 (2015), Banco de Espa˜na.[14] R.T. Rockafellar,