A Stochastic Model of Optimal Debt Management and Bankruptcy

Transcription

1 A Stochastic Model of Otimal Debt Management and Bankrutcy Alberto Bressan (, Antonio Marigonda (, Khai T. Nguyen (, and Michele Palladino ( (* Deartment of Mathematics, Penn State University University Park, PA 1680, USA. (** Diartimento di Informatica, Università di Verona, Italy (*** Deartment of Mathematics, North Carolina State University, USA Raleigh, NC 7695, USA. s: bressan@math.su.edu, antonio.marigonda@univr.it, khai@math.ncsu.edu, mu6@su.edu May, 017 Abstract A roblem of otimal debt management is modeled as a noncooerative interaction between a borrower and a ool of lenders, in infinite time horizon with exonential discount. The yearly income of the borrower is governed by a stochastic rocess. When the debt-to-income ratio x(t reaches a given size x, bankrutcy instantly occurs. The interest rate charged by the risk-neutral lenders is recisely determined in order to comensate for this ossible loss of their investment. For a given bankrutcy threshold x, existence and roerties of otimal feedback strategies for the borrower are studied, in a stochastic framework as well as in a limit deterministic setting. The aer also analyzes how the exected total cost to the borrower changes, deending on different values of x. 1 Introduction We consider a roblem of otimal debt management in infinite time horizon, modeled as a noncooerative interaction between a borrower and a ool of risk-neutral lenders. Since the debtor may go bankrut, lenders charge a higher interest rate to offset the ossible loss of art of their investment. 1

2 In the models studied in [8, 9], the borrower has a fixed income, but large values of the debt determine a bankrutcy risk. Namely, if at a given time t the debt-to-income ratio x(t is too big, there is a ositive robability that anic sreads among investors and bankrutcy occurs within a short time interval [t, t + ε]. This event is similar to a bank run. Calling T b the random bankrutcy time, this means { } Prob T b [t, t + ε] T b > t = ρ(x(t ε + o(ε. Here the instantaneous bankrutcy risk ρ( is a given, nondecreasing function. At all times t, the borrower must allocate a ortion u(t [0, 1] of his income to service the debt, i.e., aying back the rincial together with the running interest. Our analysis will be mainly focused on the existence and roerties of an otimal reayment strategy u = u (x in feedback form. In the alternative model roosed by Nuño and Thomas in [16], the yearly income Y (t is modeled as a stochastic rocess: dy (t = µy (t dt + σy (t dw. (1.1 Here µ 0 is an exonential growth rate, while W denotes Brownian motion on a filtered robability sace. Differently from [8, 9], in [16] it is the borrower himself that chooses when to declare bankrutcy. This decision will be taken when the debt-to-income ratio reaches a certain threshold x, beyond which the burden of servicing the debt becomes worse than the cost of bankrutcy. At the time T b when bankrutcy occurs, we assume that the borrower ays a fixed rice B, while lenders recover a fraction θ(x(t b [0, 1] of their outstanding caital. Here x θ(x is a nonincreasing function of the debt size. For examle, the borrower may hold an amount R 0 of collateral (gold reserves, real estate... which will be roortionally divided among creditors if bankrutcy occurs. In this case, when bankrutcy occurs each investor will receive a fraction { } R0 θ(x(t b = max x(t b, 1 (1. of his outstanding caital. Aim of the resent aer is to rovide a detailed mathematical analysis of some models closely related to [16]. We stress that these roblems are very different from a standard roblem of otimal control. Indeed, the interest rate charged by lenders is not given a riori. Rather, it is determined by the exected evolution of the debt at all future times. Hence it deends globally on the entire feedback control u(. An otimal solution for the borrower must be understood as the best trade-off between the sustainability of his debt, related to the interest rate charged by the lenders, and the need to kee the reayment rate as small as ossible. A recise descrition of our model is given in Section. Here the strategy of the borrower comrises a feedback control u = u(x, determining the fraction of income allocated to servicing the debt, and a stoing set S IR +, where bankrutcy is declared. In a way, this

3 resembles the classical roblem of stochastic control with stoing time, as in [7]. We remark that, in a naive formulation, the otimization roblem always admits the trivial solution u(x 0, S =. (1.3 This corresonds to a Ponzi scheme: no ayment is ever made, bankrutcy is never declared, and the interest on old loans is ayed by initiating more ad more new loans. This strategy guarantees zero cost, and is clearly otimal. To rule out the trivial solution and achieve a more realistic model, we assume that some uer bound x for the debt is given, beyond which bankrutcy must instantly occur. For examle, one can think of x as the maximum amount of cash that all financial markets can rovide. It can be very large, but certainly finite. In this modified setting, the otimization roblem is formulated for x [0, x ], and the stoing set S [0, x ] must contain the oint x. The main results of the aer can be summarized as follows. Given an uer bound x for the debt, we show that the otimal choice of the stoing set is S = {x }. In other words, it is never convenient for the borrower to declare bankrutcy, unless he is forced to do it. We then seek an otimal feedback control u = u (x, x [0, x ] which minimizes the exected cost to the borrower. For any value σ 0 of the diffusion coefficient in (1.1, we rove that the roblem admits at least one solution, in feedback form. In the deterministic case where σ = 0, the solution can be constructed by concatenating solutions of a system of two ODEs, with terminal data given at x = x. We then study how the exected total cost of servicing the debt together with the bankrutcy cost (exonentially discounted in time deends on the uer bound x. Let θ(x [0, 1] be the salvage rate, i.e., the fraction of outstanding caital that will be ayed back to lenders if bankrutcy occurs when the debt-to-income ratio is x. If lim s + θ(s s = +, (1.4 then, letting x +, the total exected cost to the borrower goes to zero. On the other hand, if θ(s s < +, (1.5 lim s + then the total exected cost to the borrower remains uniformly ositive as x +. Our analysis shows that, if the debtor can access a large amount x of credit, when (1.4 holds he can ostone the bankrutcy time far into the future. Due to the exonential discount, as x + his exected cost will thus aroach zero. On the other hand, when (1.5 holds, after the debt has reached a certain threshold, bankrutcy must occur within a fixed time regardless of the size of x. We remark that the assumtion (1.5 is more realistic. For examle, if (1. holds, then θ(x x = R 0 for all x large enough. The remainder of the aer is organized as follows. In Section we describe more carefully the model, deriving the equations satisfied by the value function V and the discounted bond 3

4 rice. In Sections 3 and 4 we construct equilibrium solutions in feedback form, in the stochastic case (σ > 0 and in the deterministic case (σ = 0, resectively. Finally, Sections 5 and 6 contain an analysis of how the exected cost to the borrower changes, deending on the bankrutcy threshold x. In the economics literature, some related models of debt and bankrutcy can be found in [1, 3, 8, 1, 13]. For the basic theory of otimal control and viscosity solutions of Hamilton- Jacobi equations we refer to [4, 10]. A model with stochastic growth We consider a slight variant of the model in [16]. We denote by X(t the total debt of a borrower (a government, or a rivate comany at time t. The annual income Y (t of the borrower is assumed to be a random rocess, 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 ayments with intensity (r + λe λt. Here r is the interest rate ayed on bonds, which we assume coincides with the discount rate, λ is the rate at which the borrower ays back the rincial. If no bankrutcy occurs, the ayoff for an investor will thus be 0 e rt (r + λe λt dt = 1. In case of bankrutcy, a lender recovers only a fraction θ [0, 1] of his outstanding caital. Here θ can deend on the total amount of debt at the time on bankrutcy. To offset this ossible loss, the investor buys a bond with unit nominal value at a discounted rice [0, 1]. As in [9, 16], at any time t the value (t is uniquely determined by the cometition of a ool of risk-neutral lenders. We call U(t the rate of ayments that the borrower chooses to make to his creditors, at time t. If this amount is not enough to cover the running interest and ay back art of the rincial, new bonds are issued, at the discounted rice (t. As in [16], the nominal value of the outstanding debt thus evolves according to Ẋ(t = λx(t + (λ + rx(t U(t. (.1 (t For a more detailed derivation of equation (.1 from the economic rimitives, we refer the reader to [16]. 4

5 The debt-to-income ratio is defined as x = X/Y. In view of (1.1 and (.1, Ito s formula [17, 18] yields the stochastic evolution equation [( λ + r dx(t = (t λ + σ µ x(t u(t ] dt σ x(t dw. (. (t Here u = U/Y is the fraction of the total income allocated to ay for the debt. Throughout the following we assume r > µ. In this model, the borrower has two controls. At each time t he can decide the ortion u(t of the total income which he allocates to reay the debt. Moreover, he can decide at what time T b bankrutcy is declared. Throughout the following, we assume that an uer bound x for the debt is a riori given (as an external constraint, imosed by the size of the markets, and consider strategies in feedback form. These comrise: (i a closed set S [0, x ], with x S, where bankrutcy is declared, and (ii a feedback control determining the reayment rate u = u (x [0, 1] for x [0, x ] \ S. (.3 For a given choice of the stoing set S, the bankrutcy time is thus the random variable T b. = inf { t 0 ; x(t S }. (.4 Given an initial size x 0 of the debt, the goal of the borrower is to minimize the total exected cost, exonentially discounted in time. Namely, minimize: J ( x 0, u, S [ Tb. = E e rt L ( u (x(t ] dt + e rt b B. (.5 0 x(0=x 0 Here B is a large constant, accounting for the bankrutcy cost, while L(u is the instantaneous cost to the borrower for imlementing the control u. To comlete the model, we need an equation determining the discounted bond rice in the evolution equation (.. For every x > 0, let θ(x be the salvage rate, i.e. the fraction of the outstanding caital that can be recovered by lenders, if bankrutcy occurs when the debt has size x. Given an initial debt size is x 0, the exected ayoff to a lender urchasing a couon with unit nominal value is comuted by the right hand side of [ T b ] (x 0 = E (r + λe (r+λt dt + e (r+λt b θ(x(t b. (.6 x(0=x 0 0 Assuming that the bond rice is determined by the cometition of a large ool of risk-neutral lenders, this exected ayoff should coincide with the discounted bond rice (x 0. This motivates (.6 5

6 Notice that the stoing time T b in (.4, and hence (x 0, deends on the initial state x 0, on the stoing set S, and on all values of the feedback control u (. Since the salvage rate θ( is nonincreasing, we have (x = θ(x if x S, (x [θ(x, 1] for all x [0, x ]. (.7 Having described the model, we can introduce the definition of otimal solution, in feedback form. Definition 1 (stochastic otimal feedback solution. In connection with the above model, we say that a set S [0, x ] and a air of functions u = u (x, = (x rovides an otimal solution to the roblem of otimal debt management (. (.6 if (i Given the function (, for every initial value x 0 [0, x ] the feedback control u ( with stoing time T b as in (.4 rovides an otimal solution to the stochastic control roblem (.5, with dynamics (.. (ii Given the feedback control u ( and the set S, for every initial value x 0 the discounted rice (x 0 satisfies (.6, where T b is the stoing time (.4 determined by the dynamics (.. We emhasize that, in our model, if x(t = x then bankrutcy must instantly occur. The following simle observation shows that, for the borrower, it is never convenient to voluntarily declare bankrutcy at any earlier time. Lemma.1. Let S, u (, ( be an otimal solution to the debt management roblem (. (.6. Then S = {x }. Proof. Assume that, on the contrary, there is a value x 0 < x such that x 0 S. We show that condition (i in the above definition cannot hold. Indeed, consider the otimization roblem with initial datum x(0 = x 0. If x 0 S, then bankrutcy instantly occurs at time T b = 0, and the exected cost in (.5 is J[x 0, u, S] = B. However, the alternative strategy u(t 0, with bankrutcy occurring at the first time where x(t = x, rovides the strictly smaller exected cost J ( x 0, 0, {x } [ ] = E e rt b B < B. Motivated by Lemma.1, from now on we shall always take S = {x } as the stoing set. The random stoing time is thus T b. = inf { t 0 ; x(t = x }. (.8 Concerning the cost function L in (.5, we shall assume 6

7 (A The function L is twice continuously differentiable for u [0, 1[ and satisfies L(0 = 0, L > 0, L > 0, lim L(u = +. (.9 u 1 For examle, for some c, α > 0, one may take L(u = c ln 1 1 u, or L(u = cu (1 u. α For a given function = (x, we denote by V ( the value function for the stochastic otimal control roblem (.5 with dynamics (.. Namely, V (x 0. = inf u( J ( x 0, u, {x }. (.10 Denote by H(x, ξ,. = min ω [0,1] {L(ω ξ } ( λ + r ω + λ + σ µ x ξ (.11 the Hamiltonian associated to the dynamics (. and the cost function L in (.5. Notice that, as long as > 0, the function H is differentiable with Lischitz continuous derivatives w.r.t. all arguments. By standard arguments, the value function V rovides a solution to the second order ODE with boundary conditions rv (x = H ( x, V (x, (x + (σx V (x, (.1 V (0 = 0, V (x = B. (.13 As soon as the function V is determined, the otimal feedback control is recovered by { u (x = argmin L(ω V } (x ω [0,1] (x ω. By (A this yields u (x = 0 if ( V (L 1 (x (x if V (x (x V (x (x L (0, > L (0. (.14 On the other hand, if the feedback control u = u (x is known, then by the Feynman-Kac formula ( satisfies the equation [( ] λ + r (r + λ((x 1 = (x λ + σ µ x u (x (x + (σx (x, (.15 (x 7

8 with boundary values (0 = 1, (x = θ(x. (.16 Combining (.1 and (.15, we are thus led to the system of second order ODEs rv (x = H ( x, V (x, (x + (σx V (x, (r + λ((x 1 = H ξ ( x, V (x, (x (x + (σx with the boundary conditions { V (0 = 0, V (x = B, { (0 = 1, (x = θ(x. (x, (.17 (.18 In the next section, an otimal feedback solution to the roblem (. (.6 will be obtained by solving the above system of ODEs for the value function V ( and for the discounted bond rice (. We close this section by collecting some useful roerties of the Hamiltonian function. Lemma.. Let the assumtions (A hold. Then, for all ξ 0 and 0 < 1, the function H in (.11 satisfies ( ( (λ + rx 1 λ + r + (σ λ µx ξ H(x, ξ, λ + σ µ xξ, (.19 (λ + rx 1 + (σ λ µx H ξ (x, ξ, ( λ + r λ + σ µ x. (.0 Moreover, for every x, > 0 the ma ξ H(x, ξ, is concave down and satisfies Proof. H(x, 0, = 0, (.1 ( λ + r H ξ (x, 0, = λ + σ µ x, (. ( 1 λ + r, if > λ + σ µ x, ( (.3 λ + r lim H(x, ξ, = ξ + +, if 1 λ + σ µ x. 1. Since H(x,, is defined as the infimum of a family of affine functions, it is concave down. We observe that (.11 imlies ( λ + r H(x, ξ, = λ + σ µ xξ if 0 ξ L (0. (.4 This yields the identities (.1-(.. 8

9 . Taking ω = 0 in (.11 we obtain the uer bound in (.19. By the concavity roerty, the ma ξ H ξ (x, ξ, is non-increasing. Hence (. yields the uer bound in ( Since L(w 0 for all w [0, 1], we have H(x, ξ, min { ξ } ( λ + r w + w [0,1] λ + σ µ x ξ and obtain the lower bound in (.19. On the other hand, using the otimality condition, one comutes from (.11 that where H ξ (x, ξ, = (λ + rx u (ξ, u (ξ, = argmin ω [0,1] {L(ω ξ ω } + (σ λ µx (.5 = (L 1 ( ξ < 1. Observe that, as ξ +, one has u (ξ, 1 in (.5. The non-increasing roerty of the ma ξ H ξ (x, ξ, yields the lower bound in ( To rove (.3 we observe that, in the first case, there exists ω 0 < 1 such that ( ω 0 λ + r > λ + σ µ x. Hence, letting ξ + we obtain lim H(x, ξ, lim ξ + ξ + [ L(ω 0 ω 0 ξ + ( λ + r ] λ + σ µ x ξ =. To handle the second case, we observe that, for ξ > 0 large, the minimum in (.11 is attained at the unique oint ω(ξ where L (ω(ξ = ξ/. Hence ω(ξ = 1 and lim H(x, ξ, = lim ξ + lim ξ + [ L(ω(ξ ω(ξ ( ] λ + r ξ + ξ + λ + σ µ x ξ lim L(ω(ξ = +. ξ + 9

10 Remark.3. We summarize here the main differences between the roosed model and the model resented in [16]. In [16], the borrower is a government which can control the rimary surlus ratio, the inflation rate and the time of declaring bankrutcy. The control on the inflation rate can be used by the government as a monetary olicy to temorarily deflate the actual debt value, by aying a rice in terms of welfare cost. While controlling the rimary surlus ratio is actually equivalent in our model to the choice of u(, and in both models the borrower can choose the bankrutcy time, in our model the borrower cannot choose the inflation rate r. This simlification can be motivated assuming either that the borrower is not a government, or that the monetary olicy of the government has been delegated to an indeendent central banker which acts in order to kee it constant (e.g. % in Eurozone, no matter of the consequences on the borrower s debt sustainability. In [16], the instantaneous references of the borrower are exressed by a (discounted utility function of logarithmic tye, while our analysis deals with more general cost functions L(. 3 Existence of solutions Let x > 0 be given. If a solution (V, to the boundary value roblem (.17-(.18 is found, then the feedback control u = u (x defined at (.14 and the function = (x will rovide an equilibrium solution to the debt management roblem, as in Definition 1. To construct a solution to the system (.17-(.18, we consider the auxiliary arabolic system V t (t, x = rv (t, x + H ( x, V x (t, x, (t, x + (σx V xx (t, x, t (t, x = (r + λ(1 (t, x + H ξ ( x, Vx (t, x, (t, x x (t, x + (σx with boundary conditions V (t, 0 = 0, V (t, x = B, (t, 0 = 1, (t, x = θ(x. for all t 0. xx (t, x, (3.1 Following [], the main idea is to construct a comact, convex set of functions (V, : [0, x ] [0, B] [θ(x, 1] which is ositively invariant for the arabolic evolution roblem. A toological technique will then yield the existence of a steady state, i.e. a solution to (.17-(.18. Theorem 3.1. In addition to (A, assume that σ > 0 and θ(x > 0. Then the system of second order ODEs (.17 with boundary conditions (.18 admits a C solution (V,, such that V : [0, x ] [0, B] is increasing and : [0, x ] [θ(x, 1] is decreasing. Proof. 10

11 1. For any ε > 0, consider the arabolic system V t = rv + H(x, V x, + (ε + (σx V xx, t = (r+λ(1 +H ξ (x, V x, x + (ε + (σx xx, V (0 = 0, V (x = B, (0 = 1, (x = θ(x. (3. (3.3 obtained from (3.1 by adding the terms εv xx, ε xx on the right hand sides. For any ε > 0, this renders the system uniformly arabolic, also in a neighborhood of x = 0.. Recalling [, Theorem 1], for every initial data V 0, 0 C ([0, x ], the system (3.-(3.3 with initial data V (0, x = V 0 (x, (0, x = 0 (x. (3.4 admits a unique solution V (t, x, (t, x in C ([0, T ] [0, x ] for all T > 0. Adoting a semigrou notation, let t (V (t,, (t, = S t (V 0, 0 be the solution of the system (3.-(3.3 with initial data (3.4. Consider the closed, convex set of functions in C ([0, x ] { } D = (V, : [0, x ] [0, B] [θ(x, 1] ; V, C, V x 0, x 0, and (.18 holds. (3.5 We claim that the above domain is ositively invariant under the semigrou S, namely S t (D D for all t 0. (3.6 Indeed, consider the constant functions V + (t, x = B, V (t, x = 0, + (t, x = 1, (t, x = θ(x. Recalling (.1, one easily checks that V + is a suersolution and V is a subsolution of the scalar arabolic roblem (3.. Indeed rv + + H(x, V x +, + (ε + (σx V xx + 0, V + (t, 0 0, V + (t, x B. rv + H(x, V x, + (ε + (σx V xx 0, V (t, 0 0, V (t, x B. A standard comarison rincile (see for examle Theorem 9.1 in [15] yields 0 = V (t, x V (t, x V + (t, x = B for all (t, x [0, T ] [0, x ]. 11

12 Similarly, since + is a suersolution and is a subsolution of the scalar arabolic roblem (3.3, one has that θ(x = (t, x (t, x + (t, x = 1 for all (t, x [0, T ] [0, x ]. This roves that, if the initial data V 0, 0 in (3.4 take values in the box [0, B] [θ(x, 1], then for every t 0 the solution of the system (3.-(3.3 will satisfy 0 V (t, x B, θ(x (t, x 1, (3.7 for all x [0, x ]. In turn, this imlies V x (t, 0 0, V x (t, x 0, x (t, 0 0, x (t, x 0. ( Next, we rove that the monotonicity roerties of V (t, and (t, are reserved in time. Differentiating w.r.t. x one obtains V xt = rv x + H x + H ξ V xx + H x + σ xv xx + (ε + (σx V xxx, (3.9 ( d xt = (r + λ x + dx H ξ(x, V x, x + H ξ xx + σ x xx + (ε + (σx xxx. (3.10 By (.1, for every x, one has H x (x, 0, = H (x, 0, = 0. Hence V x 0 is a subsolution of (3.9 and x 0 is a suersolution of (3.10. In view of (3.8, we obtain x (t, x 0 V x (t, x for all t 0, x [0, x ]. This concludes the roof that the set D in (3.5 is ositively invariant for the system (3.-( Thanks to the bounds (.19-(.0, we can now aly Theorem 3 in [] and obtain the existence of a steady state (V ε, ε D for the system (3.-(3.3. We recall the main argument in []. For every T > 0 the ma (V 0, 0 S T (V 0, 0 is a comact transformation of the closed convex domain D into itself in C (R. By Schauder s theorem it has a fixed oint. This yields a eriodic solution of the arabolic system (3.-(3.3, with eriod T. Letting T 0, one obtains a steady state. 1

13 5. It now remains to derive a riori estimates on this stationary solution, which will allow to take the limit as ε 0. Consider any solution to rv + H(x, V, + (ε + (σx V = 0, (3.11 (r + λ(1 + H ξ (x, V, + (ε + (σx = 0, with V increasing, decreasing, and satisfying the boundary conditions (.18. By the roerties of H derived in Lemma., we can find δ > 0 small enough and ξ 0 > 0 such that the following imlication holds: x [0, δ], [θ(x, 1], ξ ξ 0 = H(x, ξ, 0. As a consequence, if V (x > ξ 0 for some x [0, δ], then the first equation in (3.11 imlies V (x 0. We conclude that either V (x ξ 0 for all x [0, δ], or else V attains its maximum on the subinterval [δ, x ]. By the intermediate value theorem, there exists a oint ˆx [δ, x ] where V (ˆx = V (x V (δ x δ B x δ. (3.1 By (.19, the derivative V satisfies a differential inequality of the form V c 1 V + c, x [δ, x ]. (3.13 for suitable constants c 1, c. By Gronwall s lemma, from the differential inequality (3.13 and the estimate (3.1 one obtains a uniform bound on V (x, for all x [δ, ˆx] [ˆx, x ]. Relying on the first equation of (3.11, we also obtain an uniform bound on V (x, for all x [δ, x ]. 6. Similar arguments aly to. By (.0, the term H ξ (x, V, in (3.11 is uniformly bounded. For every δ > 0, by (3.11 shows that satisfies a linear ODE whose coefficients remain bounded on [δ, x ], uniformly w.r.t. ε. This yields the bound (x C δ for all x [δ, x ] (3.14 for some constant C δ, uniformly valid as ε 0. Relying on the second equation of (3.11, we also obtain an uniform bound on (x, for all x [δ, x ] To make sure that, as ε 0, the limit satisfies the boundary value (0 = 1. one needs to rovide a lower bound on also in a neighborhood of x = 0, indeendent of ε. Introduce the constant { γ =. ( } 1 λ + r min 1, (r + λ θ(x λ + σ µ. Then define (x. = 1 cx γ, 13

14 choosing c > 0 so that (x = θ(x. We claim that the convex function is a lower solution of the second equation in (3.11. Indeed, by (3.11, one has [ ( ] λ + r (r + λcx γ H ξ (x, V, cγx γ 1 (r + γ θ(x λ + σ µ γ cx γ Letting ε 0, we now consider a sequence (V ε, ε of solutions to (3.11 with boundary conditions (.18. Thanks to the revious estimates, the functions (V ε and ( ε are uniformly bounded by some constant C 1,δ > 0 on [δ, x ], and ε satisty (x ε (x 1 for all x [0, x ], ε > 0. On the other hand, since H and H ξ are uniformly bounded and uniformly Lischitz on [δ, x ] [ C 1,δ, C 1,δ ] [θ(x, 1], the functions (V ε = ε + σ x [rv ε H(x, (V ε, ε ] and ( ε = ε + σ x [(r + λ (ε 1 H ξ (x, (V ε, ε ( ε ] are also uniformly bounded and uniformly Lischitz on [δ, x ]. By choosing a suitable subsequence, we achieve the uniform convergence (V ε, ε (V,, where V, are twice continuously differentiable on the oen interval ]0, x [, and satisfy the boundary conditions (.18. Having constructed a solution (V, to the boundary value roblem (.17-(.18, a standard result in the theory of stochastic otimization imlies that the feedback control u ( in (.14 is otimal for the roblem (.5 with dynamics (.. For a roof of this verification theorem, see Theorem 4.1,.149 in [14] or Theorem 11..,.41 in [17]. As a consequence of Theorem 3.1 we thus obtain Corollary 3.. Under the same assumtions as in Theorem 3.1, the debt management roblem (. (.6 admits an otimal solution. 4 The deterministic case If σ = 0, then the stochastic equation (. reduces to the deterministic control system ( λ + r ẋ = λ µ x u. (4.1 Throughout the aer, we always assume r > µ. The deterministic Debt Management Problem can be formulated as follows. 14

15 (DMP Given an initial value x(0 = x 0 [0, x ] of the debt, minimize Tb 0 e rt L(u(t dt + e rt b B, (4. subject to the dynamics (4.1, where the bankrutcy time T b is defined as in (.4, while (t = Tb t (r+λe (r+λs 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 otimal feedback control u and the corresonding functions V, may not be smooth, a concet of equilibrium solution should be more carefully defined. Definition (deterministic otimal feedback solution. A coule of iecewise Lischitz continuous functions u = u (x and = (x rovide an equilibrium solution to the debt management roblem (DMP, with continuous value function V, if (i For every x 0 [0, x ], V is the minimum cost for the otimal control roblem subject to ẋ(t = minimize: Tb 0 e rt L(u(x(t dt + e rt b B, (4.4 ( λ + r (x(t λ µ x(t u(t (x(t, x(0 = x 0. (4.5 Moreover, every Carathéodory solution of (4.5 with u(t = u (x(t is otimal. (ii For every x 0 [0, x ], there exists at least one solution t x(t of the Cauchy roblem ( λ + r ẋ = (x λ µ x u (x (x, x(0 = x 0, (4.6 such that (x 0 = Tb with T b as in (.4. 0 (r + λe (r+λt dt + e ( r+λt b θ(x = 1 (1 θ(x e (r+λt b, (4.7 In the deterministic case, (.17 takes the form rv (x = H ( x, V (x, (x, (r + λ((x 1 ( = H ξ x, V (x, (x (x, (4.8 with Hamiltonian function (see Figure 1 H(x, ξ, = min {L(ω ξ } ( λ + r ω + ω [0,1] (λ + µ x ξ. (4.9 15

16 We consider solutions to (4.8 with the boundary condition V (0 = 0, (0 = 1, V (x = B, (x = θ(x. (4.10 Let s introduce two functions H max (x,. = su ξ 0 H(x, ξ, and ξ (x,. = argmax H(x, ξ,. ξ 0 Recalling (.5, we have H ξ (x, ξ, = (λ + rx u (x, (λ + µx (4.11 where Two cases may occur: u (ξ, = argmin {L(w ξ } w w [0,1] = (L 1 ( ξ < 1. (4.1 If (λ + rx (λ + µx 1 then the function ξ H(x, ξ, is monotone increasing and H max (x, = lim H(x, ξ, = +. (4.13 ξ In this case, we will define ξ (x,. = +. If (λ + rx (λ + µx < 1, we define u (x, = (λ + rx (λ + µx. (4.14 From (4.11 and (4.1, we have ξ (x, = L (u (x, = L ((λ + rx (λ + µx (4.15 and it yields H max (x, = H ( x, ξ (x,, ( = L (λ + rx (λ + µx. Notice that u is the control that kees the debt x constant in time. achieves the minimum in (4.9 when L ((λ + rx (λ + µx = ξ. This value u 16

17 H max (x, rv O F ξ F + ξ Figure 1: In the case where (λ + rx (λ + µx < 1, the Hamiltonian function ξ H(x, ξ, has a global maximum H max (x,. For rv H max, the values F (x, V, ξ (x, F + (x, V, are well defined. Observe that H ξξ (x, ξ, 0, H ξ (x, ξ, > 0 for all 0 ξ < ξ (x,, H ξ (x, ξ, < 0 for all ξ > ξ (x,. (4.16 We regard the first equation in (4.8 as an imlicit ODE for the function V. For every x 0 and [0, 1], if rv (x > H max (x,, then this equation has no solution. On the other hand, when 0 rv (x H max (x,, the imlicit ODE (4.8 can equivalently be written as a differential inclusion : { } V (x F (x, V,, F + (x, V,. (4.17 where F ±(x,v, are denoted by F (x, V, ξ (x, F + (x, V, and H(x, F ± (x, V,, = rv. Remark 4.1. Recalling (4.1, we observe that The value V = F + (x, V, ξ (x, corresonds to the choice of an otimal control such that ẋ < 0. The value V = F (x, V, ξ (x, corresonds to the choice of an otimal control such that ẋ > 0. When rv = H max (x,, then the value V = F + (x, V, = F (x, V, = ξ (x, corresonds to the unique control such that ẋ = 0. Since ξ H(x, ξ, is concave down, the functions F ± satisfy the following monotonicity roerties (Fig. 1 17

18 (MP For any fixed x,, the ma V F + (x, V, is decreasing, while V F (x, V, is increasing. For V = F, the second ODE in (4.8 can be written as (x = G ( x, V (x, (x, where G (x, V,. = (r + λ( 1 H ξ ( x, F (x, V,, 0. ( Construction of a solution. Consider the function W (x. = 1 r L( (r µx, (4.19 with the understanding that W (x = + if (r µx 1. Notice that W (x is the total cost of keeing the debt constantly equal to x (in which case there would be no bankrutcy and hence 1. Moreover, denote by (V B (, B ( the solution to the system of ODEs V (x = F (x, V (x, (x, (x = G (x, V (x, (x, (4.0 with terminal conditions V (x = B, (x = θ(x. (4.1 Notice that the ODE (4.0 admits a unique local solution around every oint (x 0, 0 with V (x 0, 0 = η 0 rovided that H ξ (x 0, F (x 0, η 0, 0, 0 0, i.e., F (x 0, η 0, 0 < ξ (x 0, 0 or, equivalently, rη 0 < H max (x 0, 0. On the other hand, if V B (x < W (x then H ξ (x, V B (x, B(x > 0. Assume by contradiction that Then we have H ξ (x, V B(x, B (x = 0. V B (x = 1 r Hmax (x, B (x 1 r Hmax (x, 1 = W (x and it yields a contradiction. Thus, (V B, B is uniquely defined on [x 1, x ] where the oint }. x 1 = inf {x [0, x ] ; V B (x < W (x. (4. Call V 1 ( the solution to the backward Cauchy roblem V (x = F (x, V (x, 1, x [0, x 1 ], V (x 1 = W (x 1, (4.3 18

19 B W V B V 1 0 x 1 1 x * r µ Figure : Constructing the equilibrium solution in feedback form. For an initial value of the debt x(0 x 1, the debt increases until it reaches x 1, then it is held at the constant value x 1. If the initial debt is x(0 > x 1, the debt kees increasing until it reaches bankrutcy in finite time. we will show that a feedback equilibrium solution to the debt management roblem is obtained as follows (see Figure. V 1 (x if x [0, x 1 ], V (x = (4.4 V B (x if x [x 1, x ]. 1, if x [0, x 1 ], (x = (4.5 B (x, if x ]x 1, x ]. { argmin L(ω (V } (x u ω [0,1] (x ω, if x x 1, (x = (4.6 (r µx 1, if x = x 1. Theorem 4.. Assume that the cost function L satisfies the assumtions (A, and moreover L((r µx > rb. Then the functions V,, u in (4.4 (4.6 are well defined, and rovide an equilibrium solution to the debt management roblem, in feedback form. Proof. 1. The solution of (4.0-(4.1 satisfies the obvious bounds V 0, 0, V (x B, (x [θ(x, 1]. We begin by roving that the function V B ]x 1, x ]. is well defined and strictly ositive for x 19

20 To rove that V B (x > 0 for all x ]x 1, x ], assume, on the contrary, that V B (y = 0 for some y > x 1 0. From (4.15, it holds ξ (x, C 1 > 0 for all x [y, x ], [θ(x, 1] for some ositive constant C 1. Recalling (4.16, we obtain that H ξ (x, ξ, C for all x [y, x ], [θ(x, 1], ξ [0, C 1 ] for some ositive constant C. Since H(x, 0, = 0, the mean value theorem yields H(x, F (x, V,, C F (x, V, for all x [y, x ], [θ(x, 1] rovided by F (x, V, C 1. The definition of F imlies that there exists a constant δ 1 > 0 small such that F (x, V, r C V, (4.7 for all x [y, x ], [θ(x, 1] and V [0, δ 1 ]. Hence, for any solution of (4.0, V (y = 0 imlies V (x = 0 for all x y, roviding a contradiction. Next, observe that the functions F, G are locally Lischitz continuous as long as 0 V < H max (x,. Moreover, V (x < W (x imlies V (x < W (x = H max (x, 1 H max (x, (x. Therefore, the functions V B, B are well defined on the interval [x 1, x ].. If x 1 = 0 the construction of the functions V,, u is already comleted in ste 1. In the case where x 1 > 0, we claim that the function V 1 is well defined and satisfies 0 < V 1 (x < W (x for 0 < x < x 1. (4.8 Indeed, if V 1 (y = 0 for some y > 0, the Lischitz roerty (4.7 again imlies that V 1 (x = 0 for all x y. This contradicts the terminal condition in (4.3. To comlete the roof of our claim (4.8, it suffices to show that This is true because W (x < F (x, W (x, 1 for all x ]0, x 1 ]. (4.9 W (x = r µ r L ( r µx = r µ r ξ (x, 1 < ξ (x, 1 = F ( x, H max (x, 1, 1 = F (x, W (x, 1. 0

21 3. In the remaining stes, we show that V,, u rovide an equilibrium solution. Namely, they satisfy the roerties (i-(ii in Definition. To rove (i, call V ( the value function for the otimal control roblem (4.4-(4.5. For any initial value, x(0 = x 0, in both cases x 0 [0, x 1 ] and x 0 ]x 1, x ], the feedback control u in (4.6 yields the cost V (x 0. This imlies V (x 0 V (x 0. To rove the converse inequality we need to show that, for any measurable control u : [0, + [ [0, 1], calling t x(t the solution to ( λ + r ẋ = x1 (x λ µ x u(t x1 (x, x(0 = x 0, (4.30 one has where Tb 0 e rt L(u(tdt + e rt b B V (x 0, (4.31 T b = inf { t 0 ; x(t = x } is the bankrutcy time (ossibly with T b = +. For t [0, T b ], consider the absolutely continuous function φ u (t. = t 0 e rs L(u(sds + e rt V (x(t. At any Lebesgue oint t of u(, recalling that (V, solves the system (4.8, we comute d [ ] dt φu (t = e rt L(u(t rv (x(t + (V (x(t ẋ(t (( λ + r = e [L(u(t rt rv (x(t + (V (x(t (x(t λ µ x(t u(t ] (x(t [ { e rt min L(ω (V } (x(t ω + ω [0,1] (x(t ] = e [H rt (x(t, (V (x(t, (x(t rv (x(t Therefore, ( λ + r (x(t λ µ = 0. ] x(t(v (x(t rv (x(t roving (4.31. V (x 0 = φ u (0 lim t T b φu (t = Tb 0 e rt L(u(tdt + e rt b B, 1

22 4. It remains to check (ii. The case x 0 = 0 is trivial. Two main cases will be considered. CASE 1: x 0 ]0, x 1 ]. Then there exists a solution t x(t of (4.6 such that (t = 1 and x(t ]0, x 1 ] for all t > 0. Moreover, In this case, T b = + and (4.7 holds. lim x(t = x 1. t + CASE : x 0 ]x 1, x ]. Then x(t > x 1 for all t [0, T b ]. This imlies From the second equation in (4.8 it follows ẋ(t = H ξ (x(t, V B (x(t, B (x(t. d dt (t = (x(tẋ(t = (r + λ((t 1, d ln(1 (x(t = (r + λ. dt Therefore, for every t [0, T b ] one has Letting t T b we obtain roving (4.7. (x(0 = 1 (1 (x(t e (r+λt. (x 0 = 1 (1 θ(x e (r+λt b, Remark 4.3. In general, however, we cannot rule out the ossibility that a second solution exists. Indeed, if the solution V B, B of (4.0-(4.1 can be rolonged backwards to the entire interval [0, x ], then we could relace (4.4-(4.5 simly by V (x = V B (x, (x = 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 grahs of W and V B cannot have any other intersection, in addition to 0 and x 1. Indeed, assume on the contrary that W (x = V B (x for some 0 < x < x 1 (see Figure 3. Since B (x < 1 and W (x V B (x, the inequalities rw (x = H(x, W (x, 1 < H(x, W (x, B (x H(x, V B(x, B (x = rv B (x yield a contradiction. Next, let V, be any equilibrium solution and define Then x. = su { x [0, x ] ; (x = 1 }.

23 B W V B 0 x 1 r µ x 1 x * Figure 3: By the monotonicity roerties of the Hamiltonian function H in (4.9, the grahs of V B and W cannot have a second intersection at a oint x > 0. On ]x, x ] the functions V, rovide a solution to the backward Cauchy roblem (4.0-(4.1. On ]0, x ] the function V rovides the value function for the otimal control roblem minimize: subject to the dynamics (with 1 0 e rt L(u(t dt ẋ = (r µx u, and the state constraint x(t [0, x ] for all t 0. The above imlies V (x = V B (x, if x [x, x ], V (x W (x, if x [0, x ]. Since V must be continuous at the oint x, by the revious analysis this is ossible only if x = 0 or x = x 1. 5 Deendence on the bankrutcy threshold x. In this section we study the behavior of the value function V B when the maximum size x of the debt, at which bankrutcy is declared, becomes very large. From a modeling oint of view, this amount to discuss the ossibility of the otimality of a Ponzi scheme, in which the debt is serviced by initiating more and more new loans. We 3

24 will show that under some natural assumtions on the function θ( exressing the fraction recover by lenders as a function of the debt-to-income ratio at the moment of bankrutcy. For a given x > 0, we denote by V B (, x, B (, x the solution to the system (4.0 with terminal data (4.1. Letting x, we wish to understand whether the value function V B remains ositive, or aroaches zero uniformly on bounded sets. Toward this goal, we introduce the constant M 1. = r µ max Recalling Lemma. for σ = 0, we have H(x, ξ, (r µx 1 Thus, the first equation of (4.8 imlies that rb rv B (x, x In turn, if x > M 1, this yields (r µx 1 B (x, x { 1, rb L (0 }. (5.1 ξ for all x [0, x ], ξ 0. V B(x, x x [0, x ]. V B (x, x B (x, x L (0, for all x [M 1, x ]. Calling u = u (x the otimal feedback control, by (.14 we have u (x = 0, for all x [ M 1, x ]. (5. In this case, the Hamiltonian function takes a simler form, namely H(x, V, = [ (λ + r (λ + µ ] V x, H ξ (x, V, = [ (λ + r (λ + µ ] x. Therefore, the system of ODEs (4.0 can be written as V r = [(λ + r (λ + µ]x V, ( 1 = (λ + r [(λ + r (λ + µ] x. (5.3 The second ODE of in (5.3 is equivalent to d ( (1 (x r µ dx ln (x r+λ = r + λ x. Solving backward the above ODE with the terminal data (x = θ(x, we obtain B (x, x = θ(x x x ( 1 B (x, x r µ r+λ 1 θ(x 4 for all x [ M 1, x ]. (5.4

25 Therefore, B (x, x ( θ(x x x ( θ(x x 1 + x r+λ r µ r+λ r µ for all x [ M 1, x ]. (5.5 Different cases will be considered, deending on the roerties of the function θ(. By obvious modeling considerations, we shall always assume θ(x [0, 1], θ (x x for all x 0. We first study the case where θ has comact suort. Recall that M 1 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 1. Then, for any x > M, the solution V B (, x, B (, x of (4.0-(4.1 satisfies V B (x, x = B and B (x, x = 0 for all x [ M, x ]. Proof. By (5.4 and (5.6, for every x > M one has B (x, x = 0 for all x [ M, x ]. 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 [ M, x ]. This means that bankrutcy instantly occurs if the debt reaches M. Next, we now consider that case where θ(x > 0 for all x. Lemma 5.. If x > M 1 and θ(x > 0, then θ(x > 0 for all x [0, [. (5.7 ( V B (x, x B (x, x x = B θ(x x r r µ for all x [ M 1, x ]. (5.8 In articular, for x [ M 1, x ] one has ( ( θ(x x r+λ r r+λ r µ B 1 + x ( V B (x, x x r B θ(x x r µ. (5.9 5

26 Proof. Since B (x, x solves the second equation of (5.3 and B (x, x = θ(x (0, 1, we have that x B (x, x is a strictly decreasing function of x. For a fixed value of x, let χ( : [θ(x, 1[ [0, x ] be the inverse function of B (, x. From (5.3, a direct comutation yields d d V B(χ(, x r = [(λ + r (λ + µ] χ( V B(χ(, x χ (, From (5.10 it follows d d B(χ(, x ( 1 = (λ + r [(λ + r (λ + µ] χ( χ ( = 1. d d ln V B(χ(, x = r λ + r 1 1. (5.10 Solving the above ODE with the terminal data V B (x, x = B, B (x, x = θ(x, we obtain V B (χ(, x = ( r 1 r+λ B, ( θ(x hence V B (x, x = ( 1 B (x, x r r+λ B. 1 θ(x Recalling (5.4, a direct comutation yields (5.8. The uer and lower bounds for V B (x, x in (5.9 now follow from (5.5 and the inequality B (x, x 1. Corollary 5.3. Assume that lim su x + θ(x x = +. (5.1 Then the value function V = V (x, x satisfies lim V (x, x + x = 0 for all x 0. (5.13 Indeed, for x M 1 we have V (x, x = V B (x, x, and (5.13 follows from the second inequality in (5.9. When x < M 1, since the ma x V (x, x is nondecreasing, we have 0 lim x V (x, x lim V (M 1, x = 0. x Corollary 5.4. Assume that R =. lim su x + θ(x x < +. (5.14 Then V B (x, x B Moreover, the followings holds. ( 1 + ( R x r+λ r r+λ r µ for all x > x > M 1. (5.15 6

27 (i If then θ (x θ(x + 1 x 0 and θ (x 0 for all x > 0 (5.16 inf V B(x, x = x >0 lim V B(x, x > 0 for all x M 1. (5.17 x (ii Assume that there exist 0 < δ < 1 such that δ θ (x θ(x + 1 x < 0 (5.18 for all x sufficiently large. Then, for each x > M 1, there exists an otimal value x = x (x such that V B (x, x (x = inf V B(x, x. (5.19 x 0 Proof. It is clear that (5.15 is a consequence of (5.9 and (5.14. We only need to rove (i and (ii. For a fixed x M 1, we consider the functions of the variable x alone: Y (x. = V B (x, x, q(x. = B (x, x. Using (5.8 and (5.4, we obtain Y (x Y (x = ( r q r µ (x [ θ q(x (x θ(x + 1 ], (5.0 x and This imlies q (x q(x [ q (x θ q(x (x θ(x + 1 ] x = θ (x x + θ(x + r µ ( q θ(x x r + λ (x 1 q(x + θ (x. (5.1 1 θ(x = [ r µ r+λ q(x 1 q(x + 1 ] [ θ (x θ(x + 1 ] x r µ ( (r + λ 1 + r µ r+λ q(x 1 q(x θ (x 1 θ(x. (5. If (5.16 holds, then (5.0 and (5. imly Y (x Y (x = q (x q(x [ θ (x θ(x + 1 x ] 0 for all x > x M 1. Hence the function Y is non-increasing. This roves (5.17. To rove (ii, we observe that ( 1 lim su x 1 + r µ r+λ q(x 1 q(x 1 < 0, lim x θ(x = 0. 7

28 Hence (5.18 and (5. imly [ q (x θ q(x (x θ(x 1 ] x > 0, for all x sufficiently large. By (5.0 this yields Y (x Y (x > 0 for all x large enough. Hence there exists some articular value x (x x where the function x Y (x = V B (x, x attains its global minimum. This yields ( Deendence on x in the stochastic case In this section we study how the value function deends on the bankrutcy threshold x, in the stochastic case where σ > 0. Extensions of Corollaries 5.3 and 5.4, will be roved, constructing uer and lower bounds for a solution V (, x, (, x of the system (.17-(.18, in the form where V (x V (x, x V 1 (x, 1 (x (x, x (x, (6.1 (i for any V (, with V x 0, the functions 1 ( and ( are a subsolution and a suersolution of the second equation in (3.1, resectively. (ii for any (, with [0, 1] and x 0, the function V 1 ( and V ( are a suersolution and a subsolution of the first equation in (3.1, resectively. 1. We begin by constructing a suitable air of functions V 1, 1. Let ( 1, Ṽ1 be the solution to the backward Cauchy roblem ( λ + r rṽ1(x = + σ xṽ 1, 1 Ṽ 1 (x = B, (r + λ( 1 1 = This solution satisfies ( λ + r 1 + σ x 1, 1 (x = θ(x. (6. 1 (x = θ(x x x ( σ +λ+r 1 1 (x λ+r, lim 1 θ(x 1(x = 1, (6.3 x 0+ 8

29 Ṽ 1 (x = B ( r 1 1 (x r+λ, 1 θ(x Using (6. and (6.3 one obtains ( x 1 = 1(x 1 (x + σ + r + λ x r + λ 1 1 (x = 1(x x 1 (x + σ + r + λ r + λ 1 θ(x (θ(x x r+λ r+λ+σ lim Ṽ 1 (x = 0. (6.4 x 0+ x σ r+λ+σ 1 (x λ+r λ+r+σ Since 1 is monotone decreasing, it follows that 1(x > 0 for all x ]0, x [. In turn, this yields ( λ + r (r + λ( σ x 1 + σ x 1 1 > 0. Recalling (.0, we have (r + λ(1 1 + H ξ (x, ξ, σ x. 1 > 0 for all ξ 0. (6.5 Next, differentiating both sides of the first ODE in (6., we obtain ( r σ λ + r + (λ + ( r 1 λ + r x Ṽ 1 = + σ xṽ 1 for all x ]0, x [. 1 1 This imlies 1 Recalling (.19 and (6., we obtain Ṽ 1 (x < 0 for all x ]0, x [. rṽ1 + H(x, Ṽ 1, 1 + σ x Ṽ 1 < 0. (6.6 When x 1, the ma H(x, ξ, is monotone decreasing. Defining λ+r [ ] Ṽ ( 1. for x 1 0,, r+λ r+λ V 1 (x = [ ] 1 Ṽ (x for x, r+λ x, we thus have rv 1 (x + H(x, V 1(x, q + σ x V 1 (x 0 for all q 1 (x. (6.7 9

30 . We now construct the functions V,. Defining (x a straightforward comutation yields Set (x = (x x and consider the continuous function. = 1 x For x [0, x [ one has (x = 1 and hence ( θ(x x + r µ, < 0, (x = (x x. x. = θ(x x + r µ, (6.8 (x = min { 1, (x }. (6.9 (r + λ(1 + H ξ (x, ξ, + σ x = 0. On the other hand, for x ]x, x [ and ξ 0, one has (x = (x < 1, and H ξ (x, ξ, (λ + rx 1 (x + (σ λ µx (r µx 1 (x [ ] 1 (r µ θ(x x + r µ = x r µ x > 0. (6.10 Recalling (.0, we get (r + λ(1 + H ξ (x, ξ, + σ x [ (λ + rx 1 (r + λ(1 + [ (λ + rx 1 = (r + λ(1 + (σ λ µx ] (x + σ x ] + (σ λ µx (x + σ x = (r + λ(1 [(λ + r 1 ] x + (σ λ µ + σ In articular, = 1 x (r µ = (r µθ(x x x 1 x < 0. (r + λ(1 + H ξ (x, ξ, + σ x 0 for all x ]0, x [, ξ 0. (

31 Next, define V (x For all x [0, x ], we thus have V (x = 0, and hence. = (1 (x B for all x [0, x ]. (6.1 rv + H(x, V, q + σ x V = H(x, 0, q = 0 for all q ]0, 1]. (6.13 On the other hand, for x ]x, x ] we have V (x = B (x x Recalling (.19, (6.9, (6.10 and (6.8, we estimate > 0 and V (x = B (x x. rv + H(x, V, + σ x ( (λ + rx 1 V rv + + (σ λ µx V + σ x V = B [ ( r r + λ + r 1 ] x + (σ λ µ (x σ = B for all x ]x, x [. Recalling (6.8, one has ( λ 1 x (λ + µ r = B [ λ(1 (x + (r µ (x 1 ] x > 0 (λ + rx > 1 for all x ]x, x ]. Therefore the ma H(x, V (x, is monotone decreasing on [0, 1], for all x ]x, x ]. This imlies rv + H(x, V, q + σ x V 0 for all x ]x, x ], q ]0, (x]. Together with (6.13, we finally obtain rv (x + H(x, V (x, q + σ x V (x 0 for all x ]0, x [, q ]0, (x]. (6.14 Relying on (6.5, (6.6, (6.11 and (6.14, and using the same comarison argument as in the roof of Theorem 3.1 we now rove Theorem 6.1. In addition to (A1, assume that σ > 0 and θ(x > 0. Then the system (.17 with boundary conditions (.18 admits a solution (V (, x, (, x satisfying the bounds (6.1 for all x [0, x ]. Proof. 31

32 1. Recalling D in (3.5, we claim that the domain { } D 0 = (V, D (V (x, (x [V (x, V 1 (x] [ 1 (x, (x], for all x [0, x ] (6.15 is ositively invariant for the semigrou {S t } t 0, generated by the arabolic system (3.- (3.3. Namely: S t (D 0 D 0 for all t 0. Indeed, from the roof of Theorem 3.1, we have We now observe that x (t, x 0 V x (t, x for all t > 0, x ]0, x [. (6.16 (i For any V (, with V x 0, by (6.5 the function (t, x = 1 (x is a subsolution of the second equation in (3.1. Similarly, by (6.11, the function (t, x = (x is a suersolution. (ii For any (, with [0, 1] and x 0, by (6.7 the function V (t, x = V 1 (x is a suersolution of the first equation in (3.1. Similarly, by (6.14, the function V (t, x = V (x is a subsolution. Together, (i-(ii imly the ositive invariance of the domain D 0.. Using the same argument as in ste 4 of the roof of Theorem 3.1, we conclude that the system (.17-(.18 admits a solution (V, P D 0. Corollary 6.. Let the assumtions in Theorem 6.1 hold. If lim su s + θ(s s = +, then, for all x 0, the value function V (, x satisfies lim V (x, x x = 0. (6.17 Proof. Using (4.3, (6.4 and Theorem 6.1, we have the estimate ( r 1 V (x, x 1 (x r+λ ( V 1 (x = B B 1 θ(x for all x 1 r+λ increasing, we then have x θ(x x r r+λ+σ. This imlies that (6.17 holds for all x 1 r+λ. Since x V (x, x is monotone 0 lim x V (x, x lim V x This comletes the roof of (6.17. ( 1 r + λ, x 3 = 0 for all x [ 0, 1 ]. r + λ

33 Corollary 6.3. Let the assumtions in Theorem 6.1 hold. If C 1. = lim su s + θ(s s < +, then ( lim inf V (x, x x B 1 C x where the constants C, M are defined as for all x > M, (6.18 C. = C1 + r µ. λ + µ r and M = C 1 + λ + µ r + 1. λ λ(r µ Proof. This follows from (6.9, (6.1 and Theorem Concluding remarks If the uer bound for the debt size (beyond which bankrutcy instantly occurs is allowed to be x = +, then the equations (4.8 admit the trivial solution V (x = 0, (x = 1, for all x 0. This corresonds to a Ponzi scheme, roducing a debt whose size grows exonentially, without bounds. In ractice, this is not realistic because there is a maximum amount of liquidity that the market can suly. It is interesting to understand what haens when this bankrutcy threshold x is very large. Our analysis has shown that three cases can arise, deending on the fraction θ of outstanding caital that lenders can recover. (i If lim θ(s s = +, then for the borrower it is convenient to have s + x as large as ossible. Indeed, the exected total cost for servicing the debt aroaches zero as x +. (iii If lim θ(s s < + and (5.16 holds then for the borrower it is still convenient to have s + x as large as ossible. However, as x +, the exected total cost for servicing the debt remains uniformly ositive. (iii If lim θ(s s < + and (5.18 holds, then for every initial value x 0 of the debt there s + is a value x (x 0 of the bankrutcy threshold which is otimal for the borrower. Examles corresonding to three cases (i (iii are obtained by taking { θ(s = min 1, R } 0 s α (7.1 with 0 < α < 1, α = 1, or α > 1, resectively. 33

34 In case (iii we observe that, even if the bankrutcy threshold x were not imosed by the external market but could be selected by the borrower in an otimal way, this choice of x could never time consistent. Indeed, assume that at the initial time t = 0 the borrower announces that he will declare bankrutcy when the debt reaches size x. Based on this information, the lenders determine the discounted rice of bonds. However, when the time T b comes when x(t b = x, it is never convenient for the borrower to declare bankrutcy. It is the creditors, or an external authority, that must actually enforce bankrutcy. To see this, assume that at time T b when x(t b = x the borrower announces that he has changed his mind, and will declare bankrutcy only at the later time T b when the debt reaches x(t b = x. If he chooses a control u(t = 0 for t > T b, his discounted cost will be e (T b T br B < B. This new strategy is thus always convenient for the borrower. On the other hand, it can be much worse for the lenders. Indeed, consider an investor having a unit amount of outstanding caital at time T b. If bankrutcy is declared at time T b, he will recover the amount θ(x. However, if bankrutcy is declared at the later time T b, his discounted ayoff will be T b (r + λe (r+λ(t Tb dt + e (r+λ(t b Tb θ(x. T b To areciate the difference, consider the deterministic case, assuming that θ( is the function in (7.1, with α 1, and that x M 1. By the analysis at the beginning of Section 5, we have u (x = 0 for all x [x, x ]. Relacing x with x in (5.4 we obtain that the solution to (5.3 with terminal data satisfies ( 1 B (x, x = θ(x B (x, x 1 θ(x (x = θ(x = R 0 (x α r µ r+λ < θ(x = 1 α θ(x θ(x. If the investors had known in advance that bankrutcy is declared at x = x (rather than at x = x, the bonds would have fetched a smaller rice. In conclusion, if the bankrutcy threshold x is chosen by the debtor, the only equilibrium can be x = +. In this case, the model still allows bankrutcy to occur, when total debt aroaches infinity in finite time. Acknowledgments. The authors would also like to thank the anonymous referees, whose suggestions and comments heled to imrove many sections of the aer. References [1] M. Aguiar and G. Goinath, Defaultable debt, interest rates and the current account. J. International Economics 69 (006,