A Stochastic Model of Optimal Debt Management and Bankruptcy
Alberto Bressan, Antonio Marigonda, Khai T. Nguyen, Michele Palladino
aa r X i v : . [ m a t h . O C ] S e p A Stochastic Model of Optimal Debt Management andBankruptcy
Alberto Bressan ( ∗ ) , Antonio Marigonda ( ∗∗ ) , Khai T. Nguyen ( ∗∗∗ ) , and Michele Palladino ( ∗ ) (*) Department of Mathematics, Penn State UniversityUniversity Park, PA 16802, USA.(**) Dipartimento di Informatica, Universit`a di Verona, Italy(***) Department of Mathematics, North Carolina State University, USARaleigh, NC 27695, USA.e-mails: [email protected], [email protected],[email protected], [email protected] October 8, 2018
Abstract
A problem of optimal debt management is modeled as a noncooperative game betweena borrower and a pool of lenders, in infinite time horizon with exponential discount. Theyearly income of the borrower is governed by a stochastic process. When the debt-to-income ratio x ( t ) reaches a given size x ∗ , bankruptcy instantly occurs. The interest ratecharged by the risk-neutral lenders is precisely determined in order to compensate for thispossible loss of their investment.For a given bankruptcy threshold x ∗ , existence and properties of optimal feedbackstrategies for the borrower are studied, in a stochastic framework as well as in a limitdeterministic setting. The paper also analyzes how the expected total cost to the borrowerchanges, depending on different values of x ∗ . We consider a problem of optimal debt management in infinite time horizon, modeled as anoncooperative game between a borrower and a pool of risk-neutral lenders. Since the debtormay go bankrupt, lenders charge a higher interest rate to offset the possible loss of part oftheir investment.In the models studied in [7, 8], the borrower has a fixed income, but large values of the debtdetermine a bankruptcy risk. Namely, if at a given time t the debt-to-income ratio x ( t ) is toobig, there is a positive probability that panic spreads among investors and bankruptcy occurswithin a short interval [ t, t + ε ]. This event is similar to a bank run. Calling T B the randombankruptcy time, this meansProb n T B ∈ [ t, t + ε ] (cid:12)(cid:12)(cid:12) T B > t o = ρ ( x ( t )) · ε + o ( ε ) . ρ ( · ) is a given, nondecreasing function.At all times t , the borrower must allocate a portion u ( t ) ∈ [0 ,
1] of his income to service thedebt, i.e., paying back the principal together with the running interest. Our analysis will bemainly focused on the existence and properties of an optimal repayment strategy u = u ∗ ( x )in feedback form.In the alternative model proposed by Nu˜no and Thomas in [13], the yearly income Y ( t ) ismodeled as a stochastic process: dY ( t ) = µY ( t ) dt + σY ( t ) dW. (1.1)Here µ ≥ W denotes Brownian motion on a filteredprobability space. Differently from [7, 8], in [13] it is the borrower himself that chooses whento declare bankruptcy. This decision will be taken when the debt-to-income ratio reaches acertain threshold x ∗ , beyond which the burden of servicing the debt becomes worse than thecost of bankruptcy.At the time T b when bankruptcy occurs, we assume that the borrower pays a fixed price B ,while lenders recover a fraction θ ( x ( T b )) ∈ [0 ,
1] of their outstanding capital. Here x θ ( x )is a nondecreasing function of the debt size. For example, the borrower may hold an amount R of collateral (gold reserves, real estate . . . ) which will be proportionally divided amongcreditors if bankruptcy occurs. In this case, when bankruptcy occurs each investor will receivea fraction θ ( x ( T b )) = max (cid:26) R x ( T b ) , (cid:27) (1.2)of his outstanding capital.Aim of the present paper is to provide a detailed mathematical analysis of some models closelyrelated to [13]. We stress that these problems are very different from a standard problem ofoptimal control. Indeed, the interest rate charged by lenders is not given a priori. Rather,it is determined by the expected evolution of the debt at all future times. Hence it dependsglobally on the entire feedback control u ( · ). A “solution” must be understood as a Nashequilibrium, where the strategy implemented by the borrower represents the best reply to thestrategy adopted by the lenders, and conversely.Our main results can be summarized as follows. • We first assume that value x ∗ at which bankruptcy occurs is a priori given, and seek anoptimal feedback control u = u ∗ ( x ) which minimizes the expected cost to the borrower.For any value σ ≥ σ = 0, the solution can be constructed by concatenating solutions of a system of twoODEs, with terminal data given at x = x ∗ . • We then study how the expected total cost of servicing the debt together with thebankruptcy cost (exponentially discounted in time), are affected by different choicesof x ∗ . 2et θ ( x ∗ ) ∈ [0 ,
1] be the salvage rate, i.e., the fraction of outstanding capital that willbe payed back to lenders if bankruptcy occurs when the debt-to-income ratio is x ∗ . Iflim s → + ∞ θ ( s ) s = + ∞ , (1.3)then, letting x ∗ → + ∞ , the total expected cost to the borrower goes to zero. On theother hand, if lim s → + ∞ θ ( s ) s < + ∞ , (1.4)then the total expected cost to the borrower remains uniformly positive as x ∗ → + ∞ .We remark that the assumption (1.4) is quite realistic. For example, if (1.2) holds, then θ ( x ∗ ) x ∗ = R for all x ∗ large enough. We remark that (1.4) rules out the possibility of a Ponzi scheme , where the old debt is serviced by initiating more and more new loans. Indeed,if (1.4) holds, then such a strategy will cause the total debt to blow up to infinity in finitetime.The remainder of the paper is organized as follows. In Section 2 we describe more carefully themodel, deriving the equations satisfied by the value function V and the discounted bond price p . In Sections 3 and 4 we construct equilibrium solutions in feedback form, in the stochasticcase ( σ >
0) and in the deterministic case ( σ = 0), respectively. Finally, Sections 5 and6 contain an analysis of how the expected cost to the borrower changes, depending on thebankruptcy threshold x ∗ .In the economics literature, some related models of debt and bankruptcy can be found in[1, 3, 7, 11, 12]. A general introduction to Nash equilibria and differential games can be foundin [5, 6]. For the basic theory of optimal control and viscosity solutions of Hamilton-Jacobiequations we refer to [4, 9]. We consider a slight variant of the model in [13]. We denote by X ( t ) the total debt of aborrower (a government, or a private company) at time t . The annual income Y ( t ) of theborrower is assumed to be a random process, governed by the stochastic evolution equation(1.1).The debt is financed by issuing bonds. When an investor buys a bond of unit nominal value,he receives a continuous stream of payments with intensity ( r + λ ) e − λt . Here • r is the interest rate payed on bonds, which we assume coincides with the discount rate, • λ is the rate at which the borrower pays back the principal.If no bankruptcy occurs, the payoff for an investor will thus be Z ∞ e − r ( r + λ ) e − λt dt = 1 . In case of bankruptcy, a lender recovers only a fraction θ ∈ [0 ,
1] of his outstanding capital.Here θ can depend on the total amount of debt at the time on bankruptcy. To offset this3ossible loss, the investor buys a bond with unit nominal value at a discounted price p ∈ [0 , t the value p ( t ) is uniquely determined by the competition of a poolof risk-neutral lenders.We call U ( t ) the rate of payments that the borrower chooses to make to his creditors, at time t .If this amount is not enough to cover the running interest and pay back part of the principal,new bonds are issued, at the discounted price p ( t ). The nominal value of the outstanding debtthus evolves according to ˙ X ( t ) = − λX ( t ) + ( λ + r ) X ( t ) − U ( t ) p ( t ) . (2.1)The debt-to-income ratio is defined as x = X/Y . In view of (1.1) and (2.1), Ito’s formula[14, 15] yields the stochastic evolution equation dx ( t ) = (cid:20)(cid:18) λ + rp ( t ) − λ + σ − µ (cid:19) x ( t ) − u ( t ) p ( t ) (cid:21) dt − σ x ( t ) dW. (2.2)Here u = U/Y is the portion of the total income allocated to pay for the debt. Throughoutthe following we assume r > µ . We observe that, if r < µ , then the borrower’s income growsfaster than the debt (even if no payment is ever made). In this case, with probability one thedebt-to-income ratio would approach zero as t → + ∞ .In this model, the borrower has two controls. At each time t he can decide the portion u ( t ) ofthe total income which he allocates to repay the debt. Moreover, he can decide at what time T b bankruptcy is declared. Throughout the following, we consider a control in feedback form,so that u = u ∗ ( x ) for x ∈ [0 , x ∗ ] , (2.3)while bankruptcy is declared as soon as x ( t ) reaches the value x ∗ . The bankruptcy time isthus the random variable T b . = inf (cid:8) t > x ( t ) = x ∗ (cid:9) . (2.4)At first, we let x ∗ be a given upper bound to the size of the debt. In a later section, we shallregard x ∗ as an additional control parameter, chosen by the borrower in order to minimize hisexpected cost.Given an initial size x of the debt, the total expected cost to the borrower, exponentiallydiscounted in time, is thus computed as J [ x , u ∗ , x ∗ ] = E (cid:20)Z T b e − rt L ( u ∗ ( x ( t ))) dt + e − rT b B (cid:21) x (0)= x . (2.5)Here B is a large constant, accounting for the bankruptcy cost, while L ( u ) is the instantaneouscost for the borrower to implement the control u . In the following we shall assume (A) The cost function L is twice continuously differentiable for u ∈ [0 , and satisfies L (0) = 0 , L ′ > , L ′′ > , lim u → − L ( u ) = + ∞ . (2.6)4or example, one may take L ( u ) = c ln 11 − u , or L ( u ) = cu (1 − u ) α , for some c, α > x ∗ >
0. For a given initial debt x (0) = x ∈ [0 , x ∗ ], we define the corresponding valuefunction as V ( x ) = inf u ∗ ( · ) J [ x , u ∗ , x ∗ ] . (2.7)Under the assumptions (A) we have V (0) = 0 , V ( x ∗ ) = B . (2.8)Denote by H ( x, ξ, p ) . = min ω ∈ [0 , (cid:26) L ( ω ) − ξp ω (cid:27) + (cid:18) λ + rp − λ + σ − µ (cid:19) x ξ (2.9)the Hamiltonian associated to the dynamics (2.2) and the cost function L in (2.5). Noticethat, as long as p >
0, the function H is differentiable with Lipschitz continuous derivativesw.r.t. all arguments.By a standard arguments, the value function V provides a solution to the second order ODE rV ( x ) = H (cid:0) x, V ′ ( x ) , p ( x ) (cid:1) + ( σx ) V ′′ ( x ) , (2.10)with boundary conditions (2.8). As soon as the function V is determined, the optimal feedbackcontrol is recovered by u ∗ ( x ) = argmin ω ∈ [0 , (cid:26) L ( ω ) − V ′ ( x ) p ( x ) ω (cid:27) . By (A) this yields u ∗ ( x ) = V ′ ( x ) p ( x ) ≤ L ′ (0) , ( L ′ ) − (cid:18) V ′ ( x ) p ( x ) (cid:19) if V ′ ( x ) p ( x ) > L ′ (0) . (2.11)Assuming that lenders are risk-neutral, the discounted bond price p is determined by p ( x ) = E h Z T b ( r + λ ) e − ( r + λ ) t dt + e − ( r + λ ) T b θ ( x ∗ ) i x (0)= x , (2.12)where θ denotes the salvage rate. In other words, if bankruptcy occurs when the debt-to-income ratio is x ∗ , then investors receive a fraction θ ( x ∗ ) ∈ [0 ,
1] of the nominal value of theirholding. Notice that the random variable T b in (2.4) now depends on the initial state x , thethreshold x ∗ , and on the feedback control u ∗ ( · ).5y the Feynman-Kac formula, p ( · ) satisfies the equation( r + λ )( p ( x ) −
1) = (cid:20)(cid:18) λ + rp ( x ) − λ + σ − µ (cid:19) x − u ∗ ( x ) p ( x ) (cid:21) · p ′ ( x ) + ( σx ) p ′′ ( x ) , (2.13)with boundary values p (0) = 1 , p ( x ∗ ) = θ ( x ∗ ) . (2.14)Combining (2.10) and (2.13), we are thus led to the system of second order ODEs rV ( x ) = H (cid:0) x, V ′ ( x ) , p ( x ) (cid:1) + ( σx ) · V ′′ ( x ) , ( r + λ )( p ( x ) −
1) = H ξ (cid:0) x, V ′ ( x ) , p ( x ) (cid:1) · p ′ ( x ) + ( σx ) · p ′′ ( x ) , (2.15)with the boundary conditions V (0) = 0 ,V ( x ∗ ) = B, p (0) = 1 ,p ( x ∗ ) = θ ( x ∗ ) . (2.16)We close this section by collecting some useful properties of the Hamiltonian function. Lemma 2.1.
Let the assumptions (A) hold. Then, for all ξ ≥ and p ∈ [0 , , the function H in (2.9) satisfies (cid:18) ( λ + r ) x − p + ( σ − λ − µ ) x (cid:19) ξ ≤ H ( x, ξ, p ) ≤ (cid:18) λ + rp − λ + σ − µ (cid:19) xξ, (2.17)( λ + r ) x − p + ( σ − λ − µ ) x ≤ H ξ ( x, ξ, p ) ≤ (cid:18) λ + rp − λ + σ − µ (cid:19) x. (2.18) Moreover, for every x, p > the map ξ H ( x, ξ, p ) is concave down and satisfies H ( x, , p ) = 0 , (2.19) H ξ ( x, , p ) = (cid:18) λ + rp − λ + σ − µ (cid:19) x , (2.20)lim ξ → + ∞ H ( x, ξ, p ) = −∞ , if p > (cid:18) λ + rp − λ + σ − µ (cid:19) x , + ∞ , if p ≤ (cid:18) λ + rp − λ + σ − µ (cid:19) x . (2.21) Proof. Since H ( x, · , p ) is defined as the infimum of a family of affine functions, it is concave down.We observe that (2.9) implies H ( x, ξ, p ) = (cid:18) λ + rp − λ + σ − µ (cid:19) xξ if 0 ≤ ξ ≤ pL ′ (0) . (2.22)This yields the identities (2.19)-(2.20). 6 . Taking ω = 0 in (2.9) we obtain the upper bound in (2.17). By the concavity property, themap ξ H ξ ( x, ξ, p ) is non-increasing. Hence (2.20) yields the upper bound in (2.18). Since L ( w ) ≥ w ∈ [0 , H ( x, ξ, p ) ≥ min w ∈ [0 , (cid:26) − ξp w (cid:27) + (cid:18) λ + rp − λ + σ − µ (cid:19) x ξ and obtain the lower bound in (2.17). On the other hand, using the optimality condition,one computes from (2.9) that H ξ ( x, ξ, p ) = ( λ + r ) x − u ∗ ( ξ, p ) p + ( σ − λ − µ ) x (2.23)where u ∗ ( ξ, p ) = argmin ω ∈ [0 , (cid:26) L ( ω ) − ξp ω (cid:27) = ( L ′ ) − (cid:18) ξp (cid:19) < . Observe that, as ξ → + ∞ , one has u ∗ ( ξ, p ) → ξ → H ξ ( x, ξ, p ) yields the lower bound in (2.18). To prove (2.21) we observe that, in the first case, there exists ω < ω p > (cid:18) λ + rp − λ + σ − µ (cid:19) x. Hence, letting ξ → + ∞ we obtainlim ξ → + ∞ H ( x, ξ, p ) ≤ lim ξ → + ∞ (cid:20) L ( ω ) − ω p ξ + (cid:18) λ + rp − λ + σ − µ (cid:19) x ξ (cid:21) = − ∞ . To handle the second case, we observe that, for ξ > ω ( ξ ) where L ′ ( ω ( ξ )) = ξ/p . Hence lim ξ → + ∞ ω ( ξ ) = 1 andlim ξ → + ∞ H ( x, ξ, p ) = lim ξ → + ∞ (cid:20) L ( ω ( ξ )) − ω ( ξ ) p ξ + (cid:18) λ + rp − λ + σ − µ (cid:19) x ξ (cid:21) ≥ lim ξ → + ∞ L ( ω ( ξ )) = + ∞ . Existence of solutions
Let x ∗ > V, p ) to the boundary value problem (2.15)-(2.16) is found,then the feedback control u = u ∗ ( x ) defined at (2.11) and the function p = p ( x ) provide anequilibrium solution to the debt management problem. In other words, for every initial value x of the debt, the following holds.(i) Consider the stochastic dynamics (2.2), with u ( t ) = u ∗ ( x ( t )) and p ( t ) = p ( x ( t )). Thenfor every x ∈ [0 , x ∗ ] the identity (2.12) holds, where T b is the random bankruptcy timedefined at (2.4) .(ii) Given the discounted price p = p ( x ), for every initial data x (0) = x ∈ [0 , x ∗ ] thefeedback control u = u ∗ ( x ) is optimal for the stochastic optimization problemminimize: E (cid:20)Z T b e − rt L ( u ∗ ( x ( t ))) dt + e − rT b B (cid:21) . (3.1)with stochastic dynamics (2.2), where p ( t ) = p ( x ( t )).To construct a solution to the system (2.15)-(2.16), we consider the auxiliary parabolic system V t ( t, x ) = − rV ( t, x ) + H (cid:0) x, V x ( t, x ) , p ( t, x ) (cid:1) + ( σx ) · V xx ( t, x ) ,p t ( t, x ) = ( r + λ )(1 − p ( t, x )) + H ξ (cid:0) x, V x ( t, x ) , p ( t, x ) (cid:1) · p x ( t, x ) + ( σx ) · p xx ( t, x ) , (3.2)with boundary conditions (2.16). Following [2], the main idea is to construct a compact,convex set of functions ( V, p ) : [0 , x ∗ ] [0 , B ] × [ θ ( x ∗ ) ,
1] which is positively invariant for theparabolic evolution problem. A topological technique will then yield the existence of a steadystate, i.e. a solution to (2.15)-(2.16).
Theorem 3.1.
In addition to (A) , assume that σ > and θ ( x ∗ ) > . Then the system ofsecond order ODEs (2.15) with boundary conditions (2.16) admits a C solution ( V , ¯ p ) , suchthat V : [0 , x ∗ ] → [0 , B ] is increasing and ¯ p : [0 , x ∗ ] → [ θ ( x ∗ ) , is decreasing.Proof. For any ε >
0, consider the parabolic system V t = − rV + H ( x, V x , p ) + (cid:18) ε + ( σx ) (cid:19) V xx , V (0) = 0 ,V ( x ∗ ) = B, (3.3) p t = ( r + λ )(1 − p )+ H ξ ( x, V x , p ) p x + (cid:18) ε + ( σx ) (cid:19) p xx , p (0) = 1 ,p ( x ∗ ) = θ ( x ∗ ) . (3.4)obtained from (3.2) by adding the terms εV xx , εp xx on the right hand sides. For any ε > x = 0.8 . Adopting a semigroup notation, let t ( V ( t ) , p ( t )) = S t ( V , p ) be the solution of thesystem (3.3)-(3.4), with initial data V (0 , x ) = V ( x ) , p (0 , x ) = p ( x ) . (3.5)Consider the closed, convex set of functions D = n ( V, p ) : [0 , x ∗ ] [0 , B ] × [ θ ( x ∗ ) ,
1] ;
V, p ∈ C , V x ≥ , p x ≤ , and (2.16) holds o . (3.6)We claim that the above domain is positively invariant under the semigroup S , namely S t ( D ) ⊆ D for all t ≥ . (3.7)Indeed, consider the constant functions V + ( t, x ) = B,V − ( t, x ) = 0 , p + ( t, x ) = 1 ,p − ( t, x ) = θ ( x ∗ ) . Recalling (2.19), one easily checks that V + is a supersolution and V − is a subsolution ofthe scalar parabolic problem (3.3). Indeed − rV + + H ( x, V + x , p ) + (cid:18) ε + ( σx ) (cid:19) V + xx ≤ , V + ( t, ≥ , V + ( t, x ∗ ) ≥ B. − rV − + H ( x, V − x , p ) + (cid:18) ε + ( σx ) (cid:19) V − xx ≥ , V − ( t, ≤ , V − ( t, x ∗ ) ≤ B. Similarly, p + is a supersolution and p − is a subsolution of the scalar parabolic problem(3.4).This proves that, if the initial data V , p in (3.5) take values in the box [0 , B ] × [ θ ( x ∗ ) , t ≥ ≤ V ( t, x ) ≤ B, θ ( x ∗ ) ≤ p ( t, x ) ≤ , (3.8)for all x ∈ [0 , x ∗ ]. In turn, this implies V x ( t, ≥ ,V x ( t, x ∗ ) ≥ , p x ( t, ≤ ,p x ( t, x ∗ ) ≤ . (3.9) Next, we prove that the monotonicity properties of V ( t, · ) and p ( t, · ) are preserved in time.Differentiating w.r.t. x one obtains V xt = − rV x + H x + H ξ V xx + H p p x + σ xV xx + (cid:18) ε + ( σx ) (cid:19) V xxx , (3.10)9 xt = − ( r + λ ) p x + (cid:18) ddx H ξ ( x, V x , p ) (cid:19) p x + H ξ p xx + σ xp xx + (cid:18) ε + ( σx ) (cid:19) p xxx . (3.11)By (2.19), for every x, p one has H x ( x, , p ) = H p ( x, , p ) = 0. Hence V x ≡ p x ≡ p x ( t, x ) ≤ ≤ V x ( t, x ) for all t ≥ , x ∈ [0 , x ∗ ] . This concludes the proof that the set D in (3.6) is positively invariant for the system(3.3)-(3.4). Thanks to the bounds (2.17)-(2.18), we can now apply Theorem 3 in [2] and obtain theexistence of a steady state ( V ε , p ε ) ∈ D for the system (3.3)-(3.4).We recall the main argument in [2]. For every T > V , p ) S T ( V , p ) is acompact transformation of the closed convex domain D into itself. By Schauder’s theoremit has a fixed point. This yields a periodic solution of the parabolic system (3.3)-(3.4), withperiod T . Letting T →
0, one obtains a steady state. It now remains to derive a priori estimates on this stationary solution, which will allow totake the limit as ε →
0. Consider any solution to − rV + H ( x, V ′ , p ) + (cid:18) ε + ( σx ) (cid:19) V ′′ = 0 , ( r + λ )(1 − p ) + H ξ ( x, V ′ , p ) p ′ + (cid:18) ε + ( σx ) (cid:19) p ′′ = 0 , (3.12)with V increasing, p decreasing, and satisfying the boundary conditions (2.16).By the properties of H derived in Lemma 2.1, we can find δ > ξ > x ∈ [0 , δ ] , p ∈ [ θ ( x ∗ ) , , ξ ≥ ξ = ⇒ H ( x, ξ, p ) ≤ . As a consequence, if V ′ ( x ) > ξ for some x ∈ [0 , δ ], then the first equation in (3.12) implies V ′′ ( x ) ≥
0. We conclude that either V ′ ( x ) ≤ ξ for all x ∈ [0 , x ∗ ], or else V ′ attains itsmaximum on the subinterval [ δ, x ∗ ].By the intermediate value theorem, there exists a point ˆ x ∈ [ δ, x ∗ ] where V ′ (ˆ x ) = V ( x ∗ ) − V ( δ ) x ∗ − δ ≤ Bx ∗ − δ . (3.13)By (2.17), the derivative V ′ satisfies a differential inequality of the form | V ′′ | ≤ c | V ′ | + c , x ∈ [ δ, x ∗ ] . (3.14)for suitable constants c , c . By Gronwall’s lemma, from the differential inequality (3.14)and the estimate (3.13) one obtains a uniform bound on V ′ ( x ), for all x ∈ [ δ, ˆ x ] ∪ [ˆ x, x ∗ ].10 . Similar arguments apply to p ′ . By (2.18), the term H ξ ( x, V ′ , p ) in (3.12) is uniformlybounded. For every δ >
0, by (3.12) shows that p ′ satisfies a linear ODE whose coefficientsremain bounded on [ δ, x ∗ ], uniformly w.r.t. ε . This yields the bound | p ′ ( x ) | ≤ C δ for all x ∈ [ δ, x ∗ ]for some constant C δ , uniformly valid as ε → ε →
0, the limit satisfies the boundary value p (0) = 1. one needs toprovide a lower bound on p also in a neighborhood of x = 0, independent of ε . Introducethe constant γ . = min ( , ( r + λ ) (cid:18) λ + rθ ( x ∗ ) − λ + σ − µ (cid:19) − ) . Then define p − ( x ) . = 1 − cx γ , choosing c > p − ( x ∗ ) = θ ( x ∗ ). We claim that the convex function p − is a lowersolution of the second equation in (3.12). Indeed, by (3.12), one has( r + λ ) cx γ − H ξ ( x, V ′ , p ) cγx γ − ≥ (cid:20) ( r + γ ) − (cid:18) λ + rθ ( x ∗ ) − λ + σ − µ (cid:19) γ (cid:21) cx γ ≥ . Letting ε →
0, we now consider a sequence ( V ε , p ε ) of solutions to (3.12) with bound-ary conditions (2.16). Thanks to the previous estimates, the functions V ε are uniformlyLipschitz continuous on [0 , x ∗ ], while the functions p ε are Lipschitz continuous on anysubinterval [ δ, x ∗ ] and satisfy p − ( x ) ≤ p ε ( x ) ≤ x ∈ [0 , x ∗ ] , ε > . By choosing a suitable subsequence, we achieve the uniform convergence ( V ε , p ε ) → ( V, p ),where
V, p are twice continuously differentiable on the open interval ]0 , x ∗ [, and satisfy theboundary conditions (2.16). If σ = 0, then the stochastic equation (2.2) reduces to the deterministic control system˙ x = (cid:18) λ + rp − λ − µ (cid:19) x − up . (4.1)We then consider the deterministic Debt Management Problem. (DMP) Given an initial value x (0) = x ∈ [0 , x ∗ ] of the debt, minimize Z T b e − rt L ( u ( t )) dt + e − rT b B , (4.2)11 ubject to the dynamics (4.1), where the bankruptcy time T b is defined as in (2.4), while p ( t ) = Z T b t ( r + λ ) e − ( r + λ ) s ds + e ( − r + λ )( T b − t ) · θ ( x ∗ ) = 1 − (1 − θ ( x ∗ )) e − ( r + λ )( T b − t ) . (4.3)Since in this case the optimal feedback control u ∗ and the corresponding functions V, p maynot be smooth, a concept of equilibrium solution should be more carefully defined.
Definition 4.1 (Equilibrium solution in feedback form) . A couple of piecewise Lipschitzcontinuous functions u = u ∗ ( x ) and p = p ∗ ( x ) provide an equilibrium solution to the debtmanagement problem (DMP), with continuous value function V ∗ , if(i) For every x ∈ [0 , x ∗ ], V ∗ is the minimum cost for the optimal control problemminimize: Z T b e − rt L ( u ( x ( t ))) dt + e − rT b B, (4.4)subject to ˙ x ( t ) = (cid:18) λ + rp ∗ ( x ( t )) − λ − µ (cid:19) x ( t ) − u ( t ) p ∗ ( x ( t )) , x (0) = x . (4.5)Moreover, every Carath´eodory solution of (4.5) with u ( t ) = u ∗ ( x ( t )) is optimal.(ii) For every x ∈ [0 , x ∗ ], there exists at least one solution t x ( t ) of the Cauchy problem˙ x = (cid:18) λ + rp ∗ ( x ) − λ − µ (cid:19) x − u ∗ ( x ) p ∗ ( x ) , x (0) = x , (4.6)such that p ∗ ( x ) = Z T b ( r + λ ) e − ( r + λ ) t dt + e ( − r + λ ) T b θ ( x ∗ ) = 1 − (1 − θ ( x ∗ )) · e − ( r + λ ) T b , (4.7)with T b as in (2.4).In the deterministic case, (2.15) takes the form rV ( x ) = H (cid:0) x, V ′ ( x ) , p ( x ) (cid:1) , ( r + λ )( p ( x ) −
1) = H ξ (cid:0) x, V ′ ( x ) , p ( x ) (cid:1) p ′ ( x ) , (4.8)with Hamiltonian function (see Fig. 4) H ( x, ξ, p ) = min ω ∈ [0 , (cid:26) L ( ω ) − ξp ω (cid:27) + (cid:18) λ + rp − ( λ + µ ) (cid:19) x ξ . (4.9)We consider solutions to (4.8) with the boundary condition V (0) = 0 ,V ( x ∗ ) = B, p (0) = 1 ,p ( x ∗ ) = θ ( x ∗ ) . (4.10)12ntroduce the function H max ( x, p ) . = sup ξ ≥ H ( x, ξ, p ) = L (cid:16) ( λ + r ) x − ( λ + µ ) px (cid:17) , with the understanding that H max ( x, p ) = + ∞ if ( λ + r ) x − ( λ + µ ) px ≥ . (4.11)If ( λ + r ) x − ( λ + µ ) px <
1, recalling (4.1) we define u ♯ ( x, p ) = ( λ + r ) x − ( λ + µ ) px . (4.12)Notice that u ♯ is the control that keeps the debt x constant in time. This value u ♯ achievesthe minimum in (4.9) when L ′ (cid:16) ( λ + r ) x − ( λ + µ ) px (cid:17) = ξp . This motivates the definition ξ ♯ ( x, p ) . = argmax ξ ≥ H ( x, ξ, p ) = p L ′ (cid:16) ( λ + r ) x − ( λ + µ ) px (cid:17) . (4.13)On the other hand, if ( λ + r ) x − ( λ + µ ) px ≥
1, then the function ξ H ( x, ξ, p ) is monotoneincreasing and we define ξ ♯ ( x, p ) . = + ∞ . Observe that H ξξ ( x, ξ, p ) ≤ , H ξ ( x, ξ, p ) > ≤ ξ < ξ ♯ ( x, p ) ,H ξ ( x, ξ, p ) < ξ > ξ ♯ ( x, p ) . (4.14)We regard the first equation in (4.8) as an implicit ODE for the function V . For every x ≥ p ∈ [0 , rV ( x ) > H max ( x, p ), then this equation has no solution. On the other hand,when 0 ≤ rV ( x ) ≤ H max ( x, p ) , the implicit ODE (4.8) can equivalently be written as a differential inclusion (Fig. 4): V ′ ( x ) ∈ n F − ( x, V, p ) , F + ( x, V, p ) o . (4.15) Remark . Recalling (4.1), we observe that • The value V ′ = F + ( x, V, p ) ≥ ξ ♯ ( x, p ) corresponds to the choice of an optimal controlsuch that ˙ x < • The value V ′ = F − ( x, V, p ) ≤ ξ ♯ ( x, p ) corresponds to the choice of an optimal controlsuch that ˙ x > • When rV = H max ( x, p ), then the value V ′ = F + ( x, V, p ) = F − ( x, V, p ) = ξ ♯ ( x, p )corresponds to the unique control such that ˙ x = 0.Since ξ H ( x, ξ, p ) is concave down, the functions F ± satisfy the following monotonicityproperties (Fig. 4) 13 H(x, ,p) _ F F + ξ ξ ξ rV(x,p) max Figure 1:
In the case where ( λ + r ) x − ( λ + µ ) px <
1, the the Hamiltonian function ξ H ( x, ξ, p )has a global maximum H max ( x, p ). For rV ≤ H max , the values F − ( x, V, p ) ≤ ξ ♯ ( x, p ) ≤ F + ( x, V, p )are well defined. (MP) For any fixed x, p , the map V F + ( x, V, p ) is decreasing, while V F − ( x, V, p ) isincreasing. For V ′ = F − , the second ODE in (4.8) can be written as p ′ ( x ) = G − (cid:0) x, V ( x ) , p ( x ) (cid:1) , where G − ( x, V, p ) . = ( r + λ )( p − H ξ (cid:0) x, F − ( x, V, p ) , p (cid:1) ≤ . (4.16) Consider the function W ( x ) . = 1 r L (cid:0) ( r − µ ) x (cid:1) , (4.17)with the understanding that W ( x ) = + ∞ if ( r − µ ) x ≥
1. Notice that W ( x ) is the total costof keeping the debt constantly equal to x (in which case there would be no bankruptcy andhence p ≡ V B ( · ) , p B ( · )) the solution to the system of ODEs V ′ ( x ) = F − ( x, V ( x ) , p ( x )) ,p ′ ( x ) = G − ( x, V ( x ) , p ( x )) , (4.18)with terminal conditions V ( x ∗ ) = B, p ( x ∗ ) = θ ( x ∗ ) . (4.19)Next, consider the point x . = inf n x ∈ [0 , x ∗ ] ; V B ( x ) < W ( x ) o , (4.20)14 WV B B 0 1r− µ x * x Figure 2:
Constructing the equilibrium solution in feedback form. For an initial value of the debt x (0) ≤ x , the debt increases until it reaches x , then it is held at the constant value x . If the initialdebt is x (0) > x , the debt keeps increasing until it reaches bankruptcy in finite time. and call V ( · ) the solution to the backward Cauchy problem V ′ ( x ) = F − ( x, V ( x ) , , x ∈ [0 , x ] ,V ( x ) = W ( x ) . (4.21)We will show that a feedback equilibrium solution to the debt management problem is obtainedas follows (see Fig. 4.1). V ∗ ( x ) = V ( x ) if x ∈ [0 , x ] ,V B ( x ) if x ∈ [ x , x ∗ ] . (4.22) p ∗ ( x ) = , if x ∈ [0 , x ] ,p B ( x ) , if x ∈ ] x , x ∗ ] . (4.23) u ∗ ( x ) = argmin ω ∈ [0 , (cid:26) L ( ω ) − ( V ∗ ) ′ ( x ) p ∗ ( x ) ω (cid:27) , if x = x , ( r − µ ) x , if x = x . (4.24) Theorem 4.3.
Assume that the cost function L satisfies the assumptions (A) , and moreover L (( r − µ ) x ∗ ) > rB . Then the functions V ∗ , p ∗ , u ∗ in (4.22)–(4.24) are well defined, and providean equilibrium solution to the debt management problem, in feedback form.Proof. The solution of (4.18)-(4.19) satisfies the obvious bounds V ′ ≥ , p ′ ≤ , V ( x ) ≤ B, p ( x ) ∈ [ θ ( x ∗ ) , .
15e begin by proving that the function V B is well defined and strictly positive for x ∈ ] x , x ∗ ].To prove that V B ( x ) > x ∈ ] x , x ∗ ] , assume, on the contrary, that V B ( y ) = 0 for some y > x ≥
0. By the properties of thefunction F − (see Fig. 4) it follows F − ( x, V, p ) ≤ C V , (4.25)for some constant C and all x ∈ [ y, x ∗ ], p ∈ [ θ ( x ∗ ) , V ( y ) = 0 implies V ( x ) = 0 for all x ≥ y , providing a contradiction.Next, observe that the functions F − , G − are locally Lipschitz continuous as long as 0 ≤ V < H max ( x, p ). Moreover, V ( x ) < W ( x ) implies V ( x ) < W ( x ) = H max ( x, ≤ H max ( x, p ( x )) . Therefore, the functions V B , p B are well defined on the interval [ x , x ∗ ]. If x = 0 the construction of the functions V ∗ , p ∗ , u ∗ is already completed in step . Inthe case where x >
0, we claim that the function V is well defined and satisfies0 < V ( x ) < W ( x ) for 0 < x < x . (4.26)Indeed, if V ( y ) = 0 for some y >
0, the Lipschitz property (4.25) again implies that V ( x ) = 0 for all x ≥ y . This contradicts the terminal condition in (4.21).To complete the proof of our claim (4.26), it suffices to show that W ′ ( x ) < F − ( x, W ( x ) ,
1) for all x ∈ ]0 , x ] . (4.27)This is true because W ′ ( x ) = r − µr L ′ (cid:0) r − µ ) x (cid:1) = r − µr ξ ♯ ( x, < ξ ♯ ( x, F − (cid:0) x, H max ( x, , (cid:1) = F − ( x, W ( x ) , . In the remaining steps, we show that V ∗ , p ∗ , u ∗ provide an equilibrium solution. Namely,they satisfy the properties (i)-(ii) in Definition 4.1.To prove (i), call V ( · ) the value function for the optimal control problem (4.4)-(4.5).For any initial value, x (0) = x , in both cases x ∈ [0 , x ] and x ∈ ] x , x ∗ ], the feedbackcontrol u ∗ in (4.24) yields 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 , t x ( t ) the solution to˙ x = (cid:18) λ + rp x ( x ) − λ − µ (cid:19) x − u ( t ) p x ( x ) , x (0) = x , (4.28)16ne has Z T b e − rt L ( u ( t )) dt + e − rT b B ≥ V ∗ ( x ) , (4.29)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 ( t ) . = Z t e − rs L ( u ( s )) ds + e − rt V ∗ ( x ( t )) . At any Lebesgue point t of u ( · ), we compute ddt φ u ( t ) = e − rt h L ( u ( t )) − rV ∗ ( x ( t )) + ( V ∗ ) ′ ( x ( t )) · ˙ x ( t ) i = e − rt (cid:20) L ( u ( t )) − rV ∗ ( x ( t )) + ( V ∗ ) ′ ( x ( t )) (cid:18)(cid:18) λ + rp ∗ ( x ( t )) − λ − µ (cid:19) x ( t ) − u ( t ) p ∗ ( x ( t )) (cid:19)(cid:21) ≥ e − rt (cid:20) min ω ∈ [0 , (cid:26) L ( ω ) − ( V ∗ ) ′ ( x ( t )) p ∗ ( x ( t )) ω (cid:27) + (cid:18) λ + rp ∗ ( x ( t )) − λ − µ (cid:19) x ( t )( V ∗ ) ′ ( x ( t )) − rV ∗ ( x ( t )) (cid:21) = e − rt h H (cid:0) x ( t ) , ( V ∗ ) ′ ( x ( t )) , p ∗ ( x ( t )) (cid:1) − rV ∗ ( x ( t )) i = 0 . Therefore, V ∗ ( x ) = φ u (0) ≤ lim t → T b − φ u ( t ) = Z T b e − rt L ( u ( t )) dt + e − rT b B, proving (4.29). It remains to check (ii). The case x = 0 is trivial. Two main cases will be considered. CASE 1: x ∈ ]0 , x ] . Then there exists a solution t x ( t ) of (4.6) such that p ( t ) = 1and x ( t ) ∈ ]0 , x ] for all t >
0. Moreover,lim t → + ∞ x ( t ) = x . In this case, T b = + ∞ and (4.7) holds. CASE 2: x ∈ ] x , x ∗ ] . Then x ( t ) > x for all t ∈ [0 , T b ]. This implies˙ x ( t ) = H ξ ( x ( t ) , V B ( x ( t )) , p B ( x ( t ))) . From the second equation in (4.8) it follows˙ p ( t ) = p ′ ( x ( t )) ˙ x ( t ) = ( r + λ )( p ( t ) − ,ddt ln(1 − p ( x ( t ))) = ( r + λ ) . t ∈ [0 , T b ] one has p ( x (0)) = 1 − (1 − p ( x ( t ))) · e − ( r + λ ) t . Letting t → T b we obtain p ( x ) = 1 − (1 − θ ( x ∗ ))) · e − ( r + λ ) T b , proving (4.7). Remark . The construction described at (4.22)–(4.24) uniquely determines a feedback equi-librium solution to the debt management problem.In general, however, we cannot rule out the possibility that a second solution exists. Indeed,if the solution V B , p B of (4.18)-(4.19) can be prolonged backwards to the entire interval [0 , x ∗ ],then we could replace (4.22)-(4.23) simply by V ∗ ( x ) = V B ( x ), p ∗ ( x ) = p B ( x ) for all x ∈ [0 , x ∗ ].This would yield a second solution.We claim that no other solutions can exist. This is based on the fact that the graphs of W and V B cannot have any other intersection, in addition to 0 and x . Indeed, assume on thecontrary that W ( x ) = V B ( x ) for some 0 < x < x (see Fig. 4.1). Since p B ( x ) < W ′ ( x ) ≤ V ′ B ( x ), the inequalities rW ( x ) = H ( x , W ′ ( x ) , < H ( x , W ′ ( x ) , p B ( x )) ≤ H ( x , V ′ B ( x ) , p B ( x )) = rV B ( x )yield a contradiction.Next, let V † , p † be any equilibrium solution and define x † . = sup (cid:8) x ∈ [0 , x ∗ ] ; p ( x ) = 1 (cid:9) . Then • On ] x † , x ∗ ] the functions V † , p † provide a solution to the backward Cauchy problem(4.18)-(4.19). • On ]0 , x † ] the function V † provides the value function for the optimal control problemminimize: Z ∞ e − rt L ( u ( t )) dt subject to the dynamics (with p ≡ x = ( r − µ ) x − u , and the state constraint x ( t ) ∈ [0 , x † ] for all t ≥ V † ( x ) = V B ( x ) , if x ∈ [ x † , x ∗ ] ,V † ( x ) ≤ W ( x ) , if x ∈ [0 , x † ] . Since V † must be continuous at the point x , by the previous analysis this is possible only if x = 0 or x = x . 18 V Wx xB0 1r− µ x * Figure 3:
By the monotonicity properties of the Hamiltonian function H in (4.9), the graphs of V B and W cannot have a second intersection at a point x > x ∗ . In this section we study the behavior of the value function V B when the maximum size x ∗ ofthe debt, at which bankruptcy is declared, becomes very large.For a given x ∗ >
0, we denote by V B ( · , x ∗ ), p B ( · , x ∗ ) the solution to the system (4.18) withterminal data (4.19). Letting x ∗ → ∞ , we wish to understand whether the value function V B remains positive, or approaches zero uniformly on bounded sets. Toward this goal, weintroduce the constant M . = 2 rB ( r − µ ) L ′ (0) . (5.1)From (4.8) and (4.9) it follows V ′ B ( x, x ∗ ) p B ( x, x ∗ ) ≤ rV B ( x, x ∗ )( r − µ ) x − . In turn, if x ∗ > M , this implies V ′ B ( x, x ∗ ) p B ( x, x ∗ ) ≤ L ′ (0) , for all x ∈ [ M , x ∗ ] . Calling u = u ∗ ( x ) the optimal feedback control, by (2.11) we have u ∗ ( x ) = 0 , for all x ∈ (cid:2) M , x ∗ (cid:3) . (5.2)In this case, the Hamiltonian function takes a simpler form, namely H ( x, V ′ , p ) = (cid:2) ( λ + r ) − ( λ + µ ) p (cid:3) · V ′ xp ,H ξ ( x, V ′ , p ) = (cid:2) ( λ + r ) − ( λ + µ ) p (cid:3) x . V ′ = rp [( λ + r ) − ( λ + µ ) p ] x V ,p ′ = ( λ + r ) · p ( p − λ + r ) − ( λ + µ ) p ] x . (5.3)The second ODE of in (5.3) is equivalent to ddx ln (cid:16) (1 − p ( x )) r − µ p ( x ) r + λ (cid:17) = r + λx . Solving backward the above ODE with the terminal data p ( x ∗ ) = θ ( x ∗ ), we obtain p B ( x, x ∗ ) = θ ( x ∗ ) x ∗ x · (cid:18) − p B ( x, x ∗ )1 − θ ( x ∗ ) (cid:19) r − µr + λ for all x ∈ (cid:2) M , x ∗ (cid:3) . (5.4)Therefore, p B ( x, x ∗ ) ≥ (cid:18) θ ( x ∗ ) x ∗ x (cid:19) r + λr − µ (cid:18) θ ( x ∗ ) x ∗ x (cid:19) r + λr − µ for all x ∈ (cid:2) M , x ∗ (cid:3) . (5.5)Different cases will be considered, depending on the properties of the function θ ( · ). By obviousmodeling considerations, we shall always assume θ ( x ) ∈ [0 , , θ ′ ( x ) ≤ x for all x ≥ . We first study the case where θ has compact support. Recall that M is the constant in (5.1). Lemma 5.1.
Assume that θ ( x ) = 0 for all x ≥ M , (5.6) for some constant M ≥ M . Then, for any x ∗ > M , the solution V B ( · , x ∗ ) , p B ( · , x ∗ ) of(4.18)-(4.19) satisfies V B ( x, x ∗ ) = B and p B ( x, x ∗ ) = 0 for all x ∈ (cid:2) M , x ∗ (cid:3) . Proof.
By (5.4) and (5.6), for every x ∗ > M one has p B ( x, x ∗ ) = 0 for all x ∈ (cid:2) M , x ∗ (cid:3) . Inserting this into the first ODE in (5.3), we obtain V ′ B ( x, x ∗ ) = 0 . In turn, this yields V B ( x, x ∗ ) = B for all x ∈ (cid:2) M , x ∗ (cid:3) . This means that bankruptcy instantlyoccurs if the debt reaches M . 20ext, we now consider that case where θ ( x ) > x . θ ( x ) > x ∈ [0 , ∞ [ . (5.7) Lemma 5.2. If x ∗ > M and θ ( x ∗ ) > , then V B ( x, x ∗ ) = B · (cid:18) p B ( x, x ∗ ) xθ ( x ∗ ) x ∗ (cid:19) rr − µ for all x ∈ (cid:2) M , x ∗ (cid:3) . (5.8) In particular, for x ∈ (cid:2) M , x ∗ (cid:3) one has B · (cid:18) θ ( x ∗ ) x ∗ x (cid:19) r + λr − µ ! − rr + λ ≤ V B ( x, x ∗ ) ≤ B · (cid:16) xθ ( x ∗ ) x ∗ (cid:17) rr − µ . (5.9) Proof.
Observe that x p B ( x, x ∗ ) is a strictly decreasing function of x . For a fixed value of x ∗ , let p χ ( p ) : [ θ ( x ∗ ) , [0 , x ∗ ] be the inverse function of p B ( · , x ∗ ). From (5.3), a directcomputation yields ddp V B ( χ ( p ) , x ∗ ) = rp [( λ + r ) − ( λ + µ ) p ] χ ( p ) · V B ( χ ( p ) , x ∗ ) · χ ′ ( p ) ,ddp p B ( χ ( p ) , x ∗ ) = ( λ + r ) · p ( p − λ + r ) − ( λ + µ ) p ] · χ ( p ) · χ ′ ( p ) = 1 . (5.10)From (5.10) it follows ddp ln V B ( χ ( p ) , x ∗ ) = rλ + r · p − . Solving the above ODE with the terminal data V B ( x ∗ , x ∗ ) = B , p B ( x ∗ , x ∗ ) = θ ( x ∗ ), we obtain V B ( χ ( p ) , x ∗ ) = (cid:18) − p − θ ( x ∗ ) (cid:19) rr + λ B , (5.11)hence V B ( x, x ∗ ) = (cid:18) − p B ( x, x ∗ )1 − θ ( x ∗ ) (cid:19) rr + λ B. Recalling (5.4), a direct computation yields (5.8). The upper and lower bounds for V B ( x, x ∗ )in (5.9) now follow from (5.5) and the inequality p B ( x, x ∗ ) ≤ Corollary 5.3.
Assume that lim sup x → + ∞ θ ( x ) x = + ∞ . (5.12) Then the value function V ∗ = V ∗ ( x, x ∗ ) satisfies lim x ∗ → + ∞ V ( x, x ∗ ) = 0 for all x ≥ . (5.13)Indeed, for x ≥ M we have V ( x, x ∗ ) = V B ( x, x ∗ ), and (5.13) follows from the second inequalityin (5.9). When x < M , since the map x V ( x, x ∗ ) is nondecreasing, we have0 ≤ lim x ∗ →∞ V ( x, x ∗ ) ≤ lim x ∗ →∞ V ( M , x ∗ ) = 0 . orollary 5.4. Assume that
R . = lim sup x → + ∞ θ ( x ) · x < + ∞ . (5.14) Then V B ( x, x ∗ ) ≥ B · (cid:18) Rx (cid:19) r + λr − µ ! − rr + λ for all x ∗ > x > M . (5.15) Moreover, the followings holds.(i) If θ ′ ( x ) θ ( x ) + 1 x ≥ and θ ′ ( x ) ≤ for all x > then inf x ∗ > V B ( x, x ∗ ) = lim x ∗ →∞ V B ( x, x ∗ ) > for all x ≥ M . (5.17) (ii) Assume that there exist < δ < such that δ · θ ′ ( x ) θ ( x ) + 1 x < for all x sufficiently large. Then, for each x > M , there exists an optimal value x ∗ = x ∗ ( x ) such that V B ( x, x ∗ ( x )) = inf x ∗ ≥ V B ( x, x ∗ ) . (5.19) Proof.
It is clear that (5.15) is a consequence of (5.9) and (5.14). We only need to prove (i)and (ii). For a fixed x ≥ M , we consider the functions of the variable x ∗ alone: Y ( x ∗ ) . = V B ( x, x ∗ ) , q ( x ∗ ) . = p B ( x, x ∗ ) . Using (5.8) and (5.4), we obtain Y ′ ( x ∗ ) Y ( x ∗ ) = rr − µ · (cid:18) q ′ ( x ∗ ) q ( x ∗ ) − h θ ′ ( x ∗ ) θ ( x ∗ ) + 1 x ∗ i(cid:19) , (5.20)and q ′ ( x ∗ ) q ( x ∗ ) = θ ′ ( x ∗ ) x ∗ + θ ( x ∗ ) θ ( x ∗ ) x ∗ + r − µr + λ · (cid:16) − q ′ ( x ∗ )1 − q ( x ∗ ) + θ ′ ( x ∗ )1 − θ ( x ∗ ) (cid:17) . (5.21)This implies q ′ ( x ∗ ) q ( x ∗ ) − (cid:20) θ ′ ( x ∗ ) θ ( x ∗ ) + 1 x ∗ (cid:21) =
11 + r − µr + λ · q ( x ∗ )1 − q ( x ∗ ) − · (cid:20) θ ′ ( x ∗ ) θ ( x ∗ ) + 1 x ∗ (cid:21) + r − µ ( r + λ ) (cid:16) r − µr + λ · q ( x ∗ )1 − q ( x ∗ ) (cid:17) · θ ′ ( x ∗ )1 − θ ( x ∗ ) . (5.22)If (5.16) holds, then (5.20) and (5.22) imply Y ′ ( x ∗ ) Y ( x ∗ ) = q ′ ( x ∗ ) q ( x ∗ ) − h θ ′ ( x ∗ ) θ ( x ∗ ) + 1 x ∗ i ≤ x ∗ > x ≥ M . Y is non-increasing. This proves (5.17).To prove (ii), we observe thatlim sup x ∗ →∞
11 + r − µr + λ · q ( x ∗ )1 − q ( x ∗ ) − < , lim x ∗ →∞ θ ( x ∗ ) = 0 . Hence (5.18) and (5.22) imply q ′ ( x ∗ ) q ( x ∗ ) − (cid:20) θ ′ ( x ∗ ) θ ( x ∗ ) − x ∗ (cid:21) > , for all x ∗ sufficiently large. By (5.20) this yields Y ′ ( x ∗ ) Y ( x ∗ ) > x ∗ large enough. Hence there exists some particular value x ∗ ( x ) ≥ x where the function x ∗ Y ( x ∗ ) = V B ( x, x ∗ ) attains its global minimum. This yields (5.19). x ∗ in the stochastic case In this section we study how the value function depends on the bankruptcy threshold x ∗ , in thestochastic case where σ >
0. Extensions of Corollaries 5.3 and 5.4, will be proved, constructingupper and lower bounds for the solution V ( · , x ∗ ), p ( · , x ∗ ) of the system (2.15)-(2.16), in theform V ( x ) ≤ V ( x, x ∗ ) ≤ V ( x ) , p ( x ) ≤ p ( x, x ∗ ) ≤ p ( x ) . (6.1) We begin by constructing a suitable pair of functions V , p . Let ( p , e V ) be the solutionto the backward Cauchy problem r e V ( x ) = (cid:16) λ + rp + σ (cid:17) x e V ′ , ( r + λ )( p −
1) = (cid:16) λ + rp + σ (cid:17) xp ′ , e V ( x ∗ ) = B,p ( x ∗ ) = θ ( x ∗ ) . (6.2)This solution satisfies p ( x ) = θ ( x ∗ ) x ∗ x · (cid:18) − p ( x )1 − θ ( x ∗ ) (cid:19) σ λ + rλ + r , lim x → p ( x ) = 1 , (6.3) e V ( x ) = B · (cid:18) − p ( x )1 − θ ( x ∗ ) (cid:19) rr + λ , lim x → ˜ V ( x ) = 0 . (6.4)Using (6.2) and (6.3) one obtains − p ′ ( x ) · (cid:18) xp ( x ) + σ + r + λr + λ · x − p ( x ) (cid:19) p ′ ( x ) · xp ( x ) + σ + r + λr + λ · − θ ( x ∗ )( θ ( x ∗ ) x ∗ ) r + λr + λ + σ · x σ r + λ + σ p ( x ) λ + rλ + r + σ . Since p is monotone decreasing, it follows that p ′′ ( x ) > x ∈ ]0 , x ∗ [ . In turn, thisyields ( r + λ )(1 − p ) + (cid:16) λ + rp + σ (cid:17) xp ′ + σ x p ′′ > . Recalling (2.18), we have( r + λ )(1 − p ) + H ξ ( x, ξ, p ) p ′ + σ x p ′′ > ξ ≥ . (6.5)Next, differentiating both sides of the first ODE in (6.2), we obtain (cid:18) r − σ − λ + rp + ( λ + r ) p ′ p x (cid:19) · e V ′ = (cid:18) λ + rp + σ (cid:19) x e V ′′ for all x ∈ ]0 , x ∗ [ . This implies e V ′′ ( x ) < x ∈ ]0 , x ∗ [ . Recalling (2.17) and (6.2), we obtain − r e V + H ( x, e V ′ , p ) + σ x e V ′′ < . (6.6)When x ≥ λ + r , the map p H ( x, ξ, p ) is monotone decreasing. Defining V ( x ) . = e V ( r + λ ) for x ∈ h , r + λ i , e V ( x ) for x ∈ h r + λ , x ∗ i , we thus have − rV ( x ) + H ( x, V ′ ( x ) , q ) + σ x V ′′ ( x ) ≤ q ≥ p ( x ) . (6.7) We now construct the functions V , p . Defining˜ p ( x ) . = 1 x (cid:18) θ ( x ∗ ) x ∗ + 2 r − µ (cid:19) , a straightforward computation yields˜ p ′ ( x ) = − ˜ p ( x ) x < , ˜ p ′′ ( x ) = 2 · ˜ p ( x ) x . Set x . = θ ( x ∗ ) x ∗ + 2 r − µ , (6.8)and consider the continuous function p ( x ) = min (cid:8) , ˜ p ( x ) (cid:9) . (6.9)24or x ∈ [0 , x [ one has p ( x ) = 1 and hence( r + λ )(1 − p ) + H ξ ( x, ξ, p ) p ′ + σ x p ′′ = 0 . On the other hand, for x ∈ ] x , x ∗ [ and ξ ≥
0, one has p ( x ) = ˜ p ( x ) <
1, and H ξ ( x, ξ, p ) ≥ ( λ + r ) x − p + ( σ − λ − µ ) x ≥ ( r − µ ) x − r − µ ) θ ( x ∗ ) x ∗ + 1 ≥ . (6.10)Recalling (2.18), we get( r + λ )(1 − p ) + H ξ ( x, ξ, p ) p ′ + σ x p ′′ ≤ ( r + λ )(1 − p ) + h ( λ + r ) x − p + ( σ − λ − µ ) x i p ′ ( x ) + σ x p ′′ = ( r + λ )(1 − p ) − h ( λ + r ) x − p + ( σ − λ − µ ) x i · p ( x ) x + σ p = ( r + λ )(1 − p ) − h ( λ + r ) − x + ( σ − λ − µ ) p i + σ p = 1 x − ( r − µ ) p = − ( r − µ ) θ ( x ∗ ) x ∗ x − x < . In particular,( r + λ )(1 − p ) + H ξ ( x, ξ, p ) · p ′ + σ x p ′′ ≤ x ∈ ]0 , x ∗ [ , ξ ≥ . (6.11)Next, define V ( x ) . = (1 − p ( x )) B for all x ∈ [0 , x ∗ ] . (6.12)For all x ∈ [0 , x ], we thus have V ( x ) = 0, and hence − rV + H ( x, V ′ , q ) + σ x V ′′ = H ( x, , q ) = 0 for all q ∈ ]0 , . (6.13)On the other hand, for x ∈ ] x , x ∗ ] we have V ′ ( x ) = B · p ( x ) x > V ′′ ( x ) = − B · p ( x ) x . Recalling (2.17), (6.9), (6.10) and (6.8), we estimate − rV + H ( x, V ′ , p ) + σ x V ′′ ≥ − rV + (cid:16) ( λ + r ) x − p + ( σ − λ − µ ) x (cid:17) V ′ + σ x V ′′ = B · h rp − r + (cid:16) λ + r − x + ( σ − λ − µ ) p ( x ) (cid:17) − σ p i = B · (cid:16) λ − x − ( λ + µ − r ) p (cid:17) = B · h λ (1 − p ( x )) + ( r − µ ) p ( x ) − x i > x ∈ ] x , x ∗ [.Recalling (6.8), one has ( λ + r ) x > x ∈ ] x , x ∗ ] . Therefore the map p → H ( x, V ′ ( x ) , p ) is monotone decreasing on [0 , x ∈ ] x , x ∗ ].This implies − rV + H ( x, V ′ , q ) + σ x V ′′ ≥ x ∈ ] x , x ∗ ] , q ∈ ]0 , p ( x )] . Together with (6.13), we finally obtain − rV ( x ) + H ( x, V ′ ( x ) , q ) + σ x V ′′ ( x ) ≥ x ∈ ]0 , x ∗ [ , q ∈ ]0 , p ( x )] . (6.14)Relying on (6.5), (6.6), (6.11) and (6.14), and using the same comparison argument as in theproof of Theorem 3.1 we now prove Theorem 6.1.
In addition to (A1), assume that σ > and θ ( x ∗ ) > . Then the system (2.15)with boundary conditions (2.16) admits a solution ( V ( · , x ∗ ) , p ( · , x ∗ )) satisfying the bounds (6.1)for all x ∈ [0 , x ∗ ] .Proof. Recalling D in (3.6), we claim that the domain D = n ( V, p ) ∈ D (cid:12)(cid:12)(cid:12) ( V ( x ) , p ( x )) ∈ [ V ( x ) , V ( x )] × [ p ( x ) , p ( x )] , for all x ∈ [0 , x ∗ ] o (6.15)is positively invariant for the semigroup { S t } t ≥ , generated by the parabolic system (3.3)-(3.4). Namely: S t ( D ) ⊆ D for all t ≥ . Indeed, from the proof of Theorem 3.1, we have p x ( t, x ) ≤ ≤ V x ( t, x ) for all t > , x ∈ ]0 , x ∗ [ . (6.16)We now observe that(i) For any V ( · , · ) with V x ≥
0, by (6.5) the function p ( t, x ) = p ( x ) is a subsolution ofthe second equation in (3.2). Similarly, by (6.11), the function p ( t, x ) = p ( x ) is asupersolution.(ii) For any p ( · , · ) with p ∈ [0 ,
1] and p x ≤
0, by (6.7) the function V ( t, x ) = V ( x )is a supersolution of the first equation in (3.2). Similarly, by (6.14), the function V ( t, x ) = V ( x ) is a subsolution.Together, (i)-(ii) imply the positive invariance of the domain D . Using the same argument as in step of the proof of Theorem 3.1, we conclude that thesystem (2.15)-(2.16) admits a solution ( V, P ) ∈ D .26 orollary 6.2. Let the assumptions in Theorem 6.1 hold. If lim sup s → + ∞ θ ( s ) s = + ∞ , then, for all x ≥ , the value function V ( · , x ∗ ) satisfies lim x ∗ →∞ V ( x, x ∗ ) = 0 . (6.17) Proof.
Using (4.3), (6.4) and Theorem 6.1, we have the estimate V ( x, x ∗ ) ≤ V ( x ) = B · (cid:18) − p ( x )1 − θ ( x ∗ ) (cid:19) rr + λ ≤ B · (cid:16) xθ ( x ∗ ) x ∗ (cid:17) rr + λ + σ for all x ≥ r + λ . This implies that (6.17) holds for all x ≥ r + λ . Since x V ( x, x ∗ ) is monotoneincreasing, we then have0 ≤ lim x ∗ →∞ V ( x, x ∗ ) ≤ lim x ∗ →∞ V (cid:18) r + λ , x ∗ (cid:19) = 0 for all x ∈ h , r + λ i . This completes the proof of (6.17).
Corollary 6.3.
Let the assumptions in Theorem 6.1 hold. If C . = lim sup s → + ∞ θ ( s ) s < + ∞ , then lim inf x ∗ →∞ V ( x, x ∗ ) ≥ B · (cid:18) − C x (cid:19) for all x > M , (6.18) where the constants C , M are defined as C . = C + 2 r − µ and M . = λ + µ − rλ C + 2 λ + µ − rλ ( r − µ ) + 1 . Proof.
This follows from (6.9), (6.12) and Theorem 6.1.
If we allow x ∗ = + ∞ , then the equations (4.8) admit the trivial solution V ( x ) = 0, p ( x ) = 1,for all x ≥
0. This corresponds to a Ponzi scheme, producing a debt whose size growsexponentially, without bounds. In practice, this is not realistic because there is a maximumamount of liquidity that the market can supply. It is interesting to understand what happenswhen this bankruptcy threshold x ∗ is very large.Our analysis has shown that three cases can arise, depending on the fraction θ of outstandingcapital that lenders can recover.(i) If lim s → + ∞ θ ( s ) s = + ∞ , then it is convenient to choose x ∗ as large as possible. By delayingthe time of bankruptcy, the expected cost for the borrower, exponentially discounted intime, approaches zero. 27iii) If lim s → + ∞ θ ( s ) s < + ∞ and (5.16) holds then it is still convenient to choose x ∗ as large aspossible. However,by delaying the time of bankruptcy, the cost to the borrower remainsuniformly positive.(iii) If lim s → + ∞ θ ( s ) s < + ∞ and (5.18) holds, then for every initial value x of the debt thereis a choice x ∗ ( x ) of the bankruptcy threshold which is optimal for the borrower.Examples corresponding to three cases (i)–(iii) are obtained by taking θ ( s ) = min (cid:26) , R s α (cid:27) (7.1)with 0 < α < α = 1, or α >
1, respectively.It is important to observe that the choice of the optimal bankruptcy threshold x ∗ is never“time consistent”. Indeed, at the beginning of the game the borrower announces that he willdeclare bankruptcy when the debt reaches size x ∗ . Based on this information, the lendersdetermine the discounted price of bonds. However, when the time T b comes when x ( T b ) = x ∗ ,it is never convenient for the borrower to declare bankruptcy. It is the creditors, or an externalauthority, that must enforce termination of the game.To see this, assume that at time T b when x ( T b ) = x ∗ the borrower announces that he haschanged his mind, and will declare bankruptcy only at the later time T ′ b when the debt reaches x ( T ′ b ) = 2 x ∗ . If he chooses a control u ( t ) = 0 for t > T b , his discounted cost will be e − ( T ′ b − T b ) r B < B.
This new strategy is thus always convenient for the borrower. On the other hand, it can bemuch worse for the lenders. Indeed, consider an investor having a unit amount of outstandingcapital at time T b . If bankruptcy is declared at time T b , he will recover the amount θ ( x ∗ ).However, if bankruptcy is declared at the later time T ′ b , his discounted payoff will be Z T ′ b T b ( r + λ ) e − ( r + λ )( t − T b ) dt + e − ( r + λ )( T ′ b − T b ) θ (2 x ∗ ) . To appreciate the difference, consider the deterministic case, assuming that θ ( · ) is the functionin (7.1), with α ≥
1, and that x ∗ ≥ M . By the analysis at the beginning of Section 5, we have u ∗ ( x ) = 0 for all x ∈ [ x ∗ , x ∗ ]. Replacing x ∗ with 2 x ∗ in (5.4) we obtain that the solution to(5.3) with terminal data p (2 x ∗ ) = θ (2 x ∗ ) = R (2 x ∗ ) α satisfies p B ( x ∗ , x ∗ ) = 2 θ (2 x ∗ ) · (cid:18) − p B ( x ∗ , x ∗ )1 − θ (2 x ∗ ) (cid:19) r − µr + λ < θ (2 x ∗ ) = 2 − α θ ( x ∗ ) ≤ θ ( x ∗ ) . If the investors had known in advance that bankruptcy is declared at x = 2 x ∗ (rather than at x = x ∗ ), the bonds would have fetched a smaller price.In conclusion, if the bankruptcy threshold x ∗ is chosen by the debtor, the only Nash equilibriumcan be x ∗ = + ∞ . In this case, the model still allows bankruptcy to occur, when total debtapproaches infinity in finite time. 28 eferences [1] M. Aguiar and G. Gopinath, Defaultable debt, interest rates and the current account. J. International Economics (2006), 64–83.[2] H. Amann, Invariant sets and existence theorems for semilinear parabolic equation andelliptic system, J. Math. Anal. Appl. , 432–467, 1978.[3] C. Arellano and A. Ramanarayanan, Default and the maturity structure in sovereignbonds. J. Political Economy (2102), 187–232.[4] M. Bardi and I. Capuzzo Dolcetta,
Optimal Control and Viscosity Solutions ofHamilton-Jacobi-Bellman Equations , Birkh¨auser, 1997.[5] T. Basar and G. J. Olsder,
Dynamic Noncooperative Game Theory , 2 d Edition, Aca-demic Press, London 1995.[6] A. Bressan, Noncooperative differential games.
Milan J. Math. (2011), 357–427.[7] A. Bressan and Y. Jiang, Optimal open-loop strategies in a debt management problem,submitted.[8] A. Bressan and Khai T. Nguyen, An equilibrium model of debt and bankruptcy, ESAIM; Control, Optim. Calc. Var. , to appear.[9] A. Bressan and B. Piccoli,
Introduction to the Mathematical Theory of Control , AIMSSeries in Applied Mathematics, Springfield Mo. 2007.[10] M. Burke and K. Prasad, An evolutionary model of debt.
J. Monetary Economics (2002) 1407-1438.[11] G. Calvo, Servicing the Public Debt: The Role of Expectations. American EconomicReview (1988), 647–661.[12] J. Eaton and M. Gersovitz, Debt with potential repudiation: Theoretical and empiricalanalysis. Rev. Economic Studies (1981), 289–309.[13] G. Nu˜no and C. Thomas, Monetary policy and sovereign debt vulnerability, Workingdocument n. 1517, Banco de Espa˜na Publications, 2015.[14] B. Oksendahl, Stochastic Differential Equations: An Introduction with Applications .Springer-Verlag, 2013.[15] S. E. Shreve,