American Institute of Mathematical Sciences

Advanced
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

Alberto Bressan and  Yilun Jiang

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.

Keywords: Vanishing viscosity, discontinuous feedback control, debt management.
Mathematics Subject Classification: Primary: 34B15, 35Q91, 49N70, 91A23.
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

Figure 1.  Proving that $V\leq W$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  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 $x> \bar x_i$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  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 $\xi = F^-$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Proving the convergence $V_n'\to V'$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  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 $n\to\infty$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The functions $Z^\varepsilon _n$ and $\widehat Z^\varepsilon _n$ in (4.67)-(4.68).
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617
[2]

Stefano Bianchini, Alberto Bressan. A case study in vanishing viscosity. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 449-476. doi: 10.3934/dcds.2001.7.449
[3]

Umberto Mosco, Maria Agostina Vivaldi. Vanishing viscosity for fractal sets. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1207-1235. doi: 10.3934/dcds.2010.28.1207
[4]

Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009
[5]

Hua Chen, Jian-Meng Li, Kelei Wang. On the vanishing viscosity limit of a chemotaxis model. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1963-1987. doi: 10.3934/dcds.2020101
[6]

Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial & Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161
[7]

Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021011
[8]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015
[9]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146
[10]

Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257
[11]

Thomas Strömberg. A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit. Communications on Pure & Applied Analysis, 2011, 10 (2) : 479-506. doi: 10.3934/cpaa.2011.10.479
[12]

K. T. Joseph, Manas R. Sahoo. Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2091-2118. doi: 10.3934/cpaa.2013.12.2091
[13]

Frederic Rousset. The residual boundary conditions coming from the real vanishing viscosity method. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 605-625. doi: 10.3934/dcds.2002.8.606
[14]

Giuseppe Maria Coclite, Lorenzo di Ruvo, Jan Ernest, Siddhartha Mishra. Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes. Networks & Heterogeneous Media, 2013, 8 (4) : 969-984. doi: 10.3934/nhm.2013.8.969
[15]

Andrew J. Whittle, Suzanne Lenhart, Louis J. Gross. Optimal control for management of an invasive plant species. Mathematical Biosciences & Engineering, 2007, 4 (1) : 101-112. doi: 10.3934/mbe.2007.4.101
[16]

Heinz Schättler, Urszula Ledzewicz. Perturbation feedback control: A geometric interpretation. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 631-654. doi: 10.3934/naco.2012.2.631
[17]

Eddaly Guerra, Héctor Sánchez-Morgado. Vanishing viscosity limits for space-time periodic Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 331-346. doi: 10.3934/cpaa.2014.13.331
[18]

Wenjun Wang, Lei Yao. Vanishing viscosity limit to rarefaction waves for the full compressible fluid models of Korteweg type. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2331-2350. doi: 10.3934/cpaa.2014.13.2331
[19]

Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure & Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043
[20]

Beixiang Fang, Qin Zhao. Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021066

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (107)
  • HTML views (120)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences