# American Institute of Mathematical Sciences

October  2018, 11(5): 793-824. doi: 10.3934/dcdss.2018050

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

 Department of Mathematics, Penn State University, University Park, PA 16802, USA

Received  February 2017 Revised  July 2017 Published  June 2018

Fund Project: This research was partially supported by NSF, with grant DMS-1411786: Hyperbolic Conservation Laws and Applications.

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.

Citation: Alberto Bressan, Yilun Jiang. The vanishing viscosity limit for a system of H-J equations related to a debt management problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 793-824. doi: 10.3934/dcdss.2018050
##### References:
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##### References:
 [1] H. Amann, Invariant sets and existence theorems for semilinear parabolic equation and elliptic system, J. Math. Anal. Appl., 65 (1978), 432-467.  doi: 10.1016/0022-247X(78)90192-0.  Google Scholar [2] M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar [3] G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Springer-Verlag, Paris, 1994.  Google Scholar [4] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in $R^N$, ESAIM Control Optim. Calc. Var., 19 (2013), 710-739.  doi: 10.1051/cocv/2012030.  Google Scholar [5] G. Barles and E. Chasseigne, (Almost) everything you always wanted to know about deterministic control problems in stratified domains, Netw. Heterog. Media, 10 (2015), 809-836.  doi: 10.3934/nhm.2015.10.809.  Google Scholar [6] R. Barnard and P. Wolenski, Flow invariance on stratified domains, Set-Valued Var. Anal., 21 (2013), 377-403.  doi: 10.1007/s11228-013-0230-y.  Google Scholar [7] A. Bressan and Y. Hong, Optimal control problems on stratified domains, Netw. Heter. Media, 2 (2007), 313–331 (with Y. Hong). Errata Corrige in Netw. Heter. Media, 8 (2013), 625. doi: 10.3934/nhm.2007.2.313.  Google Scholar [8] A. Bressan and Y. Jiang, Optimal open-loop strategies in a debt management problem, Analysis & Appl., 16 (2018), 133-157.  doi: 10.1142/S0219530517500038.  Google Scholar [9] A. Bressan, A. Marigonda, K. Nguyen and M. Palladino, A stochastic model of optimal debt management and bankruptcy, SIAM J. Financial Math., 8 (2017), 841-873.  doi: 10.1137/16M1095019.  Google Scholar [10] A. Bressan and K. Ngyuen, A game theoretical model of debt and bankruptcy, ESAIM: Control, Optim. Calc. Variat., 22 (2016), 953-982.  doi: 10.1051/cocv/2016030.  Google Scholar [11] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo. 2007.  Google Scholar [12] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. American Math. Soc., 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar [13] C. De Zan and P. Soravia, Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients, Interfaces Free Bound, 12 (2010), 347-368.  doi: 10.4171/IFB/238.  Google Scholar [14] M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost, Nonlinear Differential Equations Appl, 11 (2004), 271-298.  doi: 10.1007/s00030-004-1058-9.  Google Scholar [15] M. Garavello and P. Soravia, Representation formulas for solutions of the HJI equations with discontinuous coefficients and existence of value in differential games, J. Optim. Theory Appl., 130 (2006), 209-229.  doi: 10.1007/s10957-006-9099-3.  Google Scholar [16] G. Nuño and C. Thomas, Monetary policy and Sovereign Debt Vulnerability, Working document n. 1517, Banco de España Publications, 2015. Google Scholar [17] Z. Rao, A. Camilli and H. Zidani, Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations, J. Differential Equations, 257 (2014), 3978-4014.  doi: 10.1016/j.jde.2014.07.015.  Google Scholar [18] S. E. Shreve, Stochastic Calculus for Finance. Ⅱ. Continuous-time Models, Springer-Verlag, New York, 2004.  Google Scholar
Proving that $V\leq W$.
A limiting dynamics. Here the coincidence set, where $V = W$ and the dynamics is stationary, is $\Omega = \{ 0, b_1\}$. Restricted to the two open intervals $I_1 = \,]0, b_1[\,$ and $I_2 = \,]b_1,M[\,$ the dynamics is piecewise smooth. Namely, there exists intermediate points $\bar x_i\in I_i$ such that $\dot x <0$ for $x<\bar x_i$ and $\dot x > 0$ for $x> \bar x_i$.
Left: the case where $x<\hat x$. For $(r+\rho(x))V>H^{max}(x, p(x))$ the equation (4.22) has no solution. At a point where $(r+\rho(x))V<H^{max}(x, p(x))$, it determines two distinct values $F^-,F^+$ for $\xi$. Right: the case where $x>\hat x$. For any $(r+\rho(x))V>\rho(x)B$, the equation (4.22) determines a unique solution $\xi = F^-$.
Proving the convergence $V_n'\to V'$.
The functions $\zeta, \zeta_n$ in (4.31)-(4.33) and the smooth approximation $\zeta^\varepsilon$ considered at (4.54). Setting $\zeta^\varepsilon _n = \min\{\zeta^\varepsilon ,\zeta_n\}$, one has $\|\zeta^\varepsilon -\zeta^\varepsilon _n\|_{{\bf{L}}^1}\to 0$ as $n\to\infty$.
The functions $Z^\varepsilon _n$ and $\widehat Z^\varepsilon _n$ in (4.67)-(4.68).
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