The vanishing viscosity limit for a system of H-J equations related to a debt management problem

The paper studies a system of Hamilton-Jacobi equations, arising from a model of optimal debt management in infinite time horizon, with exponential discount and a bankruptcy risk. For a stochastic model with positive diffusion, the existence of an equilibrium solution is obtained by a topological argument. Of particular interest is the limit of these viscous solutions, as the diffusion parameter approaches zero. Under suitable assumptions, this (possibly discontinuous) limit can be interpreted as an equilibrium solution to a non-cooperative differential game with deterministic dynamics.


1.
Introduction. This paper is concerned with a Hamilton-Jacobi equation with discontinuous coefficients (r + ρ(x))V = H(x, p(x), V ), (1.1) together with a family of smooth viscous approximations (r + ρ(x))V n = H(x, p n (x), V n ) + σ 2 n x 2 2 V n . (1. 2) The main motivation comes from a problem of optimal debt management, in infinite time horizon, with exponential discount and in the presence of a bankruptcy risk. As in [8,9,10,16], this is modeled as a noncooperative game between a borrower and a pool of risk-neutral lenders. Here the independent variable x is the debt-toincome ratio, V is the value function for the borrower (i.e., his expected cost, under optimal play) while the function ρ(x) ≥ 0 accounts for an instantaneous bankruptcy risk, which increases with the size of the debt. A distinguished feature of this model is that the discounted bond price p(·) is is not given a priori, but depends on the entire future evolution of the system, determined by the feedback control u = u(x). Indeed, if the borrower chooses to keep the debt at a high level, then the lenders will buy the bonds at a deeply discounted price p(x) < 1, to compensate for the risk of losing their capital. The scalar equation (1.1) must thus be supplemented by an additional ODE for the function p. In particular, at points where the feedback control u(·) is discontinuous, also p(·) can have a jump.
In the first part of this paper we construct a solution to the second order equation (1.2).This can be interpreted as a Nash equilibrium solution to a differential game with stochastic dynamics. In the final sections, we study the limits V n → V and p n → p as the diffusion coefficient σ n → 0. Under suitable assumptions, we show 794 ALBERTO BRESSAN AND YILUN JIANG that this limit yields a solution to (1.1) and determines an equilibrium solution to a differential game with deterministic dynamics.
The main technical difficulties in the analysis stem from the fact that, while all functions p n are smooth, the limit p may well be discontinuous. Hamilton-Jacobi equations with discontinuous hamiltonian function have attracted much interest in recent years [4,5,6,7,13,14,15,17]. The key difference in our equations is that, since we are modeling a game, the hamiltonian function for the borrower is not given a priori, but is determined by the optimal reply adopted by the lenders. In turn, this reply depends on the feedback control chosen by the borrower.
The remainder of the paper is organized as follows. In Section 2 we explain how the equations (1.1) arise from a problem of optimal debt management, modeled as a game between a borrower and a pool of risk-neutral lenders.
In Section 3 we prove the existence of solutions to the second order boundary value problem, for σ > 0. In the case where the bankruptcy risk is this result was proved in [9]. Here we consider more general functions ρ(·), using similar ideas. The main new contribution of the present paper is the analysis of the limit σ → 0, studied in Section 4. Our main result shows the existence of a limit, for a suitable subsequence σ n → 0. Under a "generic" assumption on the limit functions V (·) and p(·), we prove that this limit yields a Nash equilibrium solution to a noncooperative game, where the total debt follows a deterministic evolution equation. The analysis relies on classical techniques from the theory of viscosity solutions [2,3,11,12], and a number of additional arguments to handle this specific problem. sk 2. Derivation of the model. The model includes the following constants: • r = discount rate, • λ = rate at which the principal is payed back, • B = bankruptcy cost to the borrower, and the variables • t = time, measured in years, • x(t) = total debt, measured as a fraction of the yearly income of the borrower, • u(t) ∈ [0, 1] = payment rate, as a fraction of the income, • p(t) ∈ [0, 1] = discounted bond price, • ρ(x) = instantaneous risk of bankruptcy, • T b = random time when bankruptcy occurs, • θ(x(T b )) ∈ [0, 1] = fraction of the outstanding capital which can be recovered by the lenders in case of bankruptcy. • L(u) = cost to the borrower for implementing the control u. We regard u as the control variable for the borrower.
Observe that, in this model, ρ is a function of x. If the total debt is large, so is the bankruptcy risk. Concerning the functions ρ, L, θ, we shall assume (A1) There exists a constant M , with rM > 1, such that the following holds. The The constant M in (A1) denotes a maximum size of the debt, beyond which bankruptcy immediately occurs. To motivate (A2), assume that the borrower owns an amount R 0 of collateral (real estate, gold reserves, etc. . .) to back up his debt. In case of bankruptcy, this will be divided among lenders. In this case the function θ will have the form In the following we define the first time when the debt reaches this maximum value. Calling T b the random time at which bankruptcy occurs, its distribution is determined as follows. If at time τ the borrower is not yet bankrupt and the total debt is x(τ ) = y, then the probability that bankruptcy will occur shortly after time τ is measured by where o(ε) denotes a higher order infinitesimal as ε → 0. The probability that the borrower is not yet bankrupt at time t > 0 is thus computed as Notice that this depends on the function τ → x(τ ). If debt is maintained at a higher level, then the probability of bankruptcy increases. This reflects lack of confidence in the ability of the borrower to eventually repay the debt. At any random time T b , panic may suddenly spread among the investors, leading to bankruptcy. The discounted bond price p = p(x) requires some explanation. If an investor purchases at time t = 0 a coupon with unit nominal value, he receives a stream of payments at all future times. The repayment rate is The primary capital is payed back at rate λ, so that the outstanding value of the loan at time t > 0 is e −λt . In addition, the borrower pays an interest r. If bankruptcy never occurs, then the total payoff for the lender (exponentially discounted in time) is Ψ = +∞ 0 e −rt (λ + r)e −λt dt = 1 . (2.7) However, if bankruptcy occurs at a random time T b , the payoff will be If θ = 1, so that the outstanding capital is recovered in full, then again Ψ = 1. In general, however, θ < 1. Assuming that lenders are risk-neutral, the coupon will thus be bought at the discounted price Here E denotes the expected value of the given quantity, depending on the random variable T b . We observe that the expected cost to the borrower can be computed by To complete the model, we still need an equation describing the evolution of the debt. Assume that the yearly income Y (t) of the borrower grows with rate µ and is subject to a stochastic perturbation: (2.11) Here W denotes standard Brownian motion. Calling U (t) the repayment rate, the nominal value of the outstanding debt evolves according to the ODĖ We define the debt-to-income ratio x . = X/Y , and set u . = U/Y . By Ito's formula, (2.11) and (2.12) yield the stochastic evolution equation (2.13) We are mainly interested in controls in feedback form: Motivated by the optimality conditions, it is convenient to define u (p, ξ) . = arg min

15)
Definition 2.1. Consider a value function V (·), a feedback control u * (·), and a price function p(·), defined for x ∈ [0, M [ . We say that the triple (V, p, u * ) yields a Nash equilibrium solution to the debt management problem if the following holds.
(i) Given the price p = p(x), one has that V (·) is the value function and u * (x) = u (p(x), V (x)) is the optimal feedback control, in connection with the stochastic control problem Here the distribution of the random time T b is determined by (2.5). (ii) Given the feedback control u = u * (x) in (2.17), for every (2.18) The theory of stochastic optimal control yields a second order ODE satisfied by the value function V . Using the Feynman-Kac formula [18] to derive an equation for p(.), we eventually obtain the system

(2.20)
This is supplemented by the boundary conditions (2.21) The optimal feedback control is then u where
2. Adopting a semigroup notation, let t → (V (t), p(t)) = S t (V 0 , p 0 ) be the solution of the parabolic system (3.16-3.17) with initial data Consider the closed, convex set of functions one easily checks that V + is a supersolution and V − is a subsolution of the first equation in (3.16). Namely, Similarly, p + is a supersolution and p − is a subsolution of the second equation in (3.16). For any initial data (V 0 , p 0 ) ∈ D, the solution of (3.16)-(3.17) will satisfy for all x ∈ [0, M ] and t ≥ 0. As a consequence, (3.21) By We thus conclude that V x = 0 is a subsolution of (3.21), proving the positive invariance of the domain D. 4. We observe that the functions H and H ξ appearing in the equation (3.14) depend Lipschitz continuously on all their arguments. Moreover, according to the bounds (3.8)-(3.10), they have sublinear growth w.r.t. the variable ξ = V x . One can thus apply Theorem 3 in [1] and obtain the existence of a steady state (V ε , p ε ) ∈ D for the parabolic problem system (3.16)- (3.17).
We remark that the analysis in [1] applies to parabolic systems in any space dimension. In this general case, by Schauder's estimates, the solution has C 2,α regularity. In our case, however, the domain [0, M ] is 1-dimensional. Solving (3.14) for V xx and p xx and using the fact that H, H ξ are Lipschitz continuous we conclude that these functions V ε , p ε are twice differentiable with Lipschitz continuous second derivatives.
We claim that p ε x (x) ≤ 0 for all x ∈ [0, M ]. Indeed, assume by contradiction that p ε x (x 0 ) > 0 at some point x 0 . Using the boundary condition (2.21) and the bounds (3.20), we can define the two points = max y; y < x 0 , p ε x (y) = 0 and p ε x (x) ≥ 0 for all x ∈ [y, x 0 ] > 0, = min y; y > x 0 , p ε x (y) = 0 and p ε x (x) ≥ 0 for all x ∈ [x 0 , y] < M. It is clear that . On the other hand, by definition we have reaching a contradiction. We thus conclude that p

5.
In the remainder of the proof, letting ε → 0 we shall obtain a limit (V, p), which provides a C 2 solution to the original boundary value problem (2.19)-(2.21). By setting the system satisfied by (V ε , p ε ) can be written as (3.23) By (3.12)-(3.13), we can find δ > 0 sufficiently small and ξ 0 > 0 such that Notice that, since u < 1 and r > µ, by (3.22) one has Roughly speaking, this means that if the debt x is large enough it will keep increasing.
Consider the first equation in (3.23). On the region where ζ ≥ 0 the first three terms on the right hand side are all nonnegative. Recalling that V ≥ 0 and 0 ≤ V ≤ B, we thus obtain The first equation in (3.23) yields an inequality of the form for suitable constants c 1 , c 2 independent of ε. By Gronwall's lemma, (3.27) and We thus conclude that V is uniformly bounded for all x ∈ [δ, M ] and thus is uniformly bounded for all A similar analysis applies to p . Since H ξ is uniformly bounded, and ρ ε (x) is uniformly bounded on x ∈ [δ, M − δ], for a given δ > 0 independent of ε, we obtain an inequality of the form for suitable constants a 1 , a 2 independent of ε.
Summarizing the previous analysis, we thus have the bounds for some constants C and C δ independent of ε.

802
ALBERTO BRESSAN AND YILUN JIANG 6. By (3.29) we can extract a subsequence ε n → 0 and achieve the convergence Since all solutions (V ε , p ε ) satisfy the boundary condtions (2.21), by uniform convergence it is clear that the same boundary conditions are satisfied by V .
To prove that also p(·) satisfies the required boundary conditions (2.21), we construct a lower solution p − and an upper solution p + of the second ODE equation in (3.23), independent of ε. Following the proof of Theorem 3.1 in [9], we have for some suitable constants c > 0, γ > 0, x 1 > 0. Next, to construct an upper solution p + , we choosex 2 such that (r−µ) −1 < x 2 < M , and take δ > 0 small enough such that M − δ > x 2 . Moreover, we introduce the constant where we used the lower bound (3.9) on H ξ . We thus conclude that p + (x) is a upper solution of the second equation in (3.23) on x ∈ [x 2 , M ]. Hence for all ε > 0. 7. By the previous analysis, the limit functions V, p in (3.30) satisfy all boundary conditions in (2.21). Observing that the convergence is uniform and the coefficient , we now show that (V, p) satisfy the system of second order ODEs (2.19). This is accomplished in three steps. By possibly extracting a further subsequence, we thus obtain the convergence It is now clear that (V, p) provides a classical solution to the system of equations (3.14). This completes the proof of Theorem 3.2. 4. The vanishing viscosity limit. We now consider a sequence of solutions (4.1) Our first result shows the existence of a limit, as σ → 0. Then there exists one pointx ∈ [0, M ] such that, by possibly extracting a subsequence, one has the uniform convergence V n → V on every domain of the form In addition, one has the pointwise convergence p n (x) → p(x) and V n (x) → V (x), for a.e. x ∈ [0, M ]. Also, the limit satisfies the H-J equation One may think ofx as a threshold for the debt size. When x <x, the optimal feedback strategy keeps the debt uniformly bounded in time. On the other hand, when x >x, the debt keeps increasing and reaches the value M in finite time, where bankruptcy instantly follows.
Our second result shows that, under suitable assumptions, the above limit (V, p) yields a Nash equilibrium solution to a problem with deterministic dynamics. Toward this goal, we first introduce an auxiliary function W : (4.3) Recalling (3.6), we can interpret W (y) as the expected cost achieved by keeping the debt at the constant value x(t) = y. This is well defined as long as the control needed to keep the debt constant satisfies On the other hand, ifū(y) ≥ 1, then W (y) = +∞. Since p(·) is monotone decreasing, the function is monotone increasing. Recalling that the instantaneous bankruptcy risk ρ(·) is a smooth function, we conclude that W has bounded variation on any subinterval [0,ȳ] where W < +∞. Downward jumps in p correspond to upward jumps in W .
We can now state the final result of this paper: In the same setting of Theorem 4.1, let W (·) be the function at Moreover assume that the following two conditions hold: (i) The set of points y where V (y) = W (y−) is finite.
(ii) If p has a jump at a pointȳ where V (ȳ) < W (ȳ−), and if the time derivativė has the same sign to the right and to the left ofȳ, then (2.14), the functions (V, p, u * ) provide an equilibrium solution to the debt management problem, according to 2.1.
Proof. A proof of Theorem 4.1 and Theorem 4.2 will be given in several steps.

-Existence of a pointwise limit.
Since each V n is increasing and each p n is decreasing, by Helly's compactness theorem we can choose a subsequence and achieve the pointwise convergence for every x ∈ [0, M ]. Clearly, V and p are monotone. The structure of this limit will be investigated in the following steps.

-Local Lipschitz continuity of V .
For every n ≥ 1, since p n (·) is non-increasing there exists a unique pointx n such that By possibly choosing a further subsequence, we can assume the convergencex n →x.
We claim that the limit function V is Lipschitz continuous on every interval of the form [0,x − ε] or [x + ε, M ]. This will be proved by showing that all derivatives V n are uniformly bounded on these intervals. To estimate V n on [x + ε, M ], let ζ n be as in (3.22), with V, p replaced by V n , p n . By construction, for any given ε > 0 there exists ζ † > 0 such that for all n ≥ 1 sufficiently large. We observe that, if the maximum of V n on [x + ε, M ] is attained at an interior point y, then rV n (y) ≥ (ζ n (y) + σ 2 n y)V n (y). (4.8) This yields the bound and we are done. If the maximum of V n is attained at x = M , then V n (M − ) ≥ 0. Hence (4.8) holds at y = M . In turn, this implies (4.9).
It remains to study the case where V n attains its maximum atx + ε. By the intermediate value theorem, there is a point y † ∈ [x + (ε/2),x + ε] such that and we are done. Or else V n attains a maximum at a point y ∈ ]y † , M ]. But in this case the bound (4.9) holds. Since the bounds (4.9), (4.11) are independent of n, they imply the Lipschitz continuity of V on the interval [x + ε, M ].
Observe that these values are uniformly bounded. Namely, there exists some constant C 0 independent of n such that ξ 0 n (x, p) ≤ C 0 . (4.14) If V n attains a local maximum at a point y ∈ [δ,x − (ε/2)] , then V n (y) = 0 and the first equation in (3.23) implies H(y, p n (y), V n (y), σ n ) = (ζ n (y) + σ 2 n y)V n (y) + L(u * n (y)) + ρ(y)B ≥ 0. (4.15) Comparing (4.15) with (4.13) and using the uniform bound (4.14), we conclude that V n (y) ≤ C 0 . (4.16) Thanks to the uniform bounds on the derivatives V n in (4.16) and (3.24), letting n → ∞ we obtain the Lipschitz continuity of the limit function V on the interval For every ε > 0, we thus conclude that on [0,x − ε] ∪ [x + ε, M ] the sequence V n converges uniformly to a Lipschitz continuous limit function V . As a consequence, V is differentiable at a.e. point x ∈ [0, M ] and satisfies the boundary conditions in (2.21).

-An upper bound on V .
We claim that, for every y ∈ [0, M ] \ {x}, one has V (y) ≤ W (y). (4.17) Intuitively, this should be obvious: V describes a minimum cost, while W is a cost achieved by one particular strategy that keeps the debt constant. For y >x, recalling (4.7) we see that W (y) = +∞ and there is nothing to prove.

-Behavior on the region where V < W .
To help the reader, we first give an overview of the forthcoming analysis. By (4.7), for x >x the debt is so large that it cannot be reduced. The dynamics is monotone increasing, and reaches the maximum value M in finite time. At that time, bankruptcy immediately occurs.
We thus focus on the region where x <x. In general, we expect that V (x) < W (x), except for a finite number of points where equality holds. These are the debt sizes at which the optimal strategy consists in keeping the debt constant in time.
In the following we consider any interval [a, b] where the strict inequality V (x) ≤ W (x)−δ holds for some δ > 0. We will show that on this interval the limit dynamics is piecewise smooth. Namely, there exists an intermediate pointx ∈ [a, b] such that V , p are continuous separately on ]a,x[ and on ]x, b[ , and moreover For technical reasons, we first prove the result assuming that all jumps in p(·) contained inside ]a, b[ satisfy the inequality (4.5). At step 6 we will show that the assumption (4.5) is only needed in the case described by condition (ii) in Theorem 4.2. We now begin working out details. If x <x and V (x) < W (x) < +∞, then as shown in [10] the equation has two solutions, which we denote by for some δ > 0 and all a < x < b. We claim that there existsx ∈ [a, b] such that (by possibly choosing a subsequence) for a.e. x <x, for a.e. x >x .
Using the fact that V n (y) → α, we can find a C 2 function ϕ and a sequence of points x n such that • V n − ϕ has a local maximum at a point x n with y < x n < z n < y + ε.
Then the case can be handled similarly. The main difference is that now we replace α+V (y) 2 byξ and choose z n < y, so the maximum becomes a minimum. The proof of the convergence V n (y) → V (y) in the case where V (y) = F − (y, V (y), p(y)) is entirely similar.
This step completes the proof of Theorem 4.1. In the remaining steps we work toward a proof of Theorem 4.2.

-Convergence of the dynamics, in the region where V < W .
By the previous analysis, for any interval I = ]a, b[ where V < W , there exists a pointx such that (4.25) holds. In particular, the feedback dynamics (4.21) induced by the limit value function V is strictly decreasing for x <x and strictly increasing for x >x.
In this step we prove that this dynamics coincides with the limit of the stochastic dynamics dx n = (ζ n (x n ) + σ 2 n x n ) dt + σ n x n dW , (4.30) where More precisely, consider an initial datum and let x(·), x n (·) be the corresponding solutions to the ODE (4.21) and to the stochastic diffusion equation (4.30), respectively. We claim that, as long as Notice that, by the convergence V n → V and p n → p, we have the pointwise a.e. convergence ζ n (x) → ζ(x). (4.33) 5.1 -lower bounds on the drift. To prove (4.32), we begin by showing that ζ n remains bounded away from zero, uniformly w.r.t. n. To fix the ideas, assume x 0 <x, so that the trajectory t → x(t) is strictly decreasing. Notice that, if we had the uniform convergence ζ n → ζ, the result would be obvious.
To fix the ideas, assume that there exists δ > 0 such that for all x ∈ [a, b] and (4.20)-(4.21) hold for a.e. x ∈ I. At this stage, we still need the assumption (4.5) on all jumps in p(.), namely at every point y ∈ [a, b] where p has a jump, there holds for all p 1 , p 2 , p 3 ∈ [p(ȳ+), p(ȳ−)]. Notice that this condition trivially holds at all points where p is continuous. In turn, by continuity and compactness, this implies that there exist ε 0 , δ 0 > 0 such that for any x,x ∈ I(y, ε 0 ), and p,p ∈ J (y, ε 0 ). Here we define By (4.34), there exists δ 1 > 0 such that at every point x ∈ ]a,x[ where V , p are well defined. We claim that a similar strict inequality also holds for the sequence of approximations. Namely, there exists a constant δ 2 > 0 such that for all x ∈ ]a,x[ and all n ≥ 1 sufficiently large. Arguing by contradiction, assume that there exists a sequence of points z n ∈]a,x[ such that ζ n (z n ) → 0.
Since V is defined a.e, for all n large enough, there exists Notice that the convex set defined in (4.40) contains the slopes of all the secant lines of V (.) on the interval I(z * , ε 0 ). We define ξ (x, p) /2.
(4.41) By the constructions of x n and z n with condition (4.37), for everyε > 0 small, there exists a C 2 function ϕ and a sequence of points y n such that • ϕ (x) − ξ 0 <ε for all x ∈ I(z * , ε 0 ).
• V n − ϕ has a local minimum at y n with z n < y n < x n .

-the distribution functions.
Using the uniform positivity of the functions ζ n and the pointwise a.e. convergence ζ n → ζ, where ζ has bounded variation, we can now prove the convergence (4.32).
To fix the ideas, assume that, for some δ > 0, (4.44) By the previous analysis, this implies for some δ 2 > 0 and all x ∈ [x + 2δ,x − 2δ] and all n sufficiently large. Let t → y(t) the solution to the Cauchy problem for the discontinuous ODĖ Even if the function ζ(·) is only measurable, the solution y is well defined. Indeed, since ζ(x) ≥ δ 2 > 0, the value y(t) is implicitly defined by We will show that the convergence (4.32) holds, for all t ∈ [0, T ]. This will be proved by establishing upper an lower bounds on the distribution functions of the random variables x n (t). For each n ≥ 1, consider the distribution function This function satisfies the parabolic equation with initial data We consider (4.50) restricted to the interval [x + δ, b − δ]. The boundary conditions for this solution are not exactly known, but we can certainly say that We claim that, for every t > 0 To prove (4.53), since the map x → ζ(x) has bounded variation, for any given ε > 0 we can find a smooth function ζ ε and a constant η = η(ε) > 0 such that (see Fig. 5) 54) and such that, calling y ε (·) the unique solution tȯ y = ζ ε (y), y(0) = y 0 − 2η . For every n ≥ 1, define ζ ε n (x) . = min ζ ε (x), ζ n (x) .
By (4.54) and the pointwise convergence ζ n → ζ, we have As shown in Fig. 6, consider the auxiliary functions Z ε n , Z ε n , defined as the solutions to the parabolic equations where ϕ 0 is a C ∞ function such that Moreover, we impose the boundary conditions Since ζ ε (·) is smooth, by the standard theory of parabolic equations it follows the existence of the limit where Z ε is the solution to the linear first order equation with the same initial data (4.60). Moreover, By a comparison argument, we have the pointwise bound In turn, this implies 5.4 -upper bounds on the probability densities. Toward the main convergence proof, we still need to estimate the difference Z ε n − Z ε n . In this subsection, we show that the derivatives satisfy a uniform bound, independent of n: for some constant K ε depending on ε but not on n. This is proved by a comparison argument. The probability densities φ ε n = Z ε n,x and φ ε n = Z ε n,x satisfy the parabolic equations (4.71) By (4.60), all these functions have the same initial data uniformly for t ∈ [0, T ]. A uniform upper bound for φ ε n , φ ε n will be proved by constructing suitable upper solutions, independent of time.
The equations in (4.71) admit infinitely many stationary solutions ψ(x). These are found by solving the equations where C 0 is an arbitrary constant. Define To construct a suitable upper bound, we consider the backward Cauchy problem Observe that the solution of (4.76) satisfies Indeed, this follows immediately from the implications Thanks to the boundary conditions (4.73)-(4.74), using a comparison argument together with (4.77) we deduce (4.78) The same argument applies to Z ε n,x , yielding (4.79) 5.5 -convergence estimates. Comparing (4.58) with (4.59), integrating by parts, and using (4.68), we obtain (4.80) Here and in the following formulas all the norms L 1 , L ∞ refer to the interval [x + δ, b − δ]. Using the bound Z ε n (t,x + δ) ≤ Z ε n (t,x + 2δ) + δ −1 Z ε n (t, ·) − Z ε n (t, ·) L 1 to estimate the last term in (4.80), and recalling (4.70), for all n sufficiently large we obtain with A = ( ζ ε x L ∞ + 1) + δ −1 (ζ max + 1), B n = ζ ε − ζ ε n L 1 K ε + (ζ max + 1) Z ε n (t,x + 2δ). By Gronwall's lemma, for all t ≤ T it follows letting n → ∞, one has B n → 0, hence the right hand side of (4.82) approaches zero as well.
Summarizing the previous analysis, for any ε > 0 we have shown that x) is defined in (4.53) and we used the notation [s] + . = max{s, 0}. A similar estimate on the distribution of Z n from below yields the complementary upper bound on the distribution of x n (t). We only sketch the argument, because it is almost identical to the previous one. Given ε > 0, we can find a smooth function ζ ε and a constant η = η(ε) such that and such that the solution y ε (·) of the Cauchy probleṁ for all t ∈ [0, T ]. We define the corresponding drifts as ζ ε n = max{ζ ε (x), ζ n (x)}. Then we consider the same parabolic equations (4.58)-(4.59) replacing the boundary conditions (4.62)-(4.63) with Z ε n (t,x + δ) = Z ε n (t,x + δ) = 0, (4.84) (4.85) As initial conditions we take (4.60), replacing (4.61) with In place of (4.83) we now obtain Indeed, assume that p has large jump at a point y ∈ ]a, b[ . Being non-increasing, the function p has bounded variation. Hence there exists ε > 0 such that all the jumps of p(·) contained inside ]y − ε, y[ ∪ ]y, y + ε[ are suitably small, so that (4.26) holds. By the previous steps, the function ζ(·) at (4.38) has a constant sign on ]y − ε, y[ and on ]y, y + ε[.
Here we relax the smallness assumption (4.5) on all the jumps of p to just the jumps under condition (ii) in Theorem 4.2, which assumes that large jumps in p can only happen at point y ∈]a, b[ where the dynamics separates or converges in its neighbourhood, namely, ζ(x) < 0 on ]y − ε, y[ and ζ(x) > 0 on ]y, y + ε[ or ζ(x) > 0 on ]y − ε, y[ and ζ(x) < 0 on ]y, y + ε[. All the other jumps should satisfy the condition (4.5).
Various cases must be considered. CASE 1. The speed ζ(x) has the same sign on the intervals ]y − ε, y[ and ]y, y + ε[. Due to the smallness assumption in p, in this case, for initial points x 0 close y, the solution t → x(t, x 0 ) of the Cauchy probleṁ depends continuously on x 0 . By the convergence (4.32), the same is true of the limit of the random variables x n in (4.30).
Recalling that p(x 0 ) is defined at (2.9) as the expected return for the investors, when the initial debt size is x 0 , the continuous dependence on x 0 implies that p is continuous near y. Therefore, no jump occurs. CASE 2. ζ(x) > 0 on ]y − ε, y[ and ζ(x) < 0 on ]y, y + ε[. Though p might have large jumps at y, this still implies that the solutions of (4.88) depend continuously on the initial point x 0 . As in the previous case, by (4.32) and (2.9) we conclude that p is continuous at the point y. and make the assumption that Ω contains finitely many points. We claim that p(y) = p (y) for all y ∈ Ω. From the previous analysis, on the finitely many open intervals where V < W we either have Since ζ(x) < 0 on I, (4.92) implies If p(·) is discontinuous at y, by construction we have ζ(x) → 0 as x → y−. Then the similar analysis shows that p(y−) = p (y) and p (y+) ≥ p (y), providing again a contradiction.
We conclude that in this case p(·) must be continuous at y and p(y) = p (y). CASE 3. There exists δ > 0 such that (4.91) holds on ]y − δ, y[ ∪ ]y, y + δ[. If p(·) is continuous at y, we obtain that ζ(x) → 0 as x → y. If p(y) = p (y), since (4.92) is satisfied on ]y, y + δ[, we must have p(y) < p (y). A similar contradiction is obtained as in CASE 2.
If p(·) is discontinuous at y, by construction we have ζ(x) → 0 as x → y−. A similar analysis now shows that p(y−) = p (y).
We conclude that in this case p(y−) = p (y) and p can have a downward jump at y.

-Behavior nearx.
By the previous analysis, V is continuous everywhere on [0, M ] except for a possible jump atx. By the definition ofx, for any x >x one has λ + r − (λ + µ)p(x) x > 1. In other words, if the debt becomes larger thanx, then it must keep increasing, reaching the bankruptcy level M in finite time.
On the other hand, for x close tox but smaller that this threshold, various cases can arise. CASE 1. ζ(x) < 0 for x ∈ ]x − ε,x[ . In this case, the evolution generated by the optimal feedback is discontinuous atx. As a result, the discounted price p will have a jump atx so that λ + r − (λ + µ)p(x−) x < 1. This guarantees that it is possible to decrease the debt, starting fromx. Notice that in this case, V may also have a jump atx. CASE 2. ζ(x) > 0 on ]x − ε,x[ and V (x−) = W (x−). In this case, we claim that V is Lipschitz continuous on ]x − ε,x[ and has a jump atx. Moreover, one has Notice that the assumption V (x−) = W (x−) implies W (x−) < +∞. In turn this implies By the assumption as in (4.36) that the jump of p atx is sufficiently small, we obtain that the solutions of (4.88) depend continuously on the initial point x 0 for any x 0 ∈]x − ε,x + ε[. By (2.9) and (2.10) we conclude that both p and V are continuous at the pointx. In this case, the evolution generated by the optimal feedback control is continuous atx and there exists somex * <x such that for any x >x * , the debt must keep increasing, reaching the bankruptcy level M in finite time.

-A limit semigroup.
By the previous analysis, if ]a, b[ is an interval where V < W , then it can be partitioned into two subintervals, where either (4.90) or (4.91) hold.
Given the limit functions V, p, and u * (x) = u (p(x), V (x)), we consider the ODĖ x = (λ + r)x − u * (x) p(x) − (λ + µ)x . to be the unique solution oḟ as long as x(t) / ∈ Ω. We then extend the definition of S by setting In other words, the evolution stops as soon as the trajectory hits Ω.

-A Nash equilibrium solution.
Finally, we claim that the limit values V, p, u * provide a Nash equilibrium solution to the debt management problem. More precisely: (i) The discounted price p(·) is consistent with the dynamics defined at (4.95)-(4.96). Indeed, for every initial point x 0 (with the possible exception of points x in (4.21) where the dynamics is discontinuous), one has and u * (x) = u (p(x), V (x)) is an optimal feedback control.
To prove (i), we observe that by construction the result is true for all x 0 in the finite set Ω where V and W coincide. The boundary conditions imply it is also true for x 0 = M .
Then we need to show that Notice that φ u is absolutely continuous except for a possible upward jump att . = sup t<T M {x(t) <x}. By the definition ofx, for any control u(·) we haveẋ(t) > 0 for all t ≥t. Indeed,x is a "point of no return": when the debt crossesx, there is no way to reduce it.