# 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:
 [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

show all references

##### 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).
 [1] Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099 [2] Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434 [3] Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 [4] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [5] Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078 [6] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 [7] Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158 [8] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [9] Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332 [10] Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 [11] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [12] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [13] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [14] Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347 [15] M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014 [16] Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011 [17] Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 [18] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [19] Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032 [20] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

2019 Impact Factor: 1.233